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Experimental quantum cloning in a pseudo-unitary system
Abstract
Deterministically cloning (copying) nonorthogonal states is forbidden in quantum mechanics, but deterministic pseudo-unitary cloning is possible in a nonunitary system. We prove and show that, for any two nonorthogonal qubit states, we can find a linear, invertible Hermitian (metric) operator to make these states mutually orthogonal with respect to the corresponding pseudo-inner product. From this metric operator, we construct a pseudo-Hermitian Hamiltonian and its evolution operator as an ideal, deterministic pseudo-unitary cloner. For applicability, suppose Alice lives in our ordinary universe and Bob lives in a universe that has the desired pseudo-unitary evolution. Alice would send her nonorthogonal states to Bob and Bob would do the cloning deterministically and send back to Alice. The quantum channel will be lossy not due to Bob's universe's properties, but rather due to problems making a channel between the two universes. We experimentally demonstrate deterministic pseudo-unitary two-qubit cloning for a photonic pseudo-unitary two-qubit system. In our universe we have to demonstrate that universe by executing a postselected gate that effects loss-based pseudo-unitary evolution. Furthermore, we introduce an algorithmic method for designing experimental realizations of a generic class of nonunitary operators for pseudo-unitary cloning.
... Over the years, pseudo-unitary quantum mechanics has rapidly grown from the interest in pseudo-Hermitian Hamiltonians [1] in early 2000 to describe PT-symmetric quantum systems [2], and recently to simulate and experiment pseudo-unitary quantum cloning and quantum deletion in optical circuits [3,4]. Furthermore, controlling Hilbert-space inner products [5] for fast quantum algorithms in PT-symmetric systems [6,7] is a direct implication of knowing how to manipulate pseudo-unitary groups. ...
... The general solution ψ(x) to the Schrödinger equation for a one-dimensional wave scattering across a localized potential is equation (1), with ψ ab (x) the solution between x = a and x = b, the region where the potential is non-zero. If the S-matrix of a system is unitary (equation (3)), then the transfer matrix (equation (4)) derived from it is known to be pseudo-unitary with respect to either the σ 3 ⊗ I m metric [8] or the I m ⊗ σ 3 metric [9], where σ 3 = |0〉〈0| − |1〉〈1| is a Pauli matrix and I m is the identity matrix with dimensions (m,m). In equations (3) and (4), t, t¢, r and r¢ are Hermitian matrices [8]. ...
... The restrictions imposed on unitary systems by No-Go theorems (no-cloning, no-deleting, no-masking) are largely studied in pseudo-unitary systems due to the possibility of flexibilizing these theorems with a set of ηorthogonal states [4,22]. These models focus on pseudo-unitary systems alone without realizing the transition from unitary systems to η-pseudo-unitary ones and vice-versa. ...
Pseudo-unitary circuits are recurring in both S-matrix theory and analysis of No-Go theorems. We propose a matrix and diagrammatic representation for the operation that maps S-matrices to T-matrices and, consequently, a unitary group to a pseudo-unitary one. We call this operation “partial inversion” and show its diagrammatic representation in terms of permutations. We find the expressions for the deformed metrics and deformed dot products that preserve physical constraints after partial inversion. Subsequently, we define a special set that allows for the simplification of expressions containing infinities in matrix inversion. Finally, we propose a renormalized-growth algorithm for the T-matrix as a possible application. The outcomes of our study expand the methodological toolbox needed to build a family of pseudo-unitary and inter-pseudo-unitary circuits with full diagrammatic representation in three
dimensions, so that they can be used to exploit pseudo-unitary flexibilization of unitary No-Go Theorems and renormalized circuits of large scattering lattices.
... It is well known for a long time that some non-Hermitian Hamiltonians have real eigenvalues. For example, Bender and his collaborators introduced a class of non-Hermitian Hamiltonians with PT -symmetry, which have infinite, discrete, and entirely real and positive spectrum [8][9][10]: H = p 2 − (ix) N , where N is a real number and not less than 2. Recently, PT -symmetry has been observed in classical optics [11][12][13][14][15][16][17][18][19][20][21][22][23][24], and non-Hermitian Hamiltonian with real eigenvalues has attracted much interest [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. In the quantum regime, some experimental results have been reported to simulate non-Hermitian system in an open conventional quantum system [29][30][31]. ...
... For example, Bender and his collaborators introduced a class of non-Hermitian Hamiltonians with PT -symmetry, which have infinite, discrete, and entirely real and positive spectrum [8][9][10]: H = p 2 − (ix) N , where N is a real number and not less than 2. Recently, PT -symmetry has been observed in classical optics [11][12][13][14][15][16][17][18][19][20][21][22][23][24], and non-Hermitian Hamiltonian with real eigenvalues has attracted much interest [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. In the quantum regime, some experimental results have been reported to simulate non-Hermitian system in an open conventional quantum system [29][30][31]. For example, an experimental investigation has been reported by using an open quantum system to simulate the PT -symmetric system as a part of the full Hermitian system [29]. ...
... In this work, the quantum deleting and cloning will be investigated in a pseudo-unitary system. We first present a pseudo-Hermitian Hamiltonian with real eigenvalues in a two-qubit system, which has recently been reported in an experimental simulation [31]. Here, we extend its theoretical part. ...
In conventional quantum mechanics, quantum no-deleting and no-cloning theorems indicate that two different and nonorthogonal states cannot be perfectly and deterministically deleted and cloned, respectively. Here, we investigate the quantum deleting and cloning in a pseudo-unitary system. We first present a pseudo-Hermitian Hamiltonian with real eigenvalues in a two-qubit system. By using the pseudo-unitary operators generated from this pseudo-Hermitian Hamiltonian, we show that it is possible to delete and clone a class of two different and nonorthogonal states, and it can be generalized to arbitrary two different and nonorthogonal pure qubit states. Furthermore, state discrimination, which is strongly related to quantum no-cloning theorem, is also discussed. Last but not least, we simulate the pseudo-unitary operators in conventional quantum mechanics with post-selection, and obtain the success probability of simulations. Pseudo-unitary operators are implemented with a limited efficiency due to the post-selections. Thus, the success probabilities of deleting and cloning in the simulation by conventional quantum mechanics are less than unity, which maintain the quantum no-deleting and no-cloning theorems.
... Chemistry functional materials, drug candidates [26,27] discrete molecular graphs virtual screening [28][29][30], evolutionary [31][32][33][34] gradient-based [35] widely used in industry & academia Quantum Experiments complex entanglement, transformations in quantum optics discrete experimental topology & continuous components highly-efficient topological search [36], evolutionary [37][38][39], reinforcement [40,41], gradient-based [42,43] PoC exp. [44][45][46][47][48][49][50][51], conceptual insights [52,53] been experimentally studied since the late 1950s; it is the basis of the ITER (International Thermonuclear Experimental Reactor) megaproject. An alternative concept called Stellarator (complexly shaped external magnetic coils for creating ring-shaped, twisted magnet fields) was conceptualised around the same time. ...
... C) High-quality experimental results for the fidelity of quantum-cloned states. Image from ref. [51] case where we could learn new concepts or ideas from solutions found by computer algorithms. This is possible, as shown in [52], where an entirely new concept for the generation of high-dimensional multi-particle entanglement has been presented. ...
... To experimentally investigate processes that depend on non-unitarity, such as dynamics involving PT-symmetry [50] and deterministic quantum cloning exploiting non-unitarity [51], the group of Xue have relied on computer algorithms for designing their experimental setups, see Fig.5A. The algorithm iteratively increases the experimental setup (in the form of Fig.5B) block by block. ...
The design of new devices and experiments in science and engineering has historically relied on the intuitions of human experts. This credo, however, has changed. In many disciplines, computer-inspired design processes, also known as inverse-design, have augmented the capability of scientists. Here we visit different fields of physics in which computer-inspired designs are applied. We will meet vastly diverse computational approaches based on topological optimization, evolutionary strategies, deep learning, reinforcement learning or automated reasoning. Then we draw our attention specifically on quantum physics. In the quest for designing new quantum experiments, we face two challenges: First, quantum phenomena are unintuitive. Second, the number of possible configurations of quantum experiments explodes combinatorially. To overcome these challenges, physicists began to use algorithms for computer-designed quantum experiments. We focus on the most mature and \textit{practical} approaches that scientists used to find new complex quantum experiments, which experimentalists subsequently have realized in the laboratories. The underlying idea is a highly-efficient topological search, which allows for scientific interpretability. In that way, some of the computer-designs have led to the discovery of new scientific concepts and ideas -- demonstrating how computer algorithm can genuinely contribute to science by providing unexpected inspirations. We discuss several extensions and alternatives based on optimization and machine learning techniques, with the potential of accelerating the discovery of practical computer-inspired experiments or concepts in the future. Finally, we discuss what we can learn from the different approaches in the fields of physics, and raise several fascinating possibilities for future research.
... Many schemes have been proposed to address this limit, which allows us to copy the original quantum information imperfectly, probabilistic [4][5][6][7][8] or with reduced fidelity by adding an auxiliary system [9][10][11][12][13][14][15]. And various physical systems have been studied to realize quantum cloning, including optical systems [13,[15][16][17][18], atomic systems [6,19], ion systems [20,21], superconducting system [22], and so on. ...
... Due to the great potential, the discussion of optical systems are relatively active in both theoretical and experimental aspects [2,15,17,23]. One reason is that photons are excellent flying bits and can be used as carriers of quantum information. ...
Quantum cloning is an essential operation in quantum information and quantum computing. Similar to the ‘copy’ operation in classical computing, the cloning of flying bits for further processing from the solid-state quantum bits in storage is an operation frequently used in quantum information processing. Here we propose a high-fidelity and controllable quantum cloning scheme between solid bits and flying bits. In order to overcome the obstacles from the no-cloning theorem and the weak phonon-photon interaction, we introduce a hybrid optomechanical system that performs both the probabilistic cloning and deterministic cloning closed to the theoretical optimal limit with the help of designed driving pulse in the presence of dissipation. In addition, our scheme allows a highly tunable switching between two cloning methods, namely the probabilistic and deterministic cloning, by simply changing the input laser pulse. This provides a promising platform for experimental executability.
... Various physical systems have been studied to realize quan-Wen-Zhao Zhang: zhangwenzhao@nbu.edu.cn tum cloning, including optical systems [13,[15][16][17][18], atomic systems [6,19], ion systems [20,21], superconducting system [22], and so on. Among them, optical systems with great potential are relatively active in both theoretical and experimental aspects [2,15,17,23]. ...
... tum cloning, including optical systems [13,[15][16][17][18], atomic systems [6,19], ion systems [20,21], superconducting system [22], and so on. Among them, optical systems with great potential are relatively active in both theoretical and experimental aspects [2,15,17,23]. Photons as excellent flying bits that can be used as carriers of quantum information, and thereby many protocols were proposed with this property, such as BB84 [24,25] and B92 [26]. Whereas photons are difficult to store, the solid bits make up for this deficiency well, exhibiting advantages such as easy storage and long decoherence times. ...
Quantum cloning is an essential operation in quantum information and quantum computing. Similar to the `copy' operation in classical computing, the cloning of flying bits for further processing from the solid-state quantum bits in storage is an operation frequently used in quantum information processing. Here we propose a high-fidelity and controllable quantum cloning scheme between solid bits and flying bits. In order to overcome the obstacles from the no-cloning theorem and the weak phonon-photon interaction, we introduce a hybrid optomechanical system that performs both the probabilistic cloning and deterministic cloning closed to the theoretical optimal limit with the help of designed driving pulse in the presence of dissipation. In addition, our scheme allows a highly tunable switching between two cloning methods, namely the probabilistic and deterministic cloning, by simply changing the input laser pulse. This provides a promising platform for experimental executability.
... NH QM often serves as an effective description of open quantum systems, typically arising from the interaction with an external environment. However, this approach raises questions about the treatment of fluctuations and the potential violation of well-established quantum theorems [14][15][16][17][18]. ...
In this paper we present a concrete example comparing the results predicted by non-Hermitian quantum mechanics with those of a more comprehensive description that considers environment-induced fluctuations. Our results highlight inaccuracies in the non-Hermitian model. Specifically, we investigate the non-Hermitian skin effect and sensor in the Hatano-Nelson model, contrasting it with a more precise Lindblad description. Our analysis reveals that these phenomena can undergo breakdown when environmental fluctuations come to the forefront, resulting in a nonequilibrium phase transition from a localized skin phase to a delocalized phase. Beyond this specific case study, we engage in a broader discussion regarding the interpretations and implications of non-Hermitian quantum mechanics. This examination serves to broaden our understanding of these phenomena and their potential consequences.
Published by the American Physical Society 2024
... Problems that are difficult to solve in Hermitian systems can be resolved in non-Hermitian systems. Recently, systems evolving under parity-time (PT ) symmetric non-Hermitian dynamics present novel features [54,55], which provide new perspec-tives and insights for solving conventional problems in quantum mechanics, both in theory [56][57][58][59][60][61] and in practice [62][63][64][65][66][67][68][69][70][71][72][73][74][75][76][77][78]. PT -symmetric systems also have widespread applications in quantum information, and there is a debate about whether they can outperform Hermitian systems [79,80]. ...
The fundamental quantum mechanical features forbid non-orthogonal quantum states to be distinguished accurately through a single-shot measurement. Using parity-time () quantum mechanics, however, a perfect state discrimination with certainty can be implemented. Here, we experimentally implement quantum state discrimination for photonic single-qubit states by executing post-selected gates and loss-based non-unitary evolutions in the -symmetric system, providing a relatively superior strategy for state discrimination in non-Hermitian systems. We also unveil the origin of quantum state discrimination by embedding the dynamics into a higher-level system under unitary dynamics. This experimental work demonstrates the key principle of quantum state discrimination and opens a door of other effective approaches in the -symmetric quantum mechanics.
... Mostafazadeh [12] showed that a non-Hermitian Hamiltonian with discrete spectra and a complete biorthonormal system of eigenvectors is pseudo-Hermitian if and only if the spectra are all real, or the complex eigenvalues come in complex conjugate pairs and the geometric multiplicity and the Jordan dimensions of the complex-conjugate eigenvalues coincide. The pseudo-Hermitian Hamiltonian which is defined by a similarity transformation, H † = WHW −1 , where W is called a metric operator for H, has developed into a noteworthy area of research, and systems evolving under pseudo-Hermitian dynamics display rich phenomena [7,9,10,22]. In fact, the metric operator for a pseudo-Hermitian Hamiltonian is not unique, and it is worthwhile to discuss the properties of the metric operator and the structure of the set of all metric operators. ...
The Hamiltonian of a conventional quantum system is Hermitian, which ensures real spectra of the Hamiltonian and unitary evolution of the system. However, real spectra are just the necessary conditions for a Hamiltonian to be Hermitian. In this paper, we discuss the metric operators for pseudo-Hermitian Hamiltonian which is similar to its adjoint. We first present some properties of the metric operators for pseudo-Hermitian Hamiltonians and obtain a sufficient and necessary condition for an invertible operator to be a metric operator for a given pseudo-Hermitian Hamiltonian. When the pseudo-Hermitian Hamiltonian has real spectra, we provide a new method such that any given metric operator can be transformed into the same positive-definite one and the new inner product with respect to the positive-definite metric operator is well defined. Finally, we illustrate the results obtained with an example.
... Over the years, pseudo-unitary quantum mechanics grew fast from the interest in pseudo-Hermitian Hamiltonians [1]: in the early 2000 to describe PT-symmetric quantum systems [2], and recently to simulate and experiment pseudo-unitary quantum cloning and quantum deletion in optical circuits ( [3], [4]). Furthermore, controlling Hilbert-space inner products [5] for fast quantum algorithms in PT-symmetric systems ( [6], [7]) is a direct implication of knowing to manipulate pseudo-unitary groups. ...
Pseudo-unitary circuits are silently recurrent in S-matrix theory. We propose a matrix and diagrammatic representation for the operation that maps S-matrices to T-matrices, and consequently a unitary group to a pseudo-unitary one. We call this operation partial inversion and show its diagrammatic representation in terms of permutations. We find the expressions for the deformed metrics and deformed dot products that preserve physical constraints after partial inversion. Subsequently, we define a special set that allows for the simplification of expressions containing infinities in matrix inversion. Finally, we proposed a renormalized-growth algorithm for the T-matrix as a possible application. Our studies furnish all the tools needed to build a family of pseudo-unitary and inter-pseudo-unitary circuits with full diagrammatic representation in three dimensions.
... In addition, the relevance of these mathematical developments is noteworthy in many topics of physics as complex scattering potentials [37][38][39], tight-binding chain [40], anisotropic XY model [41], quantum brachistochrone problem in both theoretical [42,43], and experimental [44] scenarios, coupled optomechanical systems [45], geometric phase [46], pseudochirality [47], and non-Hermitian version of Jaynes-Cummings optical model obtained from κ-deformed Dirac oscillator [48]. Furthermore, in the field of quantum information, there are many interesting investigations in the context of pseudo-Hermiticity considered for optimalspeed evolution generation [49], pseudo-Hermitian networks [50], perfect state transfer in non-Hermitian networks [51], information retrieval [52], holonomic gates [53], and an experimental quantum cloning protocol was presented in Ref. [54]. Recently, an efficient simulation scheme of a finite PTsymmetric system with LOCC was proposed in Ref. [55]. ...
In this work we present the general unified description for the unitary time evolution generated by time-dependent non-Hermitian Hamiltonians embedding the bosonic representations of su(1,1) and su(2) Lie algebras. We take into account a time-dependent Hermitian Dyson maps written in terms of the elements of those algebras with the relation between non-Hermitian and its Hermitian counterpart being independent of the algebra realization. As a direct consequence, we verify that a time-evolved state of uncoupled modes modulated by a time-dependent complex frequency may exhibit a nonzero entanglement even when the cross operators, typical of the interaction between modes, are absent. This is due the nonlocal nature of the nontrivial dynamical Hilbert space metric encoded in the time-dependent parameters of the general Hermitian Dyson map, which depend on the imaginary part of the complex frequency. We illustrate our approach by setting the
PT-symmetric case where the imaginary part of frequency is linear on time for the two-mode bosonic realization of Lie algebras.
... In addition, it is noteworthy the relevance of these mathematical developments in many topics of physics as complex scattering potentials [37][38][39], tight-binding chain [40], anisotropic XY model [41], quantum brachistochrone problem in both theoretical [42,43] and experimental [44] scenarios, coupled optomechanical systems [45], geometric phase [46], pseudochirality [47], and non-Hermitian version of Jaynes-Cummings optical model obtained from κdeformed Dirac oscillator [48]. Furthermore, in the field of quantum information, there are many interesting investigations in the context of pseudo-hermiticity considered for optimal-speed evolution generation [49], pseudo-Hermitian networks [50], perfect state transfer in non-Hermitian networks [51], information retrieval [52], holonomic gates [53], and an experimental quantum cloning protocol was presented in Ref. [54]. Recently, an efficient simulation scheme of a finite PT -symmetric system with LOCC was proposed in Ref. [55]. ...
In this work we present the general unified description for the unitary time-evolution generated by time-dependent non-Hermitian Hamiltonians embedding the bosonic representations of and Lie algebras. We take into account a time-dependent Hermitian Dyson maps written in terms of the elements of those algebras with the relation between non-Hermitian and its Hermitian counterpart being independent of the algebra realization. As a direct consequence, we verify that a time-evolved state of uncoupled modes modulated by a time-dependent complex frequency may exhibits a non-zero entanglement even when the cross-operators, typical of the interaction between modes, are absent. This is due the non-local nature of the non-trivial dynamical Hilbert space metric encoded in the time-dependent parameters of the general Hermitian Dyson map, which depend on the imaginary part of the complex frequency.
... With the interaction of the two states moving in opposite time directions, the model permits time travel, which is, of course, another interesting but controversial topic in physics. Given the model's equivalence with the pseudo-unitary model, the possibility of time travel may explain why the latter can violate certain principles of standard physics, such as the nosignaling and no-cloning laws [8,9]. One should not take the time travel and the violation of conventional principles as a failure of the models, however, as the models allow time travel only in a rigid mechanistic manner and there is no reason to believe that those conventional principles are fundamental and can survive new physics. ...
I propose a time-symmetric generalization of quantum mechanics that is inspired by scattering theory. The model postulates two interacting quantum states, one traveling forward in time and one backward in time. The interaction is modeled by a unitary scattering operator. I show that this model is equivalent to pseudo-unitary quantum mechanics.
... In particular, the development of PT-symmetric quantum theory was motivated by the observation that a Hamiltonian possesses real energy values if the Hamiltonian and its eigenvectors are invariant under an antilinear PT-symmetry. If, on the one hand PT-symmetric quantum theory has witnessed numerous theoretical [5][6][7][8][9][10][11][12] and experimental advances in the recent years [13][14][15][16][17][18][19][20][21][22], on the other hand, an operational foundation for PT-symmetric quantum theory that consistently extends standard quantum theory has not been formulated. The absence of such a consistent extension has led to disputable proposed applications of PT-symmetry that contradict established informationtheoretic principles including the no-signalling principle, faster-than-Hermitian evolution of quantum states, and the invariance of entanglement under local operations [23][24][25][26]. ...
PT-symmetric quantum theory was originally proposed with the aim of extending standard quantum theory by relaxing the Hermiticity constraint on Hamiltonians. However, no such extension has been formulated that consistently describes states, transformations, measurements and composition, which is a requirement for any physical theory. We aim to answer the question of whether a consistent physical theory with PT-symmetric observables extends standard quantum theory. We answer this question within the framework of general probabilistic theories, which is the most general framework for physical theories. We construct the set of states of a system that result from imposing PT-symmetry on the set of observables, and show that the resulting theory allows only one trivial state. We next consider the constraint of quasi-Hermiticity on observables, which guarantees the unitarity of evolution under a Hamiltonian with unbroken PT-symmetry. We show that such a system is equivalent to a standard quantum system. Finally, we show that if all observables are quasi-Hermitian as well as PT-symmetric, then the system is equivalent to a real quantum system. Thus our results show that neither PT-symmetry nor quasi-Hermiticity constraints are sufficient to extend standard quantum theory consistently.
... In reality, non-Hermitian operators can be used in many ways such as quantum open systems [20], quantum optics [21], quantum cosmology [22], and many other fields. Furthermore, pseudo-Hermitian operators belong to non-Hermitian operators, which have many applications [23,24]. ...
The theoretical framework for the uncertainty relation of Hermitian operators is perfect and has been applied in many fields. At the same time, non-Hermitian operators are also widely used in some other fields. However, the uncertainty relation of non-Hermitian operators remains to be explored. K.W. Bong and his co-workers proposed the theory of unitary uncertainty relation and verified it in the experiment [Phys. Rev. Lett. 120, 230402 (2018)]. In this work, we generalized this unitary uncertainty relation theory and proposed uncertainty relations of non-Hermitian operators. Due to the difficulties in the direct measurement of non-Hermitian operators in the uncertainty relations, we simplified the uncertainty relation of two non-Hermitian operators with pure states and proposed a realizable experimental measurement scheme by using the Mach–Zehnder interferometer. When the two non-Hermitian operators are unitary, our result can reduce to Bong et al.’s result. Furthermore, for two non-Hermitian operators but not unitary, we obtained a generalized and analogous result of theirs.
... In particular, the development of PTsymmetric quantum theory was motivated by the observation that a Hamiltonian possesses real energy values if the Hamiltonian and its eigenvectors are invariant under an antilinear PT-symmetry. If, on the one hand PTsymmetric quantum theory has witnessed numerous theoretical [5][6][7][8][9] and experimental advances in the recent years [10][11][12][13][14][15][16][17][18][19], on the other hand, an operational foundation for PT-symmetric quantum theory that consistently extends standard quantum theory has not been formulated. The absence of such a consistent extension has led to disputable proposed applications of PT-symmetry that contradict established information-theoretic principles including the no-signalling principle, faster than Hermitian evolution of quantum states and invariance of entanglement under local operations [20][21][22][23]. ...
PT-symmetric quantum theory was originally proposed with the aim of extending standard quantum theory by relaxing the Hermiticity constraint on Hamiltonians. However, no such extension has been formulated that consistently describes states, transformations, measurements and composition, which is a requirement for any physical theory. We aim to answer the question of whether a consistent physical theory with PT-symmetric observables extends the standard quantum theory. We answer this question within the framework of general probabilistic theories, which is the most general framework for physical theories. We construct the set of states of a system that result from imposing PT-symmetry on the set of observables and show that the resulting theory allows only one trivial state. We next consider the constraint of quasi-Hermiticity on observables, which guarantees the unitarity of evolution under a Hamiltonian with unbroken PT-symmetry. We show that the resultant theory in this setting has equivalent features to standard quantum theory. Finally, we show that if all observables are quasi-Hermitian as well as PT-symmetric, then the resultant theory has equivalent features to real quantum theory. Thus our results show that neither PT-symmetry nor quasi-Hermiticity constraints are sufficient to extend standard quantum theory consistently.
... These phenomena have been explained by observing that non-Hermitian Hamiltonians with unbroken PT symmetry are Hermitian with respect to a different Hilbert-space inner product [3,[15][16][17]. Changing Hilbert-space inner product is valuable for certain quantum information processing (QIP) tasks [18] such as nonorthogonal state discrimination [19], cloning [20], and quantum algorithms [21,22], but perfunctory applications have led to counter-factual conclusions [18,23,24] including violation of the no-signalling principle [25]. Our aim is to prescribe the correct procedure for changing Hilbert-space inner product and to devise an experiment to validate our prescription. ...
In quantum mechanics, physical states are represented by rays in Hilbert space H, which is a vector space imbued by an inner product 〈|〉, whose physical meaning arises as the overlap 〈ϕ|ψ〉 for |ψ〉 a pure state (description of preparation) and 〈ϕ| a projective measurement. However, current quantum theory does not formally address the consequences of a changing inner product during the interval between preparation and measurement. We establish a theoretical framework for such a changing inner product, which we show is consistent with standard quantum mechanics. Furthermore, we show that this change is described by a quantum operation, which is tomographically observable, and we elucidate how our result is strongly related to the exploding topic of PT-symmetric quantum mechanics. We explain how to realize experimentally a changing inner product for a qubit in terms of a qutrit protocol with a unitary channel.
... In the first scenario, Bob performs a POVM fΠ B 0jy ;Π B 1jy g instead of projective measurement [37][38][39][40][41][42][43] . He needs two steps to implement the two-outcome measurements. ...
The Clauser–Horne–Shimony–Holt (CHSH) inequality test is widely used as a mean of invalidating the local deterministic theories. Most attempts to experimentally test nonlocality have presumed unphysical idealizations that do not hold in real experiments, namely, noiseless measurements. We demonstrate an experimental violation of the CHSH inequality that is free of idealization and rules out local models with high confidence. We show that the CHSH inequality can always be violated for any nonzero noise parameter of the measurement. Intriguingly, less entanglement exhibits more nonlocality in the CHSH test with noisy measurements. Furthermore, we theoretically propose and experimentally demonstrate how the CHSH test with noisy measurements can be used to detect weak entanglement on two-qubit states. Our results offer a deeper insight into the relation between entanglement and nonlocality.
... Numerous variations have been developed since then. For example, genetic algorithms [11,12] coupled with neural networks [13], reinforcement-learning-based search [14], gradient-descent of a continuous experimental space [15,16] or efficient human-interpretable representations [17] and unsupervised deep generative models [18]. See a recent review about these developments [19]. ...
We demonstrate how machine learning is able to model experiments in quantum physics. Quantum entanglement is a cornerstone for upcoming quantum technologies, such as quantum computation and quantum cryptography. Of particular interest are complex quantum states with more than two particles and a large number of entangled quantum levels. Given such a multiparticle high-dimensional quantum state, it is usually impossible to reconstruct an experimental setup that produces it. To search for interesting experiments, one thus has to randomly create millions of setups on a computer and calculate the respective output states. In this work, we show that machine learning models can provide significant improvement over random search. We demonstrate that a long short-term memory (LSTM) neural network can successfully learn to model quantum experiments by correctly predicting output state characteristics for given setups without the necessity of computing the states themselves. This approach not only allows for faster search, but is also an essential step towards the automated design of multiparticle high-dimensional quantum experiments using generative machine learning models.
... Frequently, however, the design of experimental setups even for well-defined targets is challenging for the intuition of human experts, and existing systematic schemes (e.g., Ref. [18]) to date provide solutions only for specific experimental scenarios. For that reason, computational design methods for quantum optical experiments have been introduced [19], in the form of topological search augmented with machine learning [20,21], genetic algorithms [22,23], active learning approaches [24], and optimization of parametrized setups [25]. Unfortunately, due to the complexity and size of the Hilbert space as well as the breadth of quantum optical applications, those algorithms may have severe drawbacks, such as inefficient discovery rates, requirements of a huge amount of training data, or specialization on narrow sets of problems. ...
Artificial intelligence (AI) is a potentially disruptive tool for physics and science in general. One crucial question is how this technology can contribute at a conceptual level to help acquire new scientific understanding. Scientists have used AI techniques to rediscover previously known concepts. So far, no examples of that kind have been reported that are applied to open problems for getting new scientific concepts and ideas. Here, we present Theseus, an algorithm that can provide new conceptual understanding, and we demonstrate its applications in the field of experimental quantum optics. To do so, we make four crucial contributions. (i) We introduce a graph-based representation of quantum optical experiments that can be interpreted and used algorithmically. (ii) We develop an automated design approach for new quantum experiments, which is orders of magnitude faster than the best previous algorithms at concrete design tasks for experimental configuration. (iii) We solve several crucial open questions in experimental quantum optics which involve practical blueprints of resource states in photonic quantum technology and quantum states and transformations that allow for new foundational quantum experiments. Finally, and most importantly, (iv) the interpretable representation and enormous speed-up allow us to produce solutions that a human scientist can interpret and gain new scientific concepts from outright. We anticipate that Theseus will become an essential tool in quantum optics for developing new experiments and photonic hardware. It can further be generalized to answer open questions and provide new concepts in a large number of other quantum physical questions beyond quantum optical experiments. Theseus is a demonstration of explainable AI (XAI) in physics that shows how AI algorithms can contribute to science on a conceptual level.
... While quantum experiments historically have been designed by experienced human experts, their non-intuitive nature has led to the emergence of computational methods for designing quantum experiments [18][19][20][21][22][23][24][25]. However, as the dimension of state space grows exponentially with the number of photons, this approach is limited to small systems. ...
The parameters of a quantum system grow exponentially with the number of involved quantum particles. Hence, the associated memory requirement to store or manipulate the underlying wavefunction goes well beyond the limit of the best classical computers for quantum systems composed of a few dozen particles, leading to serious challenges in their numerical simulation. This implies that the verification and design of new quantum devices and experiments are fundamentally limited to small system size. It is not clear how the full potential of large quantum systems can be exploited. Here, we present the concept of quantum computer designed quantum hardware and apply it to the field of quantum optics. Specifically, we map complex experimental hardware for high-dimensional, many-body entangled photons into a gate-based quantum circuit. We show explicitly how digital quantum simulation of Boson sampling experiments can be realized.We then illustrate how to design quantum-optical setups for complex entangled photonic systems, such as high-dimensional Greenberger-Horne-Zeilinger states and their derivatives. Since photonic hardware is already on the edge of quantum supremacy and the development of gate-based quantum computers is rapidly advancing, our approach promises to be a useful tool for the future of quantum device design.
... Systems with such Hamiltonians have been a topic of interest ever since it was discussed by Bender et al [28]. A variety of exotic effects in classical set-ups like unidirectional optical transmission and single-mode lasing [29][30][31][32][33][34] and in quantum systems like the extreme acceleration of state evolution of TLS [35], quantum state discrimination [36], perfect quantum state transfer [37], violation of no-signalling theorem [38], anomalous & unconventional states in many-body systems [39][40][41][42][43][44][45] and unique correlations in their dynamics [46,47] have been explored. Recently, there has been noticeable progress in realizing physical systems which demonstrate PT-symmetry [48][49][50][51][52][53] including the many-body quantum systems [54,55]. ...
We study the dynamics of a non-Hermitian PT- symmetric Hamiltonian of a two level system (TLS) which has real eigenvalues. Within the framework of Hermitian quantum mechanics, it is known that maximal violation of Leggett–Garg inequality (LGI) is bounded by K3 = 3/2 (Luder's bound). We show that this absolute bound can be evaded when dynamics is governed by non-Hermitian PT- symmetric Hamiltonians with real eigenvalues. Moreover, the extent of violation can be optimized to asymptotically approach the algebraic maximum of , which is otherwise observed for Hermitian Hamiltonian with infinite dimensional Hilbert space. The extreme violation of LGI is shown to be directly related to the two basic ingredients: (i) the Bloch equation for the TLS having a non-linear terms which allow for accelerated dynamics of states on the Bloch sphere exceeding all known quantum speed limits of state evolution; and (ii) quantum trajectory of states lie on a great circle (geodesic path) on the Bloch sphere at all times. We demonstrate that such extreme temporal correlation of TLS can be simulated in realistic system by embedding the TLS into a higher dimensional Hilbert space such that the composite system obeys unitary dynamics. Specifically we show that a four dimensional embedding of non-Hermitian PT- symmetric TLS is enough to host K3 → 3 limit. We also discuss the effect of random noise on our results. Finally we conclude with a comparative study of our results with existing experimental realization of embedding.
... The difficulty arises from counter-intuitive quantum phenomena, which raises the question of whether human intuition is the best way to design new experiments. Several studies have therefore developed automated and machine-learning augmented approaches for the design of experiments [39][40][41][42][43][44]. The goal in our approach is to tackle this challenge in a completely different way, namely by improving the scientist's intuition about these systems. ...
Machine learning with application to questions in the physical sciences has become a widely used tool, successfully applied to classification, regression and optimization tasks in many areas. Research focus mostly lies in improving the accuracy of the machine learning models in numerical predictions, while scientific understanding is still almost exclusively generated by human researchers analysing numerical results and drawing conclusions. In this work, we shift the focus on the insights and the knowledge obtained by the machine learning models themselves. In particular, we study how it can be extracted and used to inspire human scientists to increase their intuitions and understanding of natural systems. We apply gradient boosting in decision trees to extract human-interpretable insights from big data sets from chemistry and physics. In chemistry, we not only rediscover widely know rules of thumb but also find new interesting motifs that tell us how to control solubility and energy levels of organic molecules. At the same time, in quantum physics, we gain new understanding on experiments for quantum entanglement. The ability to go beyond numerics and to enter the realm of scientific insight and hypothesis generation opens the door to use machine learning to accelerate the discovery of conceptual understanding in some of the most challenging domains of science.
... These phenomena have been explained by observing that non-Hermitian Hamiltonians with unbroken PT symmetry are Hermitian with respect to a different Hilbert-space inner product [3,[14][15][16]. Changing Hilbert-space inner-product is valuable for certain quantum information processing (QIP) tasks [17] such as non-orthogonal state discrimination [18], cloning [19] and quantum algorithms [20,21], but perfunctory applications have led to counter-factual conclusions [17,22,23] including violation of the nosignalling principle [24]. Our aim is to prescribe the correct procedure for changing Hilbert-space inner product and to devise an experiment to validate our prescription. ...
In quantum mechanics, physical states are represented by rays in Hilbert space , which is a vector space imbued by an inner product , whose physical meaning arises as the overlap for a pure state (description of preparation) and a projective measurement. However, current quantum theory does not formally address the consequences of a changing inner product during the interval between preparation and measurement. We establish a theoretical framework for such a changing inner product, which we show is consistent with standard quantum mechanics. Furthermore, we show that this change is described by a quantum channel, which is tomographically observable, and we elucidate how our result is strongly related to the exploding topic of PT-symmetric quantum mechanics. We explain how to realize experimentally a changing inner product for a qubit in terms of a qutrit protocol with a unitary channel.
... The difficulty arises from counter-intuitive quantum phenomena, which raises the question of whether human intuition is the best way to design new experiments. Several studies have therefore developed automated and machine-learning augmented approaches for the design of experiments [35][36][37][38][39][40]. The goal in our approach is to tackle this challenge in a completely different way, namely by improving the human scientist's intuition about these systems. ...
Machine learning with application to questions in the physical sciences has become a widely used tool, successfully applied to classification, regression and optimization tasks in many areas. Research focus mostly lies in improving the accuracy of the machine learning models in numerical predictions, while scientific understanding is still almost exclusively generated by human researchers analysing numerical results and drawing conclusions. In this work, we shift the focus on the insights and the knowledge obtained by the machine learning models themselves. In particular, we study how it can be extracted and used to inspire human scientists to increase their intuitions and understanding of natural systems. We apply gradient boosting in decision trees to extract human interpretable insights from big data sets from chemistry and physics. In chemistry, we not only rediscover widely know rules of thumb but also find new interesting motifs that tell us how to control solubility and energy levels of organic molecules. At the same time, in quantum physics, we gain new understanding on experiments for quantum entanglement. The ability to go beyond numerics and to enter the realm of scientific insight and hypothesis generation opens the door to use machine learning to accelerate the discovery of conceptual understanding in some of the most challenging domains of science.
... The concepts for these gates have been discovered using computer-designed quantum experiments, specifically a highly efficient version of the algorithm MELVIN [28]. Several other automated algorithms have been generated recently for the design of novel quantum-optical experiments [30,[54][55][56][57][58]. Our result indicates the possibility that computers can be used in a widely unexplored way, namely to inspire human scientists. ...
An open question in quantum optics is how to manipulate and control complex quantum states in an experimentally feasible way. Here we present concepts for transformations of high-dimensional multiphotonic quantum systems. The proposals rely on two new ideas: (i) a novel high-dimensional quantum nondemolition measurement, (ii) the encoding and decoding of the entire quantum transformation in an ancillary state for sharing the necessary quantum information between the involved parties. Many solutions can readily be performed in laboratories around the world and thereby we identify important pathways for experimental research in the near future. The concepts have been found using the computer algorithm melvin for designing computer-inspired quantum experiments. As opposed to the field of machine learning, here the human learns new scientific concepts by interpreting and analyzing the results presented by the machine. This demonstrates that computer algorithms can inspire new ideas in science, which has a widely unexplored potential that goes far beyond experimental quantum information science.
... Examples range from observations of fundamental quantum properties, such as indefinite causal orders [3] or early demonstrations of Wigner's friend paradox [4,5], high-dimensional quantum communication systems such as quantum key distribution [6,7], entanglement swapping [8] or quantum teleportation [9] and experimental quantum machine learning [10,11] and new propositions for quantum technologies [12][13][14][15][16]. While historically, quantum experiments have been designed by experienced human experts, their unintuitive nature have led to the emergence of computers for designing quantum experiments [17][18][19][20][21][22][23][24]. However, as the state space grows exponentially with the number of photons, this approach is limited to small systems. ...
The parameters of a quantum system grow exponentially with the number of involved quantum particles. Hence, the associated memory requirement goes well beyond the limit of best classic computers for quantum systems composed of a few dozen particles leading to huge challenges in their numerical simulation. This implied that verification, let alone, design of new quantum devices and experiments, is fundamentally limited to small system size. It is not clear how the full potential of large quantum systems can be exploited. Here, we present the concept of quantum computer designed quantum hardware and apply it to the field of quantum optics. Specifically, we map complex experimental hardware for high-dimensional, many-body entangled photons into a gate-based quantum circuit. We show explicitly how digital quantum simulation of Boson Sampling experiments can be realized. Then we illustrate how to design quantum-optical setups for complex entangled photon systems, such as high-dimensional Greenberger-Horne-Zeilinger states and their derivatives. Since photonic hardware is already on the edge of quantum supremacy (the limit beyond which systems can no longer be calculated classically) and the development of gate-based quantum computers is rapidly advancing, our approach promises to be an useful tool for the future of quantum device design.
... [18][19][20]) to date only provide solutions for specific experimental scenarios. For that reason, computational design methods for quantum optical experiments have been introduced [21], in the form of topological search augmented with machine learning [22,23], genetic algorithms [24][25][26], active learning approaches [27], deep recurrent neural networks [28], and optimization of parametrized setups [29]. Unfortunately, due to the complexity and size of the Hilbert space as well as the breadth of quantum optical applications, those algorithms may have severe drawbacks, such as inefficient discovery rates, requirements of a huge amount of training data or specialization on narrow sets of problems. ...
The design of quantum experiments can be challenging for humans. This can be attributed at least in part to counterintuitive quantum phenomena such as superposition or entanglement. In experimental quantum optics, computational and artificial intelligence methods have therefore been introduced to solve the inverse-design problem, which aims to discover tailored quantum experiments with particular desired functionalities. While some computer-designed experiments have been successfully demonstrated in laboratories, these algorithms generally are slow, require a large amount of data or work for specific platforms that are difficult to generalize. Here we present Theseus, an efficient algorithm for the design of quantum experiments, which we use to solve several open questions in experimental quantum optics. The algorithm' core is a physics-inspired, graph-theoretical representation of quantum states, which makes it significantly faster than previous comparable approaches. The gain in speed allows for topological optimization, leading to a reduction of the experiment to its conceptual core. Human scientists can therefore interpret, understand and generalize the solutions without performing any further calculations. We demonstrate Theseus on the challenging tasks of generating and transforming high-dimensional, multi-photonic quantum states. The final solutions are within reach of modern experimental laboratories, promising direct advances for empirical studies of fundamental questions, as well as technical applications such as quantum communication and photonic quantum information processing. In each case, the computer-designed experiment can be interpreted and conceptually understood. We argue that therefore, our algorithm contributes directly to the central aims of science.
... As illustrated in Fig. 2, our experimental setup consists of three modules: state preparation, Alice's measurement, and Bob's measurement. In the state preparation module, entangled photons are generated via type-I spontaneous parametric down-conversion (SPDC) [38][39][40][41][42][43][44][45][46][47][48]. By choosing the setting angle of the half-wave plate (HWP, H 0 ) to be cos 2χ = a, photon pairs are prepared into a family of entangled state |φ + = a |HH + √ 1 − a 2 |V V . ...
We demonstrate one-sided device-independent self-testing of any pure entangled two-qubit state based on a fine-grained steering inequality. The maximum violation of a fine-grained steering inequality can be used to witness certain steerable correlations, which certify all pure two-qubit entangled states. Our experimental results identify which particular pure two-qubit entangled state has been self-tested and which measurement operators are used on the untrusted side. Furthermore, we analytically derive the robustness bound of our protocol, enabling our subsequent experimental verification of robustness through state tomography. Finally, we ensure that the requisite no-signalling constraints are maintained in the experiment.
Uncertainty relations for Hermitian operators have been confirmed through many experiments. However, previous experiments have only tested the special case of non-Hermitian operators, i.e., uncertainty relations for unitary operators. In this study, we explore uncertainty relations for general non-Hermitian operators, which include Hermitian and unitary operators as special cases. We perform experiments with both real and complex non-Hermitian operators for qubit states, and confirm the validity of the uncertainty relations within the experimental error. Our results provide experimental evidence of uncertainty relations for non-Hermitian operators. Furthermore, our methods for realizing and measuring non-Hermitian operators are valuable in characterizing open-system dynamics and enhancing parameter estimation.
Photons are the physical system of choice for performing experimental tests of the foundations of quantum mechanics. Furthermore, photonic quantum technology is a main player in the second quantum revolution, promising the development of better sensors, secure communications, and quantum-enhanced computation. These endeavors require generating specific quantum states or efficiently performing quantum tasks. The design of the corresponding optical experiments was historically powered by human creativity but is recently being automated with advanced computer algorithms and artificial intelligence. While several computer-designed experiments have been experimentally realized, this approach has not yet been widely adopted by the broader photonic quantum optics community. The main roadblocks consist of most systems being closed-source, inefficient, or targeted to very specific use-cases that are difficult to generalize. Here, we overcome these problems with a highly-efficient, open-source digital discovery framework PyTheus, which can employ a wide range of experimental devices from modern quantum labs to solve various tasks. This includes the discovery of highly entangled quantum states, quantum measurement schemes, quantum communication protocols, multi-particle quantum gates, as well as the optimization of continuous and discrete properties of quantum experiments or quantum states. PyTheus produces interpretable designs for complex experimental problems which human researchers can often readily conceptualize. PyTheus is an example of a powerful framework that can lead to scientific discoveries – one of the core goals of artificial intelligence in science. We hope it will help accelerate the development of quantum optics and provide new ideas in quantum hardware and technology.
A quantum channel describes the general evolution of quantum systems, which is therefore a critical issue for quantum simulation and a fundamental element of quantum information. To realize quantum channels, an efficient method is to randomly implement Kraus operators, which does not require any ancillary quantum system. For open systems, it is natural to extend the method from implementation of unitary Kraus operators to nonunitary ones. However, when some Kraus operators are not proportional to unitary ones, this method becomes probabilistic even if the channel is trace preserving. In this paper, to overcome this drawback, we propose an algorithm to deterministically realize arbitrary trace-preserving channels with only one ancillary qubit and finite iterations of evolutions of the combined states of the system and the ancilla. Moreover, to show the validity of the method, we experimentally realize conventional and modified Landau-Streater channels based on the algorithm. Our results shed light on quantum simulation of quantum channels for open systems.
The manipulation of the quantum states of light in linear optical systems has multiple applications in quantum optics and quantum computation. The package QOptCraftgives a collection of methods to solve some of the most usual problems when designing quantum experiments with linear interferometers. The methods include functions that compute the quantum evolution matrix for n photons from the classical description of the system and inverse methods that, for any desired quantum evolution, will either give the complete description of the experimental system that realizes that unitary evolution or, when this is impossible, the complete description of the linear system which approximates the desired unitary with a locally minimal error. The functions in the package include implementations of different known decompositions that translate the classical scattering matrix of a linear system into a list of beam splitters and phase shifters and methods to compute the effective Hamiltonian that describes the quantum evolution of states with n photons. The package is completed with routines for useful tasks like generating random linear optical systems, computing matrix logarithms, and quantum state entanglement measurement via metrics such as the Schmidt rank. The routines are chosen to avoid usual numerical problems when dealing with the unitary matrices that appear in the description of linear systems.
Program summary
Program Title: QOptCraft
CPC Library link to program files: https://doi.org/10.17632/r24hszggf4.1
Developer's repository link: https://github.tel.uva.es/juagar/qoptcraft
Licensing provisions: Apache-2.0
Programming language: Python 3 v3.9.5:0a7dcbd, May 3 2021 17:27:52
Supplementary material:: User's manual at https://github.tel.uva.es/juagar/qoptcraft/-/blob/main/QOptCraft_V1.1_user_guide.pdf
Nature of problem: The evolution of the quantum states of light in linear optical devices can be computed from the scattering matrix of the system using a few alternative points of view. Apart from being able to compute the evolution through a known optical system, it is interesting to consider the less studied inverse problem of design: finding the optical system which gives or approximates a desired evolution. Linear optical systems are limited and can only provide a small subset of all the physically possible quantum transformations on multiple photons. Choosing the best approximation for the evolutions that cannot be achieved with linear optics is not trivial.
This software deals with the analysis of the quantum evolution of multiple photons in linear optical devices and the design of optical setups that achieve or approximate a desired quantum evolution.
Solution method: We have automated multiple computation processes regarding quantum experiments via linear optic devices. The methods rely on the properties of the groups and algebras that describe the problem of light evolution in linear system.
The library QOptCraftfor Python 3 includes known numerical methods for decomposing an optical system into beam splitters and phase shifters and methods to give the quantum evolution of system classically described by a scattering matrix using either the Heisenberg picture evolution of the states, a description based on the permanents of certain matrices or the evolution from the effective Hamiltonian of the system. It also provides methods for the design of achievable evolutions, using the adjoint representation, and for approximating quantum evolutions outside the reach of linear optics with an iterative method using Toponogov's comparison theorem from differential geometry. The package is completed with useful functions that deal with systems including losses and gain, described with quasiunitary matrices, the generation of random matrices and stable implementations of the matrix logarithm.
Additional comments including restrictions and unusual features: The package is designed to work with intermediate scale optical systems. Due to the combinatorial growth in the state space with the number of photons and modes involved, there is an upper limit on the efficiency of any classical calculation. QOptCraftserves as a design tool to explore the building blocks of photonic quantum computers, optical systems that generate useful quantum states of light for their use in metrology or other applications or the design of quantum optics experiments to probe the foundations of quantum mechanics.
Recently, impressive progress has been made in the study of non-Hermitian systems with parity-time symmetry, such as observations of topological properties of physical systems and criticality at exceptional points. A crucial aspect of parity-time symmetric nonunitary dynamics is the information flow between the system and the environment. In this paper, we use the physical quantity, distinguishability between quantum states, to uniformly quantify the information flow between low-dimensional and high-dimensional parity-time symmetric non-Hermitian systems and environments. The numerical results show that the oscillation of quantum state distinguishability and complete information retrieval and can be obtained in the parity-time-unbroken phase. However, the information decays exponentially in the parity-time-broken phase. The exceptional point marks the criticality between reversibility and irreversibility of information flow, and the distinguishability between quantum states exhibits the behavior of power-law decay. Understanding these unique phenomena in nonunitary quantum dynamics provides an important perspective for the study of open quantum systems and contributes to their application in quantum information.
Recently, impressive progress has been made in the study of non-Hermitian systems with parity-time symmetry, such as observations of topological properties of physical systems and criticality at exceptional points. A crucial aspect of parity-time symmetric nonunitary dynamics is the information flow between the system and the environment. In this paper, we use the physical quantity, distinguishability between quantum states, to uniformly quantify the information flow between low-dimensional and high-dimensional parity-time symmetric non-Hermitian systems and environments. The numerical results show that the oscillation of quantum state distinguishability and complete information retrieval and can be obtained in the parity-time-unbroken phase. However, the information decays exponentially in the paritytime-broken phase. The exceptional point marks the criticality between reversibility and irreversibility of information flow, and the distinguishability between quantum states exhibits the behavior of power-law decay. Understanding these unique phenomena in nonunitary quantum dynamics provides an important perspective for the study of open quantum systems and contributes to their application in quantum information.
By studying quantum information masking (QIM) in non-Hermitian quantum systems, we show that mutually orthogonal quantum states can be deterministically masked, while an arbitrary set of quantum states cannot be masked in non-Hermitian quantum systems. We further demonstrate that a set of linearly independent states which are mutually η -orthogonal can be deterministically masked by a pseudo-unitary operator. Moreover, we study robustness of QIM against noisy environments. The robustness of deterministic and probabilistic QIM under different quantum noise channels is analyzed in detail. Accordingly, we propose and discuss the r -uniform probabilistic QIM in multipartite systems.
Decoherence is an unavoidable effect of the quantum system coupled to an environment, which leads to the system evolving from a pure state to a mixed one. A critical issue here is that the reduction of purity is quantum or classical, which corresponds to whether or not entanglement is generated between the system and environment. We report a proof-of-principle experimental investigation of such entanglement in a simple but paradigmatic case, in which a qubit system interacts with a pure dephasing channel. The pure dephasing process is realized in a linear photonic system, and the experiment is a validation in a very controlled setting. By adopting previous methods to detect qubit-environment entanglement, we prove that such entanglement can be detected by performing local measurements on only one subsystem, even when the other subsystem in the pair cannot be detected. Our experiment paves the way for investigating features of open quantum system dynamics.
In conventional quantum mechanics, quantum no-deleting and no-cloning theorems indicate that two different and nonorthogonal states cannot be perfectly and deterministically deleted and cloned, respectively. Here, we investigate the quantum deleting and cloning in a pseudo-unitary system. We first present a pseudo-Hermitian Hamiltonian with real eigenvalues in a two-qubit system. By using the pseudo-unitary operators generated from this pseudo-Hermitian Hamiltonian, we show that it is possible to delete and clone a class of two different and nonorthogonal states, and it can be generalized to arbitrary two different and nonorthogonal pure qubit states. Furthermore, state discrimination, which is strongly related to quantum no-cloning theorem, is also discussed. Last but not least, we simulate the pseudo-unitary operators in conventional quantum mechanics with post-selection, and obtain the success probability of simulations. Pseudo-unitary operators are implemented with a limited efficiency due to the post-selections. Thus, the success probabilities of deleting and cloning in the simulation by conventional quantum mechanics are less than unity, which maintain the quantum no-deleting and no-cloning theorems.
The design of new devices and experiments has historically relied on the intuition of human experts. Now, design inspirations from computers are increasingly augmenting the capability of scientists. We briefly overview different fields of physics that rely on computer-inspired designs using a variety of computational approaches based on topological optimization, evolutionary strategies, deep learning, reinforcement learning or automated reasoning. Then we focus specifically on quantum physics. When designing new quantum experiments, there are two challenges: quantum phenomena are unintuitive, and the number of possible configurations of quantum experiments explodes exponentially. These challenges can be overcome by using computer-designed quantum experiments. We focus on the most mature and practical approaches to find new complex quantum experiments, which have subsequently been realized in the lab. These methods rely on a highly efficient topological search, which can inspire new scientific ideas. We review several extensions and alternatives based on various optimization and machine learning techniques. Finally, we discuss what can be learned from the different approaches and outline several future directions. Designing new experiments in physics is a challenge for humans; therefore, computers have become a tool to expand scientists’ capabilities and to provide creative solutions. This Perspective article examines computer-inspired designs in quantum physics that led to laboratory experiments and inspired new scientific insights.
We demonstrate one-sided device-independent self-testing of any pure two-qubit entangled state based on a fine-grained steering inequality. The maximum violation of a fine-grained steering inequality can be used to witness certain steerable correlations, which certify all pure two-qubit entangled states. Our experimental results identify which particular pure two-qubit entangled state has been self-tested and which measurement operators are used on the untrusted side. Furthermore, we analytically derive the robustness bound of our protocol, enabling our subsequent experimental verification of robustness through state tomography. Finally, we ensure that the requisite no-signaling constraints are maintained in the experiment.
Topology in quench dynamics gives rise to intriguing dynamic topological phenomena, which are intimately connected to the topology of static Hamiltonians yet challenging to probe experimentally. Here we theoretically characterize and experimentally detect momentum-time skyrmions in parity-time (PT)-symmetric non-unitary quench dynamics in single-photon discrete-time quantum walks. The emergent skyrmion structures are protected by dynamic Chern numbers defined for the emergent two-dimensional momentum-time submanifolds, and are revealed through our experimental scheme enabling the construction of time-dependent non-Hermitian density matrices via direct measurements in position space. Our work experimentally reveals the interplay of PT symmetry and quench dynamics in inducing emergent topological structures, and highlights the application of discrete-time quantum walks for the study of dynamic topological phenomena.
Photonics is a promising platform for implementing universal quantum information processing. Its main challenges include precise control of massive circuits of linear optical components and effective implementation of entangling operations on photons. By using large-scale silicon photonic circuits to implement an extension of the linear combination of quantum operators scheme, we realize a fully programmable two-qubit quantum processor, enabling universal two-qubit quantum information processing in optics. The quantum processor is fabricated with mature CMOS-compatible processing and comprises more than 200 photonic components. We programmed the device to implement 98 different two-qubit unitary operations (with an average quantum process fidelity of 93.2 ± 4.5%), a two-qubit quantum approximate optimization algorithm, and efficient simulation of Szegedy directed quantum walks. This fosters further use of the linear-combination architecture with silicon photonics for future photonic quantum processors.
We report the experimental detection of bulk topological invariants in nonunitary discrete-time quantum walks with single photons. The nonunitarity of the quantum dynamics is enforced by periodically performing partial measurements on the polarization of the walker photon, which effectively introduces loss to the dynamics. The topological invariant of the nonunitary quantum walk is manifested in the quantized average displacement of the walker, which is probed by monitoring the photon loss. We confirm the topological properties of the system by observing localized edge states at the boundary of regions with different topological invariants. We further demonstrate the robustness of both the topological properties and the measurement scheme of the topological invariants against disorder.
Universal multiport interferometers, which can be programmed to implement any linear transformation between multiple channels, are emerging as a powerful tool for both classical and quantum photonics. These interferometers are typically composed of a regular mesh of beam splitters and phase shifters, allowing for straightforward fabrication using integrated photonic architectures and ready scalability. The current, standard design for universal multiport interferometers is based on work by Reck et al. [Phys. Rev. Lett. 73, 58 (1994)PRLTAO0031-900710.1103/PhysRevLett.73.58]. We demonstrate a new design for universal multiport interferometers based on an alternative arrangement of beam splitters and phase shifters, which outperforms that by Reck et al. Our design requires half the optical depth of the Reck design and is significantly more robust to optical losses.
We propose and analyze a new approach based on parity-time ()
symmetric microcavities with balanced gain and loss to enhance the performance
of cavity-assisted metrology. We identify the conditions under which
-symmetric microcavities allow to improve sensitivity beyond what
is achievable in loss-only systems. We discuss its application to the detection
of mechanical motion, and show that the sensitivity is significantly enhanced
in the vicinity of the transition point from unbroken- to broken-
regimes. We believe that our results open a new direction for
-symmetric physical systems and it may find use in ultra-high
precision metrology and sensing.
Quantum mechanics predicts a number of at first sight counterintuitive
phenomena. It is therefore a question whether our intuition is the best way to
find new experiments. Here we report the development of the computer algorithm
Melvin which is able to find new experimental implementations for the creation
and manipulation of complex quantum states. And indeed, the discovered
experiments extensively use unfamiliar and asymmetric techniques which are
challenging to understand intuitively. The results range from the first
implementation of a high-dimensional Greenberger-Horne-Zeilinger (GHZ) state,
to a vast variety of experiments for asymmetrically entangled quantum states -
a feature that can only exist when both the number of involved parties and
dimensions is larger than 2. Additionally, new types of high-dimensional
transformations are found that perform cyclic operations. Melvin autonomously
learns from solutions for simpler systems, which significantly speeds up the
discovery rate of more complex experiments. The ability to automate the design
of a quantum experiment can be applied to many quantum systems and allows the
physical realization of quantum states previously thought of only on paper.
Entanglement plays a central role in the field of quantum information
science. It is well known that the degree of entanglement cannot be increased
under local operations. Here, we show that the concurrence of a bipartite
entangled state can be increased under the local PT -symmetric operation. This
violates the property of entanglement monotonicity. We also use the Bell-CHSH
and steering inequalities to explore this phenomenon.
We show that non-Hermitian dynamics generate substantial entanglement in
many-body systems. We consider the non-Hermitian Lipkin-Meshkov-Glick model and
show that the non-Hermitian phase transition occurs with maximum multi-particle
entanglement: there is full N-particle entanglement at the transition, in
contrast to the Hermitian case. The non-Hermitian model also exhibits more spin
squeezing than the Hermitian model, showing that non-Hermitian dynamics are
useful for quantum metrology. Experimental implementations with cavity-QED and
trapped ions are discussed.
Bender et al. [Phys. Rev. Lett. 80, 5243 (1998)] have developed PT-symmetric quantum theory as an extension of quantum theory to non-Hermitian Hamiltonians. We show that when this model has a local PT symmetry acting on composite systems, it violates the nonsignaling principle of relativity. Since the case of global PT symmetry is known to reduce to standard quantum mechanics A. Mostafazadeh [J. Math. Phys. 43, 205 (2001)], this shows that the PT-symmetric theory is either a trivial extension or likely false as a fundamental theory.
The condition of self-adjointness ensures that the eigenvalues of a Hamiltonian are real and bounded below. Replacing this condition by the weaker condition of PT symmetry, one obtains new infinite classes of complex Hamiltonians whose spectra are also real and positive. These PT symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space. This paper describes the unusual classical and quantum properties of these theories. {copyright} {ital 1998} {ital The American Physical Society}
The objective of this paper is to explain and elucidate the formalism of quantum mechanics by applying it to a well-known problem in conventional Hermitian quantum mechanics, namely the problem of state discrimination. Suppose that a system is known to be in one of two quantum states, |ψ1〉 or |ψ2〉. If these states are not orthogonal, then the requirement of unitarity forbids the possibility of discriminating between these two states with one measurement; that is, determining with one measurement what state the system is in. In conventional quantum mechanics, there is a strategy in which successful state discrimination can be achieved with a single measurement but only with a success probability p that is less than unity. In this paper, the state-discrimination problem is examined in the context of quantum mechanics and the approach is based on the fact that a non-Hermitian -symmetric Hamiltonian determines the inner product that is appropriate for the Hilbert space of physical states. It is shown that it is always possible to choose this inner product so that the two states |ψ1〉 and |ψ2〉 are orthogonal. Using quantum mechanics, one cannot achieve a better state discrimination than in ordinary quantum mechanics, but one can instead perform a simulated quantum state discrimination, in which with a single measurement a perfect state discrimination is realized with probability p.
Solving linear systems of equations is ubiquitous in all areas of science and
engineering. With rapidly growing data sets, such a task can be intractable for
classical computers, as the best known classical algorithms require a time
proportional to the number of variables N. A recently proposed quantum
algorithm shows that quantum computers could solve linear systems in a time
scale of order log(N), giving an exponential speedup over classical computers.
Here we realize the simplest instance of this algorithm, solving 2*2 linear
equations for various input vectors on a quantum computer. We use four quantum
bits and four controlled logic gates to implement every subroutine required,
demonstrating the working principle of this algorithm.
The general impossibility of determining the state of a single quantum system is proved for arbitrary measuring schemes, including a succession of measurements. Some recently proposed methods are critically examined. A scheme for tomographic measurements on a single copy of a radiation field is devised, showing that the system state is perturbed however weak the system-apparatus interaction is, due to the need of preparing the apparatus in a highly {open_quote}{open_quote}squeezed{close_quote}{close_quote} state. {copyright} {ital 1996 The American Physical Society.}
Although perfect copying of unknown quantum systems is forbidden by the laws of quantum mechanics, approximate cloning is possible. A natural way of realizing quantum cloning of photons is by stimulated emission. In this context, the fundamental quantum limit to the quality of the clones is imposed by the unavoidable presence of spontaneous emission. In our experiment, a single input photon stimulates the emission of additional photons from a source on the basis of parametric down-conversion. This leads to the production of quantum clones with near-optimal fidelity. We also demonstrate universality of the copying procedure by showing that the same fidelity is achieved for arbitrary input states.
Requiring that a Hamiltonian be Hermitian is overly restrictive. A consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but satisfies the less restrictive and more physical condition of space-time reflection symmetry (PT symmetry). One might expect a non-Hermitian Hamiltonian to lead to a violation of unitarity. However, if PT symmetry is not spontaneously broken, it is possible to construct a previously unnoticed symmetry C of the Hamiltonian. Using C, an inner product whose associated norm is positive definite can be constructed. The procedure is general and works for any PT-symmetric Hamiltonian. Observables exhibit CPT symmetry, and the dynamics is governed by unitary time evolution. This work is not in conflict with conventional quantum mechanics but is rather a complex generalization of it.
Given an initial quantum state |psi(I)> and a final quantum state |psi(F)>, there exist Hamiltonians H under which |psi(I)> evolves into |psi(F)>. Consider the following quantum brachistochrone problem: subject to the constraint that the difference between the largest and smallest eigenvalues of H is held fixed, which H achieves this transformation in the least time tau? For Hermitian Hamiltonians tau has a nonzero lower bound. However, among non-Hermitian PT-symmetric Hamiltonians satisfying the same energy constraint, tau can be made arbitrarily small without violating the time-energy uncertainty principle. This is because for such Hamiltonians the path from |psi(I)> to |psi(F)> can be made short. The mechanism described here is similar to that in general relativity in which the distance between two space-time points can be made small if they are connected by a wormhole. This result may have applications in quantum computing.
we envisage a novel quantum cloning machine, which takes an input state and produces an output state whose success branch can exist in a linear superposition of multiple copies of the input state and the failure branch exist in a superposition of composite state independent of the input state. We prove that unknown non-orthogonal states chosen from a set can evolve into a linear superposition of multiple clones by a unitary process if and only if the states are linearly independent. We derive a bound on the success probability of the novel cloning machine. We argue that the deterministic and probabilistic clonings are special cases of our novel cloning machine.
We derive a tight upper bound for the fidelity of a universal N to M qubit cloner, valid for any M \geq N, where the output of the cloner is required to be supported on the symmetric subspace. Our proof is based on the concatenation of two cloners and the connection between quantum cloning and quantum state estimation. We generalise the operation of a quantum cloner to mixed and/or entangled input qubits described by a density matrix supported on the symmetric subspace of the constituent qubits. We also extend the validity of optimal state estimation methods to inputs of this kind.
We establish the best possible approximation to a perfect quantum cloning machine which produces two clones out of a single input. We analyze both universal and state-dependent cloners. The maximal fidelity of cloning is shown to be 5/6 for universal cloners. It can be achieved either by a special unitary evolution or by a novel teleportation scheme. We construct the optimal state-dependent cloners operating on any prescribed two non-orthogonal states, discuss their fidelities and the use of auxiliary physical resources in the process of cloning. The optimal universal cloners permit us to derive a new upper bound on the quantum capacity of the depolarizing quantum channel. Comment: 30 pages (RevTeX), 2 figures (epsf), further results and further authors added, to appear in Physical Review A
We study certain linear and antilinear symmetry generators and involution operators associated with pseudo-Hermitian Hamiltonians and show that the theory of pseudo-Hermitian operators provides a simple explanation for the recent results of Bender, Brody and Jones (quant-ph/0208076) on the CPT-symmetry of a class of PT-symmetric non-Hermitian Hamiltonians. We present a natural extension of these results to the class of diagonalizable pseudo-Hermitian Hamiltonians H with a discrete spectrum. In particular, we introduce generalized parity (), time-reversal (), and charge-conjugation () operators and establish the - and -invariance of H. Comment: 25 pages, slightly revised version, accepted for publication in J. Math. Phys
We show that a diagonalizable (non-Hermitian) Hamiltonian H is
pseudo-Hermitian if and only if it has an antilinear symmetry, i.e., a symmetry
generated by an invertible antilinear operator. This implies that the
eigenvalues of H are real or come in complex conjugate pairs if and only if H
possesses such a symmetry. In particular, the reality of the spectrum of H
implies the presence of an antilinear symmetry. We further show that the
spectrum of H is real if and only if there is a positive-definite inner-product
on the Hilbert space with respect to which H is Hermitian or alternatively
there is a pseudo-canonical transformation of the Hilbert space that maps H
into a Hermitian operator.
We give a necessary and sufficient condition for the reality of the spectrum
of a non-Hermitian Hamiltonian admitting a complete set of biorthonormal
eigenvectors.
We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudo-supersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler-DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum. Comment: Revised version, to appear in J. Math. Phys
The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose +complex conjugate) is replaced by the physically transparent condition of space–time reflection ( ) symmetry. If H has an unbroken symmetry, then the spectrum is real. Examples of -symmetric non-Hermitian quantum-mechanical Hamiltonians are and . Amazingly, the energy levels of these Hamiltonians are all real and positive!
Does a -symmetric Hamiltonian H specify a physical quantum theory in which the norms of states are positive and time evolution is unitary? The answer is that if H has an unbroken symmetry, then it has another symmetry represented by a linear operator . In terms of , one can construct a time-independent inner product with a positive-definite norm. Thus, -symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution.
The Lee model provides an excellent example of a -symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm 'ghost' state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. The correct interpretation of the ghost is simply that the non-Hermitian Lee-model Hamiltonian is -symmetric. The operator for the Lee model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.
We experimentally simulate nonunitary quantum dynamics using a single-photon interferometric network and study the information flow between a parity-time- (PT-)symmetric non-Hermitian system and its environment. We observe oscillations of quantum-state distinguishability and complete information retrieval in the PT-symmetry-unbroken regime. We then characterize in detail critical phenomena of the information flow near the exceptional point separating the PT-unbroken and PT-broken regimes, and demonstrate power-law behavior in key quantities such as the distinguishability and the recurrence time. We also reveal how the critical phenomena are affected by symmetry and initial conditions. Finally, introducing an ancilla as an environment and probing quantum entanglement between the system and the environment, we confirm that the observed information retrieval is induced by a finite-dimensional entanglement partner in the environment. Our work constitutes the first experimental characterization of critical phenomena in PT-symmetric nonunitary quantum dynamics.
Signaled by nonanalyticities in the time evolution of physical observables, dynamic quantum phase transitions (DQPTs) emerge in quench dynamics of topological systems and possess an interesting geometric origin captured by dynamic topological order parameters (DTOPs). In this Letter, we report the experimental study of DQPTs using discrete-time quantum walks of single photons. We simulate quench dynamics between distinct Floquet topological phases using quantum-walk dynamics and experimentally characterize DQPTs and the underlying DTOPs through interference-based measurements. The versatile photonic quantum-walk platform further allows us to experimentally investigate DQPTs for mixed states and in parity-time-symmetric nonunitary dynamics for the first time. Our experiment directly confirms the relation between DQPTs and DTOPs in quench dynamics of topological systems and opens up the avenue of simulating emergent topological phenomena using discrete-time quantum-walk dynamics.
Topological matter exhibits exotic properties yet phases characterized by large topological invariants are difficult to implement, despite rapid experimental progress. A promising route toward higher topological invariants is via engineered Floquet systems, particularly in photonics, where flexible control holds the potential of extending the study of conventional topological matter to novel regimes. Here we implement a one-dimensional photonic quantum walk to explore large winding numbers. By introducing partial measurements and hence loss into the system, we detect winding numbers of three and four in multistep nonunitary quantum walks, which agree well with theoretical predictions. Moreover, by probing statistical moments of the walker, we identify locations of topological phase transitions in the system and reveal the breaking of pseudounitary near topological phase boundaries. As the winding numbers are associated with nonunitary time evolution, our investigation enriches understanding of topological phenomena in nonunitary settings.
We experimentally investigate topological phenomena in one-dimensional discrete-time photonic quantum walks using a combination of methods. We first detect winding numbers of the quantum walk by directly measuring the average chiral displacement, which oscillates around quantized winding numbers for finite-step quantum walks. Topological phase transitions can be identified as changes in the center of oscillation of the measured chiral displacement. The position of topological phase transition is then confirmed by measuring the moments of the walker probability distribution. Finally, we observe localized edge states at the boundary of regions with different winding numbers. We also confirm the robustness of edge states against chiral-symmetry-preserving disorder.
In rotating ring resonators, resonant frequencies are split because of the Sagnac effect. The rotation sensitivity of the frequency splitting characterizes the sensitivity of resonator-based optical gyroscopes. In this paper, it is shown that the sensitivity of frequency splitting can be significantly enhanced in a ring resonator operating at an exceptional point (EP), which is a non-Hermitian degeneracy where two eigenvalues and the corresponding eigenmodes coalesce. As an example, a ring resonator with a periodic structure is proposed and theoretically and numerically studied. It is numerically demonstrated that in the resonator operating near an EP, the rotation-induced frequency splitting can be more than two orders greater than that in conventional ring resonators. In addition, this paper discusses the influence of the resonator loss on the measurement sensitivity of the frequency splitting and a method of rotation detection based on rotation-induced changes of eigenmodes near an EP.
The study of non-Hermitian systems with parity–time (PT) symmetry is a rapidly developing frontier. Realized in recent experiments, PT-symmetric classical optical systems with balanced gain and loss hold great promise for future applications. Here we report the experimental realization of passive PT-symmetric quantum dynamics for single photons by temporally alternating photon losses in the quantum walk interferometers. The ability to impose PT symmetry allows us to realize and investigate Floquet topological phases driven by PT-symmetric quantum walks. We observe topological edge states between regions with different bulk topological properties and confirm the robustness of these edge states with respect to PT-symmetry-preserving perturbations and PT-symmetry-breaking static disorder. Our results contribute towards the realization of quantum mechanical PT-synthetic devices and suggest exciting possibilities for the exploration of the topological properties of non-Hermitian systems using discrete-time quantum walks.
By studying the information flow between a general parity-time (PT) symmetric non-Hermitian system and its environment, we find that the complete information retrieval from the environment can be achieved in the PT-unbroken phase, whereas no retrieval can be made in the PT-broken phase. Thus the PT transition point also marks the reversible/irreversible transition of the information flow. To understand the physics behind this criticality, we embed a PT-symmetric system into a larger Hilbert space in which the total system obeys unitary dynamics, and show that the dimension of the ancilla required for the extension diverges at the exceptional point where the recurrence time of the information retrieval exhibits critical behavior.
A general theory of sensors based on the detection of splittings of resonant frequencies or energy levels operating at so-called exceptional points is presented. Exploiting the complex-square-root topology near such non-Hermitian degeneracies has a great potential for enhanced sensitivity. Passive and active systems are discussed. The theory is specified for whispering-gallery microcavity sensors for particle detection. As example, a microdisk with two holes is studied numerically. The theory and numerical simulations demonstrate a sevenfold enhancement of the sensitivity.
Several types of sensors used in physics are based on the detection of splittings of resonant frequencies or energy levels. We propose here to operate such sensors at so-called exceptional points, which are degeneracies in open wave and quantum systems where at least two resonant frequencies or energy levels and the corresponding eigenstates coalesce. We argue that this has great potential for enhanced sensitivity provided that one is able to measure both the frequency splitting as well as the linewidth splitting. We apply this concept to a microcavity sensor for single-particle detection. An analytical theory and numerical simulations prove a more than threefold enhanced sensitivity. We discuss the possibility to resolve individual linewidths using active optical microcavities.
If a photon of definite polarization encounters an excited atom, there is typically some nonvanishing probability that the atom will emit a second photon by stimulated emission. Such a photon is guaranteed to have the same polarization as the original photon. But is it possible by this or any other process to amplify a quantum state, that is, to produce several copies of a quantum system (the polarized photon in the present case) each having the same state as the original? If it were, the amplifying process could be used to ascertain the exact state of a quantum system: in the case of a photon, one could determine its polarization by first producing a beam of identically polarized copies and then measuring the Stokes parameters1. We show here that the linearity of quantum mechanics forbids such replication and that this conclusion holds for all quantum systems.
It is shown that in principle a device exists which would duplicate a quantum system within a class of quantum states if and only if those quantum states are mutually orthogonal. The possible existence of a related noiseless photon amplifier is also established.
A recent proposal to achieve faster-than-light communication by means of an EPR-type experimental set-up is examined. We demonstrate that such superluminal communication is not possible. The crucial role of the linearity of the quantum mechanical evolution laws in preventing causal anomalies is stressed.
We analyze to what extent it is possible to copy arbitrary states of a two-level quantum system. We show that there exists a "universal quantum copying machine", which approximately copies quantum mechanical states in such a way that the quality of its output does not depend on the input. We also examine a machine which combines a unitary transformation with a selective measurement to produce good copies of states in a neighborhood of a particular state. We discuss the problem of measurement of the output states. Comment: RevTex, 26 pages, to appear in Physical Review A
An algorithmic proof that any discrete finite-dimensional unitary operator can be constructed in the laboratory using optical devices is given. Our recursive algorithm factorizes any N×N unitary matrix into a sequence of two-dimensional beam splitter transformations. The experiment is built from the corresponding devices. This also permits the measurement of the observable corresponding to any discrete Hermitian matrix. Thus optical experiments with any type of radiation (photons, atoms, etc.) exploring higher-dimensional discrete quantum systems become feasible.
We show that, given a general mixed state for a quantum system, there are no physical means for {\it broadcasting\/} that state onto two separate quantum systems, even when the state need only be reproduced marginally on the separate systems. This result generalizes and extends the standard no-cloning theorem for pure states. Comment: 11 pages, formatted in RevTeX
The promise of tremendous computational power, coupled with the development of robust error-correcting schemes, has fuelled extensive efforts to build a quantum computer. The requirements for realizing such a device are confounding: scalable quantum bits (two-level quantum systems, or qubits) that can be well isolated from the environment, but also initialized, measured and made to undergo controllable interactions to implement a universal set of quantum logic gates. The usual set consists of single qubit rotations and a controlled-NOT (CNOT) gate, which flips the state of a target qubit conditional on the control qubit being in the state 1. Here we report an unambiguous experimental demonstration and comprehensive characterization of quantum CNOT operation in an optical system. We produce all four entangled Bell states as a function of only the input qubits' logical values, for a single operating condition of the gate. The gate is probabilistic (the qubits are destroyed upon failure), but with the addition of linear optical quantum non-demolition measurements, it is equivalent to the CNOT gate required for scalable all-optical quantum computation.
We construct a probabilistic quantum cloning machine by a general unitary-reduction operation. With a postselection of the measurement results, the machine yields faithful copies of the input states. It is shown that the states secretly chosen from a certain set \=\left\{\left| \Psi_1\right> ,\left| \Psi_2\right> ,... ,\left| \Psi_n\right> \right\} % \left| \Psi_1\right>, \left| \Psi_2\right>, ... , and \left| \Psi_n\right>n+1n$ linearly independent states. The optimal efficiencies for this type of measurement are obtained. Comment: Extension of quant-ph/9705018, 12pages, latex, to appear in Phys. Rev. Lett