János A. Bergou

City University of New York - Hunter College | Hunter CUNY · Department of Physics and Astronomy

We determine the optimal measurement that maximizes the average information gain about the state of a qubit system. The qubit is prepared in one of two known states with known prior probabilities. To treat the problem analytically we employ the formalism developed for the maximum confidence quantum state discrimination strategy and obtain the POVM...

Quantum entanglement assisted with measurements provides various pathways to communicate information to parties within a network. In this work, we generalize a previous broadcasting protocol and present schemes to broadcast product and multipartite entangled quantum states, where in the latter case the sender can remotely add phase gates or abort d...

Quantum entanglement assisted with measurements provides various pathways to communicate information to parties within a network. In this work, we generalize a previous broadcasting protocol and present schemes to broadcast product and multi-partite entangled quantum states, where in the latter case the sender can remotely add phase gates or abort...

Quantum information processing using linear optics is challenging due to the limited set of deterministic operations achievable without using complicated resource-intensive methods. While techniques such as the use of ancillary photons can enhance the information processing capabilities of linear optical systems, they are technologically demanding....

The no-cloning theorem forbids the distribution of an unknown state to more than one receiver. However, if the sender knows the state, and the state is chosen from a restricted set of possibilities, a procedure known as remote state preparation can be used to broadcast a state. Here we examine a remote state preparation protocol that can be used to...

The no-cloning theorem forbids the distribution of an unknown state to more than one receiver. However, if the sender knows the state, and the state is chosen from a restricted set of possibilities, a procedure known as remote state preparation can be used to broadcast a state. Here we examine a remote state preparation protocol that can be used to...

Quantum information processing using linear optics is challenging due to the limited set of deterministic operations achievable without using complicated resource-intensive methods. While techniques such as the use of ancillary photons can enhance the information processing capabilities of linear optical systems they are technologically demanding....

Entanglement is an essential feature of quantum mechanics. If two systems are entangled, there are not individual quantum states describing each system, only a global state describing both. Systems that are entangled can exhibit strong correlations, and it is possible to make use of those correlations in quantum communication protocols. In this cha...

Photons are the only viable approach for long distance transmission of quantum information. A capability that has been impressively demonstrated with the distribution of quantum keys and entanglement via satellite. It is therefore natural to ask if qubits encoded in photons could also be used for quantum computation.

We have already come across the Shannon and von Neumann entropies in the chapter on entanglement. The Shannon entropy assesses the information content of a classical probability distribution while the von Neumann entropy does the same for its quantum counterpart, the quantum state (density matrix) of a quantum system. We have also come across the t...

Measurements are an integral part of quantum information processing. Reading out the quantum information at the end of the processing pipeline is equivalent to learning what final state the system is in at the output since information is encoded in the state. In fact, information is the state itself. Since finding out the state of a system can be d...

Quantum communication is the most advanced area of quantum information processing and quantum computing. This is where the most fundamental features of quantum mechanics are only a short step away from spectacular practical applications. We have already seen two such applications: Dense coding and teleportation. In this chapter we shall deal with w...

In the remaining chapters of the book we will survey the leading approaches for implementing quantum computation. This is a vast, and very active field of research. The presentation will not be comprehensive, but will strive to explain the basic physical principles underlying the most promising approaches, as well as providing a snapshot of the cur...

In this chapter we shall look at a number of quantum algorithms. We are going to compare their performance, in terms of number of steps, to classical algorithms that accomplish the same task.

In this chapter we will study qubits encoded in superconducting circuits and semiconductor quantum dots. In contrast to the qubits discussed so far these approaches do not involve optics or lasers and are therefore closer to the technologies that are widely used today for information processing. On the other hand maintaining and controlling quantum...

Time evolution in textbook quantum mechanics is represented by unitary maps |ψ〉→ U|ψ〉 and ρ → UρU†, where U = e−itH. This is not the most general evolution possible. We can couple our system to another one, evolve both with a unitary operator that will, in general, create entanglement between the two systems, and then trace out the second system. T...

If we are given quantum information in the form of qubits in a particular state, we have seen that we can process that information by sending the qubits through different sequences of gates. A particular collection of gates constitutes a quantum machine that manipulates the information encoded in the qubits in a particular way. A quantum machine ca...

The stabilizer formalism provides another method of constructing quantum error-correcting codes. However, we are going to be interested in it primarily for a different reason; it allows us to prove the Gottesman–Knill theorem. That theorem serves as a useful warning. Just because you are manipulating qubits with quantum gates does not guarantee tha...

One of the biggest problems in building a quantum computer is noise or decoherence. Qubits are coupled to other systems whether we want them to be or not, e.g. atoms couple to the electromagnetic field and spins couple to other spins via dipole-dipole interactions. These unwanted couplings can cause errors, and we need to protect quantum informatio...

We are going to require a more general description of a quantum state than that given by a state vector. The density matrix provides such a description. Its use is required when we are discussing an ensemble of pure states, or when we are describing a subsystem of a larger system.

We study the role of average concurrence in entanglement swapping in quantum networks. We begin with qubit pure states, and there is a very simple rule governing the propagation of average concurrence in multiple swaps. We find a similarly simple rule for average concurrence when creating a Greenberger-Horne-Zeilinger state from three entangled pai...

We study the role of average concurrence in entanglement swapping in quantum networks. We begin with qubit pure states, and there is a very simple rule governing the propagation of average concurrence in multiple swaps. We look at examples of mixed qubit states, and find the relation for pure states gives an upper bound on what is possible with mix...

This new edition of a well-received textbook provides a concise introduction to both the theoretical and experimental aspects of quantum information at the graduate level. While the previous edition focused on theory, the book now incorporates discussions of experimental platforms. Several chapters on experimental implementations of quantum informa...

We present wave-particle-duality relations involving the relative entropy coherence measure, which plays a prominent role in the resource theory of coherence. The main input in these relations is an entropic bound for the which-way information, which we derive in this paper. We show that this latter crucially depends on the choice of the measuremen...

Quantum retrodiction is a time-symmetric approach to quantum mechanics with applications in a number of important problems. One of the major challenges to its more widespread applicability is the restriction of its symmetric formalism to unbiased sources. The main result of this paper is to develop a general theory yielding a symmetric formalism fo...

We study sequential state discrimination measurements performed on the same qubit by subsequent observers. Specifically, we focus on the case when the observers perform a kind of a minimum-error type state discriminating measurement where the goal of the observers is to maximize their joint probability of successfully guessing the state that the qu...

We present wave-particle duality relations involving the relative entropy coherence measure, which plays a prominent role in the resource theory of coherence. The main input in these relations is an entropic bound for the which-way information, which we derive in this letter. We show that this latter crucially depends on the choice of the measureme...

In a recent paper [Phys. Rev. Lett. 111, 100501 (2013)], a scheme was proposed where subsequent observers can extract unambiguous information about the initial state of a qubit, with finite joint probability of success. Here, we generalize the problem for arbitrary preparation probabilities (arbitrary priors). We discuss two different schemes: one...

In a recent paper [Phys. Rev. Lett. 111, 100501 (2013)], a scheme was proposed where subsequent observers can extract unambiguous information about the initial state of a qubit, with finite joint probability of success. Here, we generalize the problem for arbitrary preparation probabilities (arbitrary priors). We discuss two different schemes: one...

To set the record straight, we present a brief history of interferometric complementarity relations, including those recently rediscovered by other groups, and results for multi-path interferometers and finite groups employing l1 and entropic coherence measures.

We show that the protocol known as quantum state separation can be used to transfer information between the phase and path of a particle in an interferometer. When applied to a quantum eraser, this allows us to erase some, but not all, of the path information. We can control how much path information we wish to erase.

Wave-particle duality relations express the fact that knowledge about the path a particle took suppresses information about its wave-like properties, in particular, its ability to generate an interference pattern. Recently, duality relations in which the wave-like properties are quantified by using measures of quantum coherence have been proposed....

We show that the protocol known as quantum state separation can be used to transfer information between the phase and path of a particle in an interferometer. When applied to a quantum eraser, this allows us to erase some, but not all, of the path information. We can control how much path information we wish to erase.

We give operational meaning to wave-particle duality in terms of discrimination games. Duality arises as a constraint on the probability of winning these games. The games are played with the aid of an n-port interferometer, and involve 3 parties, Alice and Bob, who cooperate, and the House, who supervises the game. In one game called ways they atte...

Perfect cloning of a known set of states with arbitrary prior probabilities is possible if we allow the cloner to sometimes fail completely. In the optimal case the probability of failure is at its minimum allowed by the laws of quantum mechanics. Here we show that it is possible to lower the failure rate below that of the perfect probabilistic clo...

We find two relations between coherence and path-information in a multi-path interferometer. The first builds on earlier results for the two-path interferometer, which used minimum-error state discrimination between detector states to provide the path information. For visibility, which was used in the two-path case, we substitute a recently defined...

Probabilistic quantum state transformations can be characterized by the degree of state separation they provide. This, in turn, sets limits on the success rate of these transformations. We consider optimum state separation of two known pure states in the general case where the known states have arbitrary a priori probabilities. The problem is formu...

We consider specific signatures of squeezing for time-modulated light fields and propose the scheme of an optical parametric oscillator driven by a continuously modulated pump field. The application of a periodically modulated driving field instead of a continuous wave field drastically improves the degree of quadrature integral squeezing. This qua...

We find a relation between coherence and path-information in a multi-path interferometer. This work builds on earlier results for the two-path interferometer, which used minimum-error state discrimination between detector states to provide the path information. For visibility, which was used in the two-path case, we substitute a recently defined me...

We discuss a state discriminator that unambiguously distinguishes between two quantum registers prepared with multiple copies of two unknown qubits. This device achieves the optimal performance by von Neumann measurement and general POVM in different ranges of the input preparation probabilities, respectively, and in the limit of very large program...

We solve the long-standing problem of making n perfect clones from m copies of one of two known pure states with minimum failure probability in the general case where the known states have arbitrary a priori probabilities. The solution emerges from a geometric formulation of the problem. This formulation also reveals a deeper connection between clo...

In the photon number distribution p(θ,k) of the micromaser, distinct structures are formed by the ridges connecting the peaks in the θ-k space. We refer to these structures as phases. By a simple condition we can distinguish between "semiclassical" and "quantum" regimes of operation near and far above threshold, respectively. In the semiclassical r...

We discuss sequential unambiguous state-discrimination measurements performed on the same qubit. Alice prepares a qubit in one of two possible states. The qubit is first sent to Bob, who measures it, and then on to Charlie, who also measures it. The object in both cases is to determine which state Alice sent. In an unambiguous state discrimination...

Measurements are an integral part of quantum information processing. Reading out the quantum information at the end of the processing pipeline is equivalent to learning what final state the system is in at the output since information is encoded in the state. In fact, information is the state itself. Since finding out the state of a system can be d...

Time evolution in textbook quantum mechanics is represented by unitary maps \(\vert \psi \rangle \rightarrow U\vert \psi \rangle\) and \(\rho \rightarrow U\rho {U}^{\dag }\), where \(U ={ \mathrm{e}}^{-itH}\). This is not the most general evolution possible. We can couple our system to another one, evolve both with a unitary operator that will, in...

Quantum communication is the most advanced area of quantum information processing and quantum computing. This is where the most fundamental features of quantum mechanics are only a short step away from spectacular practical applications. We have already seen two such applications: dense coding and teleportation. In this chapter we shall deal with w...

In this chapter we shall look at a number of quantum algorithms. We are going to compare their performance, in terms of number of steps, to classical algorithms that accomplish the same task.

One of the biggest problems in building a quantum computer is noise or decoherence. Qubits are coupled to other systems whether we want them to be or not, e.g., atoms couple to the electromagnetic field and spins couple to other spins via dipole–dipole interactions. These unwanted couplings can cause errors, and we need to protect quantum informati...

We are going to require a more general description of a quantum state than that given by a state vector. The density matrix provides such a description. Its use is required when we are discussing an ensemble of pure states or when we are describing a subsystem of a larger system.

We begin with some definitions. Consider a quantum state in a tensor product Hilbert space, \(\mathcal{H} = \mathcal{H}_{A} \otimes \mathcal{H}_{B}\).

We present an experimentally feasible method, based on currently available cavity QED technology, to generate n-partite linear cluster and graph states in external degree of freedom of atoms. The scheme is based on first tagging n two-level atoms with the respective cavity fields in momentum space. Later on an effective Ising interaction between su...

Preface.- Introduction.- The Density Matrix.- Entanglement.- Generalized Quantum Dynamics.- Quantum Measurement Theory.- Quantum Cryptography.- Quantum Algorithms.- Quantum Machines.- Decoherence and Quantum Error Correction.- Index.

A complete geometric view is presented for the optimal unambiguous discrimination among N>2 pure states. A single intuitive picture contains all aspects of the problem: linear independence of the states, positivity of the detection operators, and a graphic method for finding and classifying the optimal solutions. The method is illustrated on the ex...

In this paper we present the solution to the problem of optimally discriminating among quantum states, i.e., identifying the states with maximum probability of success when a certain fixed rate of inconclusive answers is allowed. By varying the inconclusive rate, the scheme optimally interpolates between Unambiguous and Minimum Error discrimination...

We study the probabilistic cloning of three symmetric states. These states are defined by a single complex quantity, the inner product among them. We show that three different probabilistic cloning machines are necessary to optimally clone all possible families of three symmetric states. We also show that the optimal cloning probability of generati...

We propose a quantum circuit implementation of the unambiguous quantum state discriminator. The circuit is made entirely of standard logical quantum gates, and provides an optimal implementation of the Positive Operator Valued Measurement (POVM) for the unambiguous discrimination of quantum states. We also propose an actual experimental setup of th...

In quantum information processing and quantum computing protocols the carrier of information is a quantum system and information is encoded in the state of a quantum system. After processing the information it has to be read out what is equivalent to determining the final state of the system. When the possible final states are not orthogonal this i...

Multiphoton states are widely applied in quantum information technology. By the methods presented in this paper, the structure of a multiphoton state in the form of multiple single-photon qubit products can be mapped to a single-photon qudit, which could also be in a separable product with other photons. This makes possible the manipulation of such...

Detailed theoretical and experimental study of the atom–field interaction starting from first principles was made possible by the realization of the single-atom micromaser. The situation realized in a micromaser is very close to the ideal case of a single two-level atom interacting with a single quantized mode of a superconducting cavity. This syst...

Quantum key distribution schemes which employ encoding on vacuum-one-photon qubits are capable of transferring more information bits per particle than the standard schemes employing polarization or phase coding. We calculate the maximum number of classical bits per particle that can be securely transferred when the key distribution is performed wit...

We present a simple scheme using two identical cross-phase modulation processes in decoherence environment to generate superpositions of two coherent states with the opposite phases, which are known as cat states. The scheme is shown to be robust against decoherence due to photon absorption losses and other errors, and the design of its experimenta...

DOI:https://doi.org/10.1103/PhysRevA.78.069902

We address the problem of unambiguously identifying the state of a probe qudit with the state of one of d reference qudits. The reference states are assumed pure and linearly independent but we have no knowledge of them. The state of the probe qudit is assumed to coincide equally likely with either one of the d unknown reference states. We derive t...

We present a quantum repeater protocol that generates the elementary segments of entangled photons through the communication of qubus in coherent states. The input photons at the repeater stations can be in arbitrary states to save the local state preparation time for the operations. The flexibility of the scheme accelerates the generation of the e...

This paper has been withdrawn by the authors, due a oversimplified decoherence model. It will be substituted by a new work.

We present a local unitary operation strategy to realize the transformations between bi-partite entangled pure states without any communication between the sharing parties. It also saves the interaction with an ancilla in implementing the transformations.

We propose a setup that transforms a photon pair in arbitrary rank-four mixed state, which could also be unknown, to a Bell state. The setup involves two linear optical circuits processing the individual photons and a parity gate working with weak cross-Kerr nonlinearity. By the photon number resolving detection on one of the output quantum bus or...

In quantum information processing and quantum computing protocols information is encoded in the state of a quantum system. After processing the information it has to be read out what is equivalent to determining the final state of the system. When the possible final states are not orthogonal this is a highly nontrivial task that constitutes the gen...

We present a general linear optics based approach to implement contractive transformations that map products of N coherent states to products of M coherent states (M≤N) and apply it to nondestructive quantum database search.

To implement a positive-operator-valued measurement (POVM), which is defined on the dS-dimensional Hilbert space of a physical system, one has to extend the Hilbert space to include dA additional dimensions (called the ancilla). This is done via either the tensor product extension (TPE) or the direct sum extension (DSE). The implementation of a POV...

This two-volume book is a great addition to the growing number of books devoted to the field. It is very clearly written by classroom professionals, always with the students in mind. The tutorial presentation is supplemented with a number of exercises whose solutions are also given at the end of each volume. The first volume can serve as a textbook...

We consider complementarity in a bipartite quantum system of arbitrary dimensions. Single-partite and bipartite properties turn out as mutually exclusive quantities. The single-partite properties can be related to a generalized predictability and visibility which compose two complementary realities for themselves. These properties combined become m...

Determining the state of a quantum system is a central task in quantum information processing since it encompasses the read-out problem. Very often the optimized state discrimination strategy is a generalized measurement (Positive Operator Valued Measure, POVM). Therefore, we begin with a brief introduction to the theory of generalized measurements...

We propose an experimental setup that is capable of unambiguously discriminating any pair of linearly independent single photon polarization qubits, about which we do not have any knowledge except that an extra pair of these unknown states are provided as the reference. This setup, which is constructed with optical controlled-NOT (CNOT) gates, weak...

We present a quantum circuit that implements a nondemolition measurement of complementary single- and bipartite properties of a two-qubit system: entanglement and single-partite visibility and predictability. The system must be in a pure state with real coefficients in the computational basis, which allows a direct operational interpretation of tho...

State discrimination constitutes the read-out stage of quantum information processing. Optimized discrimination strategies often involve generalized measurements (POVMs). The talk will review recent progress in mixed state discrimination, optical implementation of POVMs and selected applications. Article not available.

We give a scheme for a physical implementation of the programmable state discriminator that unambiguously discriminates between two unknown qubit states with optimal probability of success. One copy of each of the unknown states is provided as input, or program, for the two program registers, and the data state, which is guaranteed to be prepared i...

We present a quantum circuit that implements a non-demolition measurement of complementary single- and bi-partite properties of a two-qubit system: entanglement and single-partite visibility and predictability. The system must be in a pure state with real coefficients in the computational basis, which allows a direct operational interpretation of t...

We present two different (but from a performance point of view) equivalent physical implementations for a programmable-state discriminator in terms of quantum circuits. The circuits that perform a positive-operator valued measure are built entirely of known logical quantum gates. The first, more-complicated circuit leads to a simpler readout, where...

The discrimination of any pair of unknown quantum states is performed by devices processing three parts of inputs: copies of the pair of unknown states we want to discriminate are respectively stored in two program systems and copies of data, which is guaranteed to be one of the unknown states, in a third system. We study the efficiency of such pro...

We discuss a state discriminator that unambiguously distinguishes two quantum registers prepared with multiple copies of two unknown qubits. This device achieves the optimal performance by von Neumann measurement and general positive operator value measurement (POVM) in different ranges of the input preparation probabilities, respectively, and in t...

We show that a general linear transformation from one single photon qudit to another, the dimension of which can be either equal or unequal to that of the first one, can be implemented by linear optics. As an application of the scheme we elaborate a method to deterministically realize any finite-element Positive Operator Value Measure (POVM) on sin...

The problem of operator/gate testing in quantum information is closely related to that of discriminating between unknown quantum states. We present generalized measurements and their linear optical implementations that accomplish this task optimally.

Complementarity relations in composite bipartite quantum systems of arbitrary dimensions are proposed. Genuine bipartite quantum properties whose information content is quantified by the generalized concurrence mutually exclude the single-partite properties of the subsystems. The single-partite properties are determined by generalized predictabilit...

We present a general scheme to realize the POVMs for the unambiguous discrimination of quantum states. For any set of pure states it enables us to set up a feasible linear optical circuit to perform their optimal discrimination, if they are prepared as single-photon states. An example of unknown states discrimination is discussed as the illustratio...

We describe a class of programmable devices that can discriminate between two quantum states. We consider two cases. In the first, both states are unknown. One copy of each of the unknown states is provided as input, or program, for the two program registers, and the data state, which is guaranteed to be prepared in one of the program states, is fe...

We show how to optimally unambiguously discriminate between two subspaces of a Hilbert space. In particular we suppose that we are given a quantum system in either the state \psi_{1}, where \psi_{1} can be any state in the subspace S_{1}, or \psi_{2}, where \psi_{2} can be any state in the subspace S_{2}, and our task is to determine in which of th...

Starting from a Hamiltonian model of the coupled three-level/two-mode system, a nonlinear quantum theory of the quantum beat laser is developed. The theory is valid under the special conditions that lead to correlated spontaneous emission laser (CEL) operation in the linear theory. It is shown that vanishing of the diffusion constant for the relati...

In quantum information and quantum computing, the carrier of information is a quantum system and information is encoded in its state. Secure communication schemes employ non‐orthogonal states that can not be discriminated with 100% probability of success. We briefly review various possible optimized state discrimination strategies since they provid...

Based on our previous publication [U. Herzog and J. A. Bergou, Phys.Rev. A 71, 050301(R) (2005)] we investigate the optimum measurement for the unambiguous discrimination of two mixed quantum states that occur with given prior probabilities. Unambiguous discrimination of nonorthogonal states is possible in a probabilistic way, at the expense of a n...

A new quantum communication scheme is introduced which is the quantum realization of the classical Kish-Sethuraman (KS) cipher. First the message is bounced back with additional encryption by the Receiver and the original encryption is removed and the message is resent by the Sender. The mechanical analogy of this operation is using two padlocks; o...