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We report preparation and characterization of coherent superposition states t[0>+alpha]1> of the electromagnetic field by conditional measurements on a beam splitter. This state is generated in one of the beam splitter output channels if a coherent state [alpha> and a single-photon Fock state [1> are present in the two input ports and a single photon is registered in the other beam splitter output. The single photon thus plays a role of a "catalyst:" it is explicitly present in both the input and the output channels of the interaction yet facilitates generation of a nonclassical state of light.

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... Non-Gaussian operation is one such technique which has been employed in different quantum protocols such as squeezing and entanglement distillation [19][20][21][22][23], quantum teleportation [24][25][26][27][28][29][30][31], quantum key distri- * manali.verma.31@gmail.com † chandan.quantum@gmail.com ...

... The measurement operator corresponding to difference intensity detection,Ô =â † 1â 1 − a † 2â 2 , which is also called as photon number difference operator. For the evaluation of phase uncertainty (19), we need to compute the mean and variance of the oper-atorÔ. The operatorÔ 2 can be written aŝ ...

... The quantity M n2,m2 n1,m1 is similar to the moment generating function and different moments of symmetrically ordered operators in the phase uncertainty expression can be evaluated using this. By setting appropriate values of the parameters n 1 , m 1 , n 2 , and m 2 , we obtain analytical expression of phase uncertainty Eq. (19) for different NGSSV states. Now we show the plot of ∆ϕ optimized over the transmissivity τ of the beam splitter for different NGSSV states as a function of squeezing parameter r in Fig. 2. We have set d x = 10 and d p = 0, which corresponds to the phase matching condition of the coherent plus SSV state [56,57]. ...

We investigate the benefits of probabilistic non-Gaussian operations in phase estimation using difference-intensity and parity detection-based Mach-Zehnder interferometers (MZI). We consider an experimentally implementable model to perform three different non-Gaussian operations, namely photon subtraction (PS), photon addition (PA), and photon catalysis (PC) on a single-mode squeezed vacuum (SSV) state. In difference-intensity detection-based MZI, two PC operation is found to be the most optimal, while for parity detection-based MZI, two PA operation emerges as the most optimal process. We have also provided the corresponding squeezing and transmissivity parameters yielding best performance, making our study relevant for experimentalists. Further, we have derived the general expression of moment-generating function, which shall be useful in exploring other detection schemes such as homodyne detection and quadratic homodyne detection.

... To generate a novel non-Gaussian state from a Gaussian or non-Gaussian state without changing the number of photons in the original state, we can use a special method called quantum-optical catalysis [22]. It was introduced by Lvovsky and Mlynek [23]. Since then, this operation has been applied on many Gaussian states, e.g., the TMSVS [22,24], the entangled coherent state [25], the squeezed coherent state [26]. ...

... The quantum-optical catalysis was first conceived by Dakna et al. [38]. It was later improved and defined by Lvovsky and Mlynek [23]. The catalysis is engineered by a beam splitter and a photon-number-resolving photo-detector. ...

... The EPR correlation in(23) as a function of |ξ | with several values of k and l with q = 0 for in (a) t 1 = t 2 = 0.4, in (b) t 1 = t 2 = 0.6, and in (c) t 1 = t 2 = 0.9. The red lines correspond to the PCS and the other curves are the MCPCSs The EPR correlation coefficient V EP R in (23) as functions of t 1 and t 2 with several values of k and l as well as |ξ | with q = 0. ...

Quantum-optical catalysis is a quantum state engineering that can convert states (Gaussian and/or non-Gaussian) into non-Gaussian states without changing the number of photons in the original states. This technique has been applied on many two-mode Gaussian states. In this paper, we introduce new two-mode non-Gaussian entangled states, called multiphoton catalytic pair coherent states (MCPCSs), based on the study of operating multiphoton quantum catalysis on two-mode non-Gaussian states, which are the pair coherent states (PCSs). By investigating linear entropy, Einstein-Podolsky-Rosen (EPR) correlation and EPR steering, it is shown that these properties in the new states can be enhanced compared with the PCSs by increasing the coherent parameter amplitude |ξ|. In the small regions of |ξ| and in the transmission coefficients space (t1,t2), the enhanced regions of the degree of entanglement are enlarged with increasing the numbers of catalytic photons, whereas in contrast to the EPR correlation and the EPR steering. Using the MCPCSs as entanglement resources to teleport a coherent state via the Braunstein and Kimble protocol, we calculate the average fidelity Fav of the teleportation process. The investigated results show that the average fidelity is enhanced in the large regions of the transmission coefficients t1,t2 as small values of |ξ|. In particular, for zero-photon catalysis, Fav is improved in the case of |ξ| high.

... In Section III, we extend the concept of "transformability" to define ZPS-based nonclassicality criteria, and in Section IV we describe a method to predict which input states will statistically transform under ZPS. In Section V we consider two more rich examples of ZPS input states-the displaced squeezed state [29] and the catalyzed coherent state [30]-which illustrate the concept of transformability over a limited range of input state parameter space. Finally, in Section VI we consider the effects of realistic detectors on experimentally observing these ZPS-based statistical transformations. ...

... Here we consider two rich examples within this context-displaced squeezed states, as considered by Dodonov et al. [29], and catalyzed coherent states [30]. ...

... Catalyzed coherent states (CCS) can be generated by mixing a coherent state α⟩ with a single-photon Fock state at a beamsplitter with reflectance Λ, and conditioning the output on the detection of exactly one photon in the auxiliary output mode [30]. As first proposed by Xu and Yuan, a similar catalysis procedure can be implemented with the signal and idler modes of an optical parametric amplifier (OPA) with an equivalent catalysis parameter Λ = 1 − 1 g 2 , where g is the gain of the OPA [37][38][39]. ...

Zero-photon subtraction (ZPS) is a conditional measurement process that can reduce the mean photon number of quantum optical states without physically removing any photons. Here we show that ZPS can also be used to transform certain super-Poissonian states into sub-Poissonian states, and vice versa. Combined with a well-known "no-go" theorem on conditional measurements, this effect leads to a new set of non-classicality criteria that can be experimentally tested through ZPS measurements.

... 3 The process that gives rise to such two-mode states of light via beam splitting is known as multiphoton interference 4-6 , and serves as a critical element in several applications including quantum optical interferometry 7 , and quantum state engineering where beam splitters and conditional measurements are utilized to perform post-selection techniques such as photon subtraction [8][9][10] , photon addition 11 , and photon catalysis. [12][13][14] In spite of its name, "multiphoton interference" does not involve the interference of photons. Rather, as has been emphasized by Glauber 15 , it is always the addition of the quantum amplitudes (themselves being complex numbers) associated with these states that give rise to interference effects. ...

... An implication of the above results is that a measurement of output component state |2, 2⟩ 12 for the input state |1, 3⟩ (in) 12 will exhibit complete destructive interference. However, a measurement of output component state |2, 2⟩ 12 for the input state |2, 2⟩ In a realistic experiment one has to contend with the prospects of imperfect detector efficiency, the influence of the photon mode functions in wavepackets, and potential time delay between photon detection in the output ports of the BS. ...

When quantum state amplitudes interfere, surprising non-classical features emerge which emphasis the roles of indistinguishability and discreteness in quantum mechanics. A famous example in quantum optics is the Hong Ou Mandel interference effect,a major ingredient in current quantum information processing using photonics. Traditionally the HOM features interference between amplitudes for two one-photon number states. Surprisingly, interference can be manifested when one amplitude represents that most classical of light field states, the coherent state, provided the partner state is non-classical (eg a single photon state or an odd photon number state). Imposing such nonclassical features on an otherwise classical state is the focus of this article. Recently, the HOM effect has been generalized to the multi-photon case, termed the extended HOM effect by the authors.The implication of the extended HOM effect is that if an odd parity state, comprising only odd numbers of photons, enters one input port of a 50:50 beam splitter, then regardless of the state entering the other input port, be it pure or mixed, there will no output coincident counts. In this work, we explain the extended HOM as arising from a sequence of pairwise HOM-like complete destructive interferences occurring simultaneously in the multicomponent amplitude for the output coincidence counts. We first demonstrate this diagrammatically in order to build physical intuition, before developing a general analytical proof. We then examine the case of a single photon interacting with a coherent state (and idealized laser), and consider prospects for experimental detection by including the effect of imperfect detection efficiency. This work highlights the importance of the non-classicality of light, and in particular the interference effects stemming from the discreteness of photon quanta.

... Related methods for the practical generation of nonclassical states encompass conditional measurements utilizing beam splitters (BSs) [11,[18][19][20][21][22][23], parametric amplifiers [14][15][16], down-conversion [24], and four-wave mixing approaches [18,25]. In particular, within the field of quantum state engineering, conditional measurements stand out as a versatile and promising approach. ...

... This approach has also been explored using squeezed vacuum states [20,26], thermal states [27], binomial states, and Fock states [18]. Similarly, extensive investigations have been conducted regarding coherent states [21][22][23] and squeezed vacuum states [21] interfered with a few photons in a BS. ...

The recent advent of integrated waveguide systems with reconfigurable propagation constants and coupling coefficients has opened the door to using waveguide detuning as a resource for readily tailoring the quantum properties of light states. Here we theoretically demonstrate that waveguide mode detuning can be used for molding the nonclassical properties of two interacting quantum optical fields in integrated waveguide couplers. In particular, we explore the states that are generated by conditional measurements when one of the input ports of the waveguide coupler is excited by coherent states, squeezed vacuum states, and thermal states, while the other port is excited by a single-photon Fock state. We explore the detuning range required to attain nonclassical states. Our findings could pave the way for a robust integrated-optics protocol, providing enhanced control and engineering capabilities over multiphoton quantum states.

... It is well known that nonclassical states have played an important role in quantum optics and quantum information processing, including their preparation, manipulation, and measurement [1][2][3][4][5]. A recent study by Lvovsky and Mlynek (2002) involved the coherent superposition states | ñ 0 and | ñ 1 were generated by a low transmittance BS [6]. Their specific preparation scheme is that the BS is fed by a weak coherent state and a single photon Fock state, followed by one-photon detection in one output port, then the desired state is obtained at the other output ports [7,8]. ...

... A recent study by Lvovsky and Mlynek (2002) involved the coherent superposition states | ñ 0 and | ñ 1 were generated by a low transmittance BS [6]. Their specific preparation scheme is that the BS is fed by a weak coherent state and a single photon Fock state, followed by one-photon detection in one output port, then the desired state is obtained at the other output ports [7,8]. In a follow-up study, Scheel et al found nonlinear operators are realized by singlephoton sources and appropriate measurements of auxiliary modes in linear optical elements, such as BS [9]. ...

We use the quadrature measurement to generate the novel nonclassical states via the beam splitter with two input states, i.e., a Fock state and a vacuum state. It is interesting to find that the desired target states are the Hermite polynomial excited vacuum states. Our results have shown that the zero-position detection for the position detector, the little photon number in the input state, and the high transmittance of the beam splitter (BS) are beneficial to improve the detection efficiency of finding the output states. The proposed states quantum statistical properties and squeezing effects are also studied in detail via different criteria. Our numerical analysis demonstrates that the output quantum states are new nonclassical states. Compared with the method of photon catalysis, position detection is easier to realize in experiments. Therefore, the results in this paper shall provide theoretical support for the experimental generation of several new nonclassical states.

... Recently, quantum catalysis operations have been studied [20,59,[68][69][70][71][72][73] as an alternative solution to resolve this issue. In particular, zero-photon catalysis (ZPC) operation is noiseless and can be implemented with existing technologies [74]. ...

... In particular, zero-photon catalysis (ZPC) operation is noiseless and can be implemented with existing technologies [74]. For example, quantum catalysis operations have been employed in the generation of a nonclassical state of light [69], continuous variable entanglement [72], performance improvements of Gaussian-modulated CV-QKD [68,75], performance improvements of four-state MDI-CV-QKD [59], entanglement improvements [71], nonclassicality of a two-mode, non-Gaussian entangled state [73], and generation of a non-Gaussian state [70]. Since the eight-state QKD protocol uses an increased number of states to encode the information compared to the four-state protocol, it provides a larger optimal variance, a higher level of security, distance enhancement, and a higher SKR. ...

Zero-photon catalysis (ZPC) introduces noiseless attenuation and can be implemented by existing technologies in quantum key distribution (QKD) protocols. In this paper, we present a ZPC-based eight-state measurement-device-independent continuous-variable QKD (MDI-CV-QKD) combined with discrete modulation and reverse reconciliation. This ZPC-involved eight-state protocol shows better efficiency in terms of optimal modulation variances, secret key rates, transmission distances, tolerable excess noises, and reconciliation efficiency compared to the eight-state protocol without ZPC, the four-state protocol without ZPC, and the four-state protocol with ZPC, at a low signal-to-noise ratio (SNR)

... Recently, quantum catalysis operations have been studied [20,59,[68][69][70][71][72][73] as an alternative solution to resolve this issue. Especially, Zero-Photon Catalysis (ZPC) operation is noiseless and can be implemented with existing technologies [74]. ...

... Especially, Zero-Photon Catalysis (ZPC) operation is noiseless and can be implemented with existing technologies [74]. For example, quantum catalysis operations have been employed in the generation of non-classical state of light [69], continuous variable entanglement [72], performance improvement of Gaussian modulated CV-QKD [68,75], performance improvement of four-state MDI-CV-QKD [59], entanglement improvement [71], nonclassicality of two-mode non-Gaussian entangled state [73], and generation of non-Gaussian state [70]. Since the eight-state quantum key distribution protocol uses an increased number of states to encode the information compared to the four-state protocol. ...

Zero-photon catalysis (ZPC) introduces noiseless attenuation and can be implemented by existing technologies in quantum key distribution (QKD) protocols. In this paper, we present a ZPC-based eight-state measurement-device-independent continuous-variable QKD (MDI-CV-QKD) combined with discrete modulation and reverse reconciliation. This ZPC-involved eight-state protocol shows better efficiency in terms of optimal modulation variances, secret key rates, transmission distances, tolerable excess noises, and reconciliation efficiency compared to the eight-state protocol without ZPC, the four-state protocol without ZPC, and the four-state protocol with ZPC, at a low signal-to-noise ratio (SNR).

... Recently, quantum catalysis operations have been studied [20,59,[68][69][70][71][72][73] as an alternative solution to resolve this issue. In particular, zero-photon catalysis (ZPC) operation is noiseless and can be implemented with existing technologies [74]. ...

... In particular, zero-photon catalysis (ZPC) operation is noiseless and can be implemented with existing technologies [74]. For example, quantum catalysis operations have been employed in the generation of a nonclassical state of light [69], continuous variable entanglement [72], performance improvements of Gaussian-modulated CV-QKD [68,75], performance improvements of four-state MDI-CV-QKD [59], entanglement improvements [71], nonclassicality of a two-mode, non-Gaussian entangled state [73], and generation of a non-Gaussian state [70]. Since the eight-state QKD protocol uses an increased number of states to encode the information compared to the four-state protocol, it provides a larger optimal variance, a higher level of security, distance enhancement, and a higher SKR. ...

Zero-photon catalysis (ZPC) introduces noiseless attenuation and can be implemented by existing technologies in quantum key distribution (QKD) protocols. In this paper, we present a ZPC-based eight-state measurement-device-independent continuous-variable QKD (MDI-CV-QKD) combined with discrete modulation and reverse reconciliation. This ZPC-involved eight-state protocol shows better efficiency in terms of optimal modulation variances, secret key rates, transmission distances, tolerable excess noises, and reconciliation efficiency compared to the eight-state protocol without ZPC, the four-state protocol without ZPC, and the four-state protocol with ZPC, at a low signal-to-noise ratio (SNR).

... Both experimental methods include conditional measurements. Considering these approaches, a similar non-Gaussian operation, photon catalysis, was also studied by A. I. Lvovsky and J. Mlynek [34]. Photon catalysis has been demonstrated to enhance the * ljr1996@tamu.edu ...

... Quantum entanglement improvement based on non-Gaussian operations are investigated from various perspectives, in recent years [8,[19][20][21][22]34]. Such operations are shown to be useful for entanglement distillation [6,23], quantum commutation [24,25] and quantum teleportation [8,22,26,27]. ...

Enhancing quantum entanglement is important for many quantum information processing applications. In this paper, we consider a protocol for entanglement enhancing in a two-mode squeezed vacuum state (TMSVS), attained based on photon subtraction, photon catalysis, and photon addition. Central to such an operation is the task of mixing and detecting number states with each mode of TMSVS. We analyze various settings and find an optimal setup for improving the entanglement of the state.

... The generation of two-mode entangled states of light can be accomplished by mixing nonclassical single-mode states of light at a beam splitter (BS) [1]. The process that gives rise to such two-mode states of light via beam splitting is known as multiphoton interference [2], and serves as a critical element in several applications including quantum optical interferometry [3], and quantum state engineering where beam splitters and conditional measurements are utilized to perform post-selection techniques such as photon subtraction [4][5][6], photon addition [7], and photon catalysis [8][9][10]. ...

... Ou [17] and Kuzmich et al. [18] studied only the case where one photon is mixed with a CV state at a beam splitter. Though they noted the important interference effects, they did not explicitly describe the existence of a nodal line of zeros representing complete destructive interference in the joint photon number distribution for the output fields as was done by BMG [8]. Furthermore, the former authors did not extend their considerations to mixing CV states of light with 2 or more photons, as was done by BMG [15]. ...

We show that the parity (evenness or oddness) of a nonclassical state of light has a dominant influence on the interference effects at a balanced beam splitter, irrespective of the state initially occupying the other input mode. Specifically, the parity of the nonclassical state gives rise to destructive interference effects that result in deep valleys in the output joint number distribution of which the Hong-Ou-Mandel (HOM) effect is a limiting case. The counter-intuitive influence of even a single photon to control the output of a beam splitter illuminated by any field, be it a coherent or even a noisy thermal field, demonstrates the extraordinary power of non-classicality. The canonical example of total destructive interference of quantum amplitudes leading to the absence of coincidence counts from a 50/50 beam splitter is the celebrated HOM effect, characterized by the vanishing of the joint probability of detecting singe photons in each of the output beams. We show that this is a limiting case of more general input states upon which a 50/50 BS can create total, or near total, destructive interference of quantum amplitudes. For odd photon number input Fock states of arbitrary value n>0 we show that the joint photon number probabilities vanish when detecting identical photon numbers in each output beams. We examine the mixing of photon number states of n = 1, 2, and 3 with a CV state, such as a coherent state of arbitrary amplitude, and a thermal state. These vanishing joint probabilities form what we call a central nodal line -- a contiguous set of zeros representing complete destructive interference of quantum amplitudes. For odd or even photon number Fock states with n>1 there will be additional off-diagonal pseudo-nodal curves along which the joint photon number probabilities are either zero, or near zero, which constitute a near, but not complete, destructive interference.

... 尽管光子扣除有上述优势, 但是在优化调 制方差的情况下, 执行减光子操作的成功概率却低 于0.25, 这使得它在提升量子密钥分发性能方面也 存在某种缺陷 [20] . 为了克服这种缺陷, 量子催化操 作 [21,22] 是一种可行的和较为成功的方案. 在催化 过程中, 辅助模的光子似乎看起来没有变化, 但是 却能够促进主通道模之间的量子态转换, 从而避免 了通信双方的信息量丢失. ...

... 关于更详细的量子催化等效算符的推导可参考文 献 [9,20,21,23]. 因此, 输入-输出量子态的关系式 ...

Compared with discrete variable quantum key distribution (DVQKD), continuous variable (CV) QKD has high security bit rate and other advantages, which, however, are slightly insufficient in secure transmission distance. In addition, the application of quantum catalysis has significantly improved the performance of Gaussian modulated (GM) CVQKD, especially in secure transmission distance. Recently, the application of quantum catalysis has significantly improved the performance of GM-CVQKD. However, whether it can be used to improve the performance of discrete modulated (DM) CVQKD protocol is still ambiguous. Therefore, a scheme of DM CVQKD protocol based on quantum catalysis is proposed in this paper to further improve the performance of the proposed protocol in terms of secure key rate, secure transmission distance and maximum tolerable noise. Our results show that under the same parameters, when the transmittance T introduced by quantum catalysis is optimized, the proposed scheme can effectively further improve the performance of QKD system compared with the original four-state modulation CVQKD scheme. In particular, when the tolerable excess noise is 0.002, the use of quantum catalysis can break the safe communication distance of 300 km with a key rate of 10–8 bits/pulse. However, if this noise is too large, the improvement in the effect of quantum catalysis on protocol performance will be restrained. In addition, in order to highlight the advantages of the use of quantum catalysis, the ultimate limit PLOB (Pirandola-Laurenza-Ottaviani-Banchi) bound of point-to-point quantum communication is given in this paper. The simulation results indicate that although neither the original scheme nor the proposed scheme can break the bound, compared with the former, the latter can be close to the boundary in long-distance transmission. These results provide theoretical basis for achieving the ultimate goal of global quantum security communication.

... Quadrature squeezing reaches its minimum in the unit transmissivity limit, where D Consequently, the 1-PC operation lacks the capacity to distill squeezing. This is in contrast with the fact that 1-PC operation can distill squeezing from an ensemble of mixed states[53,54]. ...

We consider distillation of squeezing in single mode squeezed vacuum state using three different probabilistic non-Gaussian operations: photon subtraction (PS), photon addition (PA) and photon catalysis (PC). To achieve this, we consider a practical model to implement these non-Gaussian operations and derive the Wigner characteristic function of the resulting non-Gaussian states. Our result shows that while PS and PC operations can distill squeezing, PA operation cannot. Furthermore, we delve into the success probabilities associated with these non-Gaussian operations and identify optimal parameters for the distillation of squeezing. Our current analysis holds significant relevance for experimental endeavors concerned with squeezing distillation.

... Lvovsky and co-workers reported the creation of the superposition of qubits by merging the single-photon Fock state and coherent state through the two input ports of BS. Followed by the single photon projection with small transmissivity at the output port of BS [24]. Since the single photon is involved and detected at the one output port, this process is called Quantum Optical Catalysis. ...

Creating non-Gaussian photonic states in the continuous variable regime, with high fidelity is essential for implementing universal quantum computation. However, this is a challenging task to achieve the essential nonlinearity. Alternatively, various non-Gaussian states of light can be created by the use of a simple linear setup called Quantum Optical Catalysis (QOC). In this work, we attempt to bring out the salient features of the multi-photon QOC process in terms of state preparation and characterization. The notion of state preparation from the QOC is achieved by expressing the output state as Displaced Qudits (DQ). The obtained superposition coefficients facilitate the characterization of states and carve a path to get desired non-Gaussian states. Moreover, the figures of merit of the prepared states are employed through Hilbert Schmidt Distance, Wigner negativity, and quadrature squeezing. From the results, it is inferred that the creation of individual displaced number states plays a predominant role in non-Gaussianity amongst the states derived. Meanwhile, the superposition of number states remains effective in achieving a significant degree of squeezing. In addition, the non-ideal preparation of DQ under realistic experimental conditions is investigated by incorporating imperfect photon detectors and mixed photon sources. The calculated success probability also attests to the potential for state generation.

... [27,28] that can affect the photonic qubits. It may be interesting to study loss suppression on arbitrary non-Gaussian, nonclassical states with non-Gaussian conditional measurements photon catalysis, for example, that have been useful in quantum key distribution [9,[74][75][76][77][78]. More generally, better error suppression techniques can be explored for arbitrary noise channels on diverse platforms [31]. ...

It is known that multiphoton states can be protected from decoherence due to a passive loss channel by applying noiseless attenuation before and noiseless amplification after the channel. In this work, we propose the combined use of multiphoton subtraction on four-component cat codes and teleamplification to effectively suppress errors under detection and environmental losses. The back-action from multiphoton subtraction modifies the encoded qubit encoded on cat states by suppressing the higher photon numbers, while simultaneously ensuring that the original qubit can be recovered effectively through teleamplification followed by error correction, thus preserving its quantum information. With realistic photon subtraction and teleamplification-based scheme followed by optimal error-correcting maps, one can achieve a worst-case fidelity (over all encoded pure states) of over 93.5% (82% with only noisy teleamplification) at a minimum success probability of about 3.42%, under a 10% environmental-loss rate, 95% detector efficiency and sufficiently large cat states with the coherent-state amplitudes of 2. This sets a promising standard for combating large passive losses in quantum-information tasks in the noisy intermediate-scale quantum (NISQ) era, such as direct quantum communication or the storage of encoded qubits on the photonic platform.

... Notably, the pioneering proposal for quantum computing in ion traps [42] (see also [43,44]) considers two spin qubits that become entangled via an interaction with a cavity that can be considered catalytic [3]. Another relevant direction is that of "multiphoton catalysis" (see, e.g., [45][46][47][48]), which is a heralded catalytic process, where the catalyst is returned only with some probability. In contrast, our catalytic protocol is deterministic. ...

Catalysis plays a key role in many scientific areas, most notably in chemistry and biology. Here we present a catalytic process in a paradigmatic quantum optics setup, namely the Jaynes-Cummings model, where an atom interacts with an optical cavity. The atom plays the role of the catalyst, and it allows for the deterministic generation of nonclassical light in the cavity. Considering a cavity prepared in a “classical” coherent state, and choosing appropriately the atomic state and the interaction time, we obtain an evolution with the following properties. First, the state of the cavity has been modified, and it now features nonclassicality, as witnessed by sub-Poissonian statistics or Wigner negativity. Second, the process is catalytic in the sense that the atom is deterministically returned to its initial state exactly, and it can be reused multiple times. What is more, we also show that our findings are robust under dissipation and can be applied to scenarios featuring cavity loss and atomic decay. Finally, we investigate the mechanism of this catalytic process, in particular highlighting the key role of correlations and quantum coherence.
Published by the American Physical Society 2024

... A conditional separation of timescales unfolds in the course of a particular unraveling dictated by the selected quadrature amplitude of the signal field. We have visualized the time-asymmetric fluctuations of the field amplitude triggered by recordable photon-counting sequences and have also explored the possibility of an operational determination [65][66][67][68][69] of phase-space quasiprobability distributions and photon-state superpositions [70][71][72] for the different unravelings. ...

We discuss the conditional measurement of field amplitudes by a nonclassical photon sequence in the Jaynes-Cummings (JC) model under multiphoton operation. We do so by employing a correlator of immediate experimental relevance to reveal a distinct nonclassical evolution in the spirit of Foster []. The correlator relies on the complementary nature of the pictures obtained from different unravelings of a JC source master equation. We demonstrate that direct photodetection entails a conditioned separation of timescales, a quantum beat, and a semiclassical oscillation, produced by the coherent light-matter interaction in its strong-coupling limit. We single out the quantum beat in the analytical expression for the waiting-time distribution, pertaining to the particle nature of the scattered light, and find a negative spectrum of quadrature amplitude squeezing, characteristic of its wave nature for certain operation settings. Finally, we jointly detect the dual aspects through the wave-particle correlator, showing an asymmetric regression of fluctuations to the steady state which depends on the quadrature amplitude being measured.
Published by the American Physical Society 2024

... The quantum optical catalysis scheme was first proposed, and the preparation and characterization of coherent superposition states via conditional measurements on a beam splitter were then reported. [20] Bartley et al. successfully generated nonclassical multiphoton states by using a beam splitter to perform single-photon quantum catalysis. [21] Interestingly, similar to a beam splitter, nonlinear manipulation of the two-mode squeezing process can enhance nonclassicality and entanglement. ...

A new kind of non-Gaussian quantum catalyzed state is proposed via multiphoton measurements and two-mode squeezing as an input of thermal state. The characteristics of generating multiphoton catalysis output state depends on the thermal parameter, catalyzed photon number and squeezing parameter. We then analyze the nonclassical properties by examining the photon number distribution, photocount distribution and partial negativity of the Wigner function. Our findings indicate that nonclassicality can be achieved through the implementation of multiphoton catalysis operations, and modulated by the thermal parameter, catalyzed photon number and squeezing parameter.

... It may be interesting to study loss suppression on arbitrary nongaussian, nonclassical states with other nongaussian conditional measurements photon catalysis, for example, that have been useful in quantum key distribution [65][66][67][68][69]. More generally, better error suppression techniques can be explored for arbitrary noise channels on diverse platforms [29]. ...

It is known that multiphoton states can be protected from decoherence due to a passive loss channel by applying noiseless attenuation before and noiseless amplification after the channel. In this work, we propose the combined use of multiphoton subtraction on four-component cat codes and teleamplification to effectively suppress errors under detection and environmental losses. The back-action from multiphoton subtraction modifies the encoded qubit encoded on cat states by suppressing the higher photon numbers, while simultaneously ensuring that the original qubit can be recovered effectively through teleamplification followed by error correction, thus preserving its quantum information. With realistic photon subtraction and teleamplification-based scheme followed by optimal error-correcting maps, one can achieve a worst-case fidelity (over all encoded pure states) of over 93.5% (82% with only noisy teleamplification) at a minimum success probability of about 3.42%, under a 10% environmental-loss rate, 95% detector efficiency and sufficiently large cat states with the coherent-state amplitudes of 2. This sets a promising standard for combating large passive losses in quantum-information tasks in the noisy intermediate-scale quantum (NISQ) era, such as direct quantum communication or the storage of encoded qubits on the photonic platform.

... [25][26][27] A PB based single photon source is usually divided into three types: conventional photon blockade (CPB), which emerges due to the nonlinearity of the system eigenvalues, [13] unconventional photon blockade (UPB), which occurs due to destructive interferences between different transition pathways, [28] and photon blockade induced by truncation of Hilbert space, as observed in quantum linear scissors. [29] UPB was found first by Liew and Savona, [30,31] and has been studied in a optical cavity with a quantum dot, [32,33] weakly nonlinear photonicmolecules, [34] coupled optomechanical systems, [35] coupled cavities with second-order nonlinearity, [36] coupled cavities with thirdorder nonlinearity, [37] spinning anharmonic resonator. [38] Recently, UPB in two-photon transition processes have triggered much attention, [39][40][41][42] and other multi-photon processes have been discovered in these processes. ...

In a two-frequency cavity driving and atom driving atom-cavity system, we find the photon blockade effect. In a truncated eigenstates space, we calculate the zero-delay second-order correlation function of the cavity mode analytically and obtain an optimal condition for the photon blockade. By including three transition pathways, we find the higher excitations of the cavity mode can be further suppressed and the zero-delay second-order correlation function can be reduced additionally. Based on the master equation, we simulate the system evolution and find the analytical solutions match well with the numerical results. Our scheme is rubust with small fluctuations of parameters and maybe used as a new type of single photon source.

... In other words, one might ask whether non-Gaussian states can serve as useful catalysts. This approach was investigated in Ref. (Lvovsky and Mlynek, 2002) who considered a phenomenon they termed "quantum optical catalysis", which has been further generalized to "multi-photon catalysis" in later works (Bartley et al., 2012;Hu et al., 2017;Scheel et al., 2003;Xu, 2015), see also Refs. (Birrittella et al., 2018;Zhang et al., 2021;Zhou et al., 2018). ...

Catalysts open up new reaction pathways that can speed up chemical reactions while not consuming the catalyst. A similar phenomenon has been discovered in quantum information science, where physical transformations become possible by utilizing a quantum degree of freedom that returns to its initial state at the end of the process. In this review, a comprehensive overview of the concept of catalysis in quantum information science is presented and its applications in various physical contexts are discussed.

... Another relevant direction is that of "multiphoton catalysis" (see e.g. [46][47][48][49]) which is a heralded catalytic process, where the catalyst is returned only with some probability. In contrast, our catalytic protocol is deterministic. ...

Catalysis plays a key role in many scientific areas, most notably in chemistry and biology. Here we present a catalytic process in a paradigmatic quantum optics setup, namely the Jaynes-Cummings model, where an atom interacts with an optical cavity. The atom plays the role of the catalyst, and allows for the deterministic generation of non-classical light in the cavity. Considering a cavity prepared in a "classical'' coherent state, and choosing appropriately the atomic state and the interaction time, we obtain an evolution with the following properties. First, the state of the cavity has been modified, and now features non-classicality, as witnessed by sub-Poissonian statistics or Wigner negativity. Second, the process is catalytic, in the sense that the atom is deterministically returned to its initial state exactly, and could then in principle be re-used multiple times. We investigate the mechanism of this catalytic process, in particular highlighting the key role of correlations and quantum coherence.

... For values of θ ≠ π, coherences (ρ 01 ,ρ 10 ) appearing in the density matrices translate into phase-dependent Wigner functions 29,30 . While scanning θ across π, these coherences flip sign, together with the measured mean value of the X = X ϕ=0 quadrature (Fig. 4a). ...

Engineering the quantum states of freely propagating light is of paramount importance for quantum technologies. As yet, the experimental generation of photonic states with negative Wigner functions has relied intrinsically on probabilistic schemes, heralded by the projection of a quantum measurement. Here we describe the fully deterministic preparation of freely propagating quantum states of light with negative Wigner functions, obtained by mapping the internal state of an intracavity Rydberg superatom onto an optical qubit encoded as a superposition of 0 and 1 photons. This approach enables us to reach a 60% photon generation efficiency in a well-controlled spatiotemporal mode while maintaining strong photon antibunching. By changing the qubit rotation angle, we observe an evolution from quadrature squeezing to Wigner negativity. Our experiment demonstrates this new technique as a viable method for deterministically generating non-Gaussian photonic resources, lifting several major roadblocks in optical quantum engineering.

... In this paper, we propose a scheme to generate the multi-mode entangled catalysis squeezed vacuum states (MECSVS). Due to the fact that the multiphoton catalysis operation [47,48] can improve the fidelity in quantum teleportation [49,50], extend the transmission distance in continuous variable quantum key distribution [51,52], and enhance the sensitivity of phase estimation for a single-phase estimation [20] and undo the noise effect of the channel [53], we also propose ascheme to improve the multiparameter estimation precision by using the MECSVS. Our results clearly show that the multiphoton catalytic operation can further improve the precision of phase estimation compared with the result with ordinary ESVS as the probe state. ...

We propose a method to generate the multi-mode entangled catalysis squeezed vacuum states (MECSVS) by embedding the cross-Kerr nonlinear medium into the Mach-Zehnder interferometer. This method realizes the exchange of quantum states between different modes based on Fredkin gate. In addition, we study the MECSVS as the probe state of multi-arm optical interferometer to realize multi-phase simultaneous estimation. The results show that the quantum Cramer-Rao bound (QCRB) of phase estimation can be improved by increasing the number of catalytic photons or decreasing the transmissivity of the optical beam splitter using for photon catalysis. In addition, we also show that even if there is photon loss, the QCRB of our photon catalysis scheme is lower than that of the ideal entangled squeezed vacuum states (ESVS), which shows that by performing the photon catalytic operation is more robust against photon loss than that without the catalytic operation. The results here can find important applications in quantum metrology for multiparatmeter estimation.

... Due to the influence of some environmental factors in actual scenarios, the entangled state will degenerate, thereby reducing fidelity. Fortunately, we can compensate for the above environmental influences by using entanglement enhancing approaches, such as photon catalysis, which is one of the non-Gaussian operations [28] that has been used in quantum communications for performance improvement [29][30][31][32][33]. ...

Continuous-variable quantum teleportation (CVQT) plays a vital role in practical quantum communications. However, the degradation of quantum signals, usually caused by the absorption and scattering of light in actual scenarios, has an influence on the transmission performance and hence hinders its implementations. In this paper, we propose a non-Gaussian approach to improve the performance of CVQT via photon catalysis through a lossy optical fiber channel. The photon catalysis-enabled scheme can be performed on both sides of Einstein–Podolsky–Rosen state prepared by the receiver, where it gives birth to the enhancement of entanglement of the system. Numerical simulations show that the performance of the photon catalysis-enabled CVQT has been improved in terms of both fidelity and maximal transmission distance. Moreover, the larger the squeezing parameter results in the more enhancement of the CVQT system. The results may provide a useful insight for the practical implementation of CVQT.

... For values of θ = π, coherences (ρ 01 ,ρ 10 ) appearing in the density matrices translate into phase-dependent Wigner functions [25,26]. While scanning θ across π, these coherences flip sign, together with the measured mean value of the X = X φ=0 quadrature ( Fig.4(a)). ...

Engineering quantum states of free-propagating light is of paramount importance for quantum technologies. Coherent states ubiquitous in classical and quantum communications, squeezed states used in quantum sensing, and even highly-entangled cluster states studied in the context of quantum computing can be produced deterministically, but they obey quasi-classical optical field statistics described by Gaussian, positive Wigner functions. Fully harnessing the potential of many quantum engineering protocols requires using non-Gaussian Wigner-negative states, so far produced using intrinsically probabilistic methods. Here we describe the first fully deterministic preparation of non-Gaussian Wigner-negative free-propagating states of light, obtained by mapping the internal state of an intracavity Rydberg superatom onto an optical qubit encoded as a superposition of 0 and 1 photons. This approach allows us to reach a 60% photon generation efficiency in a well-controlled spatio-temporal mode, while maintaining a strong photon antibunching. By changing the qubit rotation angle, we observe an evolution from quadrature squeezing to Wigner negativity. Our experiment sets this new technique as a viable method to deterministically generate non-Gaussian photonic resources, lifting several major roadblocks in optical quantum engineering.

... However, the success probability for implementing single PS is less than 0.25, which might lead to the limited improvement of CV-QKD [10]. To avoid this problem, a kind of non-Gaussian operation, quantum catalysis first proposed by Lvovsky [35], has been successfully used to enhance the nonclassicality and entanglement properties of quantum states, improving the quantum coherence and expanding the transmission distance of QKD. In Ref. ...

The entanglement improvement is theoretically investigated when applying a single-side quantum scissors (SSQS) with a local squeezing operation and two-asymmetrical beam splitters (BSs) to one mode of an input two-mode squeezed vacuum state (TMSV). It is found that the gain factor can be significantly enhanced with the increasing of local squeezing parameter at the expense of the success probability. The entanglement can also be further improved adjusting the local-squeezing or the transmissivity of BSs in a small initial squeezing region. In addition, our scheme is robust against the photon loss in TMSV. The improved effect becomes more obvious due to the presence of local squeezing. However, the case is not true for a more realistic SSQS. In both cases, the asymmetric BSs play a positive role for the entanglement improvement. These results suggest that the squeezing-based SSQS at single-photon level is beneficial to effectively improve the entanglement, which may have potential applications in quantum communication.

... To extend the above work, the nonclassicality and decoherence of the PS-based squeezed thermal state (STS) [24], the PA-based STS [34] and the CS-based STS [35] were also proposed successively. Except for the traditional non-Gaussian operations described above, the quantum catalysis [36][37][38][39] and the quantum scissor [40,41], which can be seen as novel non-Gaussian operations, were also used to produce the non-classical states, even the entangled states. These results indicate that applying non-Gaussian operations into Gaussian states is an effective way of generating strongly non-classical states, which may have more potential applications in quantum metrology [42,43], quantum key distribution [44][45][46][47][48] and quantum illumination [49]. ...

The nonclassicality is the prerequisite for quantum states to be applied into quantum information, especially for quantum metrology. Here we theoretically investigate the non-classical properties of the non-Gaussian state generated by repeatedly operating a number-conserving generalized superposition of products (GSP), i.e., (s 1 aa † + t 1 a † a)m with ${s}_{1}^{2}+{t}_{1}^{2}=1,$ on the squeezed thermal state (STS), in terms of second-order correlation function, Mandel’s Q parameter, quadrature squeezing and Wigner function (WF). It is shown that, compared to the cases of the STS, the GSP-STS with the high-order GSP operations (m > 1) at the small-squeezing levels can be beneficial to the existence of the photon-antibunching effect, the sub-Poissonian distribution and the partial negativity of the WF, apart from the quadrature squeezing. In addition, for the case of m = 1, we also compare with the non-classical properties of several different non-Gaussian resources, involving the photon-subtracted-then-added (PSTA) STS, the GSP-STS and the photon-added-then-subtracted (PATS) STS. It is found that the PSTA-STS with respect to the sub-Poissonian distribution and the partial negativity of the WF has a better performance than others. Significantly, the generated GSP-STS has an obvious advantage of showing the photon-antibunching effects, compared to the PSTA-STS and the PATS-STS, which means that our scheme may have an excellent guidance for the practical implementations in quantum information.

... In this respect, it has been shown that continuous variable entanglement distillation from noisy Gaussian entangled states cannot be effected by Gaussian operations alone [17,18], therefore, some non-Gaussian operations such as photon counting are required. In this regard, some recursive degaussifying schemes have been considered for Gaussian states such as quantum catalysis, photon subtraction, symmetric photon replacement and purifying distillation [19][20][21][22][23]. The mentioned schemes regaussify the final output and yield highly of entangled or highly pure Gaussian states [24,25]. ...

We develop a linear optics network for the generation of photonic entangled states via designing a quantum circuit consisting of optical elements, i.e., beam splitters and birefringent crystals. To achieve the purpose, at first we introduce non-entangled single-photon states with their Gaussian spectral amplitude functions as the inputs of the circuit. Then, we show that the outcome of the circuit is an entangled Gaussian photonic state characterized with its covariance matrix. The quantum optical Gaussian states constitute an important class of robust quantum states which are manipulatable by the existing technologies. Meanwhile, we investigate the generation of biphoton entangled states, in detail. Also, we evaluate the concurrence (as a measure of entanglement) and also the probability density function (PDF) corresponding to biphoton states. In the continuation, we study other possible applications of such quantum circuits. We demonstrate that how one can estimate the position of outcome, i.e., the probability of finding entangled photons in a confidence ellipsoid. Our numerical results show that the entanglement of biphoton states strongly depends on their correlation matrix. As an outstanding feature, the PDF of the output state of the circuit provides an elegant criterion to identify the entangled photonic states from their separable counterparts. The designed quantum circuit and the obtained results may be implemented in the development of quantum information and communication protocols with continuous variables, besides their practical importance in realizing more complicated quantum networks.

... The amount of NLSQ observed here is higher than can be achieved with arbitrary Gaussian states, showing that the generated non-Gaussian states are indeed appropriate approximations as ancillary states for the cubic phase gate. Note that there are already various demonstrations of generation of superposition of Fock states [21][22][23][24][25][26][27][28], where the superposition of up to three photons [28] has already been realized. However, the superpositions of Fock states demonstrated before are not sufficient to observe NLSQ, despite their nonclassical aspects as negative Wigner functions. ...

Quantum non-Gaussian gate is a missing piece to the realization of continuous-variable universal quantum operations in an optical system. In a measurement-based implementation of the cubic phase gate, a lowest-order non-Gaussian gate, non-Gaussian ancillary states that have a property we call nonlinear squeezing are required. This property, however, has never been experimentally verified. In this paper, we generate a superposition between a vacuum state and a single-photon state whose nonlinear squeezing is maximized by the optimization of the superposition coefficients. The nonlinear squeezing is observed via real-time quadrature measurements, meaning that the generated states are compatible with real-time feedforward and are suitable as ancillary states for the cubic phase gate in the time domain. Moreover, by increasing the number of photons, it is expected that nonlinear squeezing can be further improved. The idea presented here can be readily extended to higher-order phase gates [P. Marek et al., Phys. Rev. A 97, 022329 (2018)]. As such, this work presents an important step to extend continuous-variable quantum information processing from Gaussian regime to non-Gaussian regime.

... Then, we can find a setup that consists of squeezed light sources, linear optics, and photon number detectors that herald |ψ target when particular results are detected by the photon number detectors. Based on this idea, generations of various non-Gaussian states have been explored [18,22,[29][30][31][32][33][34][35][36][37][38][39][40][41]. In this approach, however, the generation system is highly dependent on the target states and the generation systems tend to become more complex when the maximum photon number increases. ...

We propose and analyze a setup to tailor the wave functions of the quantum states. Our setup is based on the quantum teleportation circuit, but instead of the usual two-mode squeezed state, two-mode non-Gaussian entangled state is used. Using this setup, we can generate various classes of quantum states such as Schr\"odinger cat states, four-component cat states, superpositions of Fock states, and cubic phase states. These results demonstrate the versatility of our system as a state generator and suggest that conditioning using homodyne measurements is an important tool in the generations of the non-Gaussian states in complementary to the photon number detection.

... Despite the aforementioned advantages, the success probability of implementing photonsubtracted operations is less than 0.25, which would lead to limited performance improvements [35,36]. To eliminate this drawback, recently, quantum catalysis (QC) [35][36][37][38][39][40][41] has been viewed as another useful method to extend the maximal transmission distance of the MDI-CVQKD system [42], compared with the single-photon subtraction case. Unfortunately, a major problem, common to the aforementioned GM CVQKD, is that the reconciliation efficiency β is low, especially in long-distance transmission with a low signal-to-noise ratio. ...

Discrete modulation can make up for the shortage of transmission distance in measurement-device-independent continuous-variable quantum key distribution (MDI-CVQKD), providing a unique advantage against all side-channel attacks but also creating a challenge for further performance improvement. Here we suggest a quantum catalysis (QC) approach for enhancing the performance of the discrete-modulated (DM) MDI-CVQKD in terms of the achievable secret key rate and lengthening the maximal transmission distance. The numerical simulation results show that the QC-based MDI-CVQKD with discrete modulation, involving a zero-photon catalysis (ZPC) operation, can not only obtain a higher secret key rate than the original DM protocol, but also contribute to a reasonable increase of the corresponding optimal variance. As for the extreme asymmetric and symmetric cases, the secret key rate and maximal transmission distance of the ZPC-involved DM MDI-CVQKD system can be further improved under the same parameters. This approach enables the system to tolerate lower reconciliation efficiency, which may provide excellent potential for practical implementations with state-of-art technology.

Catalysts open up new reaction pathways that can speed up chemical reactions while not consuming the catalyst. A similar phenomenon has been discovered in quantum information science, where physical transformations become possible by utilizing a quantum degree of freedom that returns to its initial state at the end of the process. In this review, a comprehensive overview of the concept of catalysis in quantum information science is presented and its applications in various physical contexts are discussed.

In this study, we explore an approach aimed at enhancing the transmission or reflection coefficients of absorbing materials through the utilization of joint measurements of entangled photon states. On the one hand, through the implementation of photon catalysis in the reflected channel, we can effectively modify the state of the transmission channel, leading to a notable improvement in the transmission ratio. Similarly, this approach holds potential for significantly amplifying the reflection ratio of absorbing materials, which is useful for detecting cooperative targets. On the other hand, employing statistical counting methods based on the technique of heralding on zero photons, we evaluate the influence of our reflection enhancement protocol for detecting noncooperative targets, which is validated through Monte Carlo simulations of a quantum radar setup affected by Gaussian white noise. Our results demonstrate a remarkable enhancement in the signal-to-noise ratio of imaging, albeit with an increase in mean-square error. These findings highlight the potential practical applications of our approach in the implementation of quantum radar.

Non-Gaussian quantum states are crucial to fault-tolerant quantum computation with continuous-variable systems. Usually, generation of such states involves trade-offs between success probability and quality of the resultant state. For example, injecting squeezed light into a multimode interferometer and postselecting on certain patterns of photon-number outputs in all but one mode, a fundamentally probabilistic task, can herald the creation of cat states, Gottesman-Kitaev-Preskill (GKP) states, and more. We consider the addition of a non-Gaussian resource state, particularly single photons, to this configuration and show how it improves the qualities and generation probabilities of desired states. With only two modes, adding a single-photon source improves GKP-state fidelity from 0.68 to 0.95 and adding a second then increases the success probability eightfold; for cat states with a fixed target fidelity, the probability of success can be improved by factors of up to four by adding single-photon sources. These demonstrate the usefulness of additional commonplace non-Gaussian resources for generating desirable states of light.

Zero-photon subtraction (ZPS) is a conditional measurement process that can reduce the mean photon number of quantum optical states without physically removing any photons. Here we show that ZPS can also be used to transform certain super-Poissonian states into sub-Poissonian states and vice versa. Combined with a well-known “no-go” theorem on conditional measurements, this effect leads to a set of nonclassicality criteria that can be experimentally tested through ZPS measurements.

We propose a method to generate the multi-mode entangled catalysis squeezed vacuum states (MECSVS) by embedding the cross-Kerr nonlinear medium into the Mach—Zehnder interferometer. This method realizes the exchange of quantum states between different modes based on Fredkin gate. In addition, we study the MECSVS as the probe state of multi-arm optical interferometer to realize multi-phase simultaneous estimation. The results show that the quantum Cramer—Rao bound (QCRB) of phase estimation can be improved by increasing the number of catalytic photons or decreasing the transmissivity of the optical beam splitter using for photon catalysis. In addition, we also show that even if there is photon loss, the QCRB of our photon catalysis scheme is lower than that of the ideal entangled squeezed vacuum states (ESVS), which shows that by performing the photon catalytic operation is more robust against photon loss than that without the catalytic operation. The results here can find applications in quantum metrology for multiparatmeter estimation.

Based on the conditional double interferometers proposed by Paris, we shall generate a class of finite-dimensional nonclassical states by inputting several different states including coherent state(CS), squeezed vacuum state(SVS), and thermal state(TS). It is shown that the double interferometers with two single-photons inputs and two single-photon detections are seen as an equivalent projection operator composed of vacuum state, one-photon state and two-photon state. Then we study the detection probability of output states and the fidelity between the corresponding input state and the output state, respectively. In addition, the nonclassical characteristics of three kinds of output states has also been analyzed and compared in detail by means of intensity gain, Mandel’s Q parameter, as well as the Wigner function. These results show that for the case of the input SVS, the nonclassical characteristics of its output state is independent of phase shift and is only related to the initial squeezed parameter. However, for the cases of the input CS and TS, the output states can present strong non-classicality by modulating the input initial parameters and the internal phase shifter of interferometer.

The ability to manipulate light at the level of single photons, its elementary excitation quanta, has recently made it possible to produce a rich variety of tailor-made quantum states and arbitrary quantum operations, of high interest for fundamental science and applications. Here we present a concise review of the progress made over the last few decades in the engineering of quantum light states. Although far from exhaustive, this review aims at providing a sufficiently wide and updated introduction that may serve as the entry point to such a fascinating and rapidly evolving field.

We propose an efficient way to implement new family of continuous variable (CV) states of definite parity. Measurement induced CV states of definite parity states are realized after subtraction of an arbitrary number of photons from the initial single-mode squeezed vacuum (SMSV) state using a photon number resolving (PNR) detector. Optical design requires irreducible number of optical elements for implementation of the CV states of definite parity. The potential of using the CV states in optical quantum information processing can be high. As an example, we show the possibility of using a family of the CV states of definite parity for quantum engineering of optical even/odd Schrödinger cat states (SCSs). In particular, we report the possibility of implementing the CV states of definite parity that approximate even/odd SCSs of amplitude slightly greater than 4 with fidelity prevailing 0,99 after subtraction of 50,51 photons from original SMSV. The success probability being the third key parameter of the optical design, decreases with an increase in the number of photons, but generally remains at an acceptable level for further use in quantum information processing in the case of a small number of subtracted photons.

We investigate linear optical noiseless quantum amplifiers driven by auxiliary Fock states and photon number measurements. We seek optimal interferometric schemes that maximize the probability of success of such noiseless quantum amplifiers and we focus on fixed linear optical interferometric setups that do not involve feed-forward. We first investigate noiseless amplifiers based on the quantum-scissors scheme and then extend our analysis to general interferometric couplings. We find that the more general interferometric couplings can outperform the quantum scissors. For noiseless amplifiers driven by an auxiliary single-photon state we recover the single-photon catalysis as an efficient noiseless amplification configuration for setups without feed-forward. We complement our analysis of linear optical noiseless quantum amplifiers by discussion of the general form of single-mode linear optical quantum operations driven by auxiliary N-photon state.

Enhancing quantum entanglement is important for many quantum information processing applications. In this paper, we consider a protocol for entanglement enhancing in a two-mode squeezed vacuum state (TMSVS), attained based on photon subtraction, photon catalysis, and photon addition. Central to such an operation is the task of mixing and detecting number states with each mode of TMSVS. We analyze various settings and find an optimal setup for improving the entanglement of the state.

Non‐hermitian spectral degeneracies, known as exceptional points (EPs), feature the simultaneous coalescence of both eigenvalues and the associated eigenstates of a system. A host of intriguing EP effects and their applications have been revealed in the classical realm, such as loss‐induced lasing, single‐mode laser, and EP‐enhanced sensing. Here, it is shown that a purely quantum effect, known as single‐photon blockade, emerges in a Kerr microring resonator due to EP‐induced asymmetric coupling between the optical modes and the nonlinearity‐induced anharmonic energy‐level spacing. A striking feature of this photon blockade is that it emerges at two‐photon resonance which in Hermitian systems will only lead to photon‐induced tunneling but not to photon blockade. By tuning the system towards or away from an EP, one can control quantum correlations, implying the potential use of their system for frequency tunable single‐photon generation and an antibunching‐to‐bunching light switch. The work sheds new light on EP‐engineered purely quantum effects, providing unique opportunities for making and utilizing various single‐photon quantum EP devices. Chiral exceptional points (EPs) can emerge by controlling the relative angular position of two nanotips placed near an optical Kerr resonator. The interplay of EPs and Kerr nonlinearity leads to the counterintuitive effect of two‐photon resonance antibunching. Also, frequency‐tunable photon blockade can be achieved in such a quantum non‐Hermitian device.

We show that the parity (evenness or oddness) of a nonclassical state of light has a dominant influence on the interference effects at a balanced beam splitter, irrespective of the state initially occupying the other input mode. Specifically, the parity of the nonclassical state gives rise to destructive interference effects that result in deep valleys in the output joint number distribution of which the Hong-Ou-Mandel (HOM) effect is a limiting case. The counterintuitive influence of even a single photon to control the output of a beam splitter illuminated by any field, be it a coherent or even a noisy thermal field, demonstrates the extraordinary power of nonclassicality. The canonical example of total destructive interference of quantum amplitudes leading to the absence of coincidence counts from a 50:50 beam splitter (BS) is the celebrated HOM effect, characterized by the vanishing of the joint probability of detecting singe photons in each of the output beams. We show that this is a limiting case of more general input states upon which a 50:50 BS can create total, or near total, destructive interference of quantum amplitudes. For the case of an odd photon-number input Fock state of arbitrary value n>0 we show that the joint photon-number probabilities vanish when detecting identical photon numbers in each output beams. We specifically examine the mixing of photon-number states of n=1, 2, and 3 with a continuous-variable state, such as a coherent state of arbitrary amplitude, and a thermal state. These vanishing joint probabilities form what we call a central nodal line: A contiguous set of zeros representing complete destructive interference of quantum amplitudes. We further show that with odd or even photon-number Fock states n, with n>1, there will be additional off-diagonal curves along which the joint photon-number probabilities are either zero, or near zero, which we call pseudonodal curves, which constitute a near, but not complete, destructive interference pattern in the photon-number space. We interpret all of these interference effects as an extension of the HOM effect. We explain the origin of these effects and explore the experimental prospects for observing them with currently available number-resolving detectors in the presence of a small amount of noise.

We theoretically investigate the generation of a novel non-Gaussian entangled state by using two single-photon catalytic quantum scissors (CQS) on an input two-mode squeezed vacuum state, and derive the effective operator of the CQS which is closely associated with the transmittance of beam splitters. The output state of this setup composes of the zero-, single- and two-photon entangled states. The results show that both larger success probability of achieving such event and higher fidelity between the input and output states can be attained by taking appropriate parameters of the transmittance and the initial squeezing parameters. Moreover, the entanglement properties of the generated non-Gaussian entangled state are discussed in terms of the degree of entanglement, the Einstein–Podolsky–Rosen (EPR) correlation and the two-mode squeezing effect. Our simulation results indicate that, in the small initial squeezing levels, there are always two improved areas of entanglement at the low and high transmittance ranges, respectively. Comparing to the improved areas of the entanglement, we find that both EPR correlation and two-mode squeezing effect rather than the entanglement degree are more rigorous in the measurement of the improved entanglement. These results show that the CQS operation is able to efficiently enhance the entanglement, which may be useful for the continuous-variable quantum communication.

We propose and analyze a setup to tailor the wave functions of quantum states. Our setup is based on the quantum teleportation circuit, but instead of the usual two-mode squeezed state, a two-mode non-Gaussian entangled state is used. Using this setup, we can generate various classes of quantum states such as Schrödinger cat states, four-component cat states, superpositions of Fock states, and cubic phase states. These results demonstrate the versatility of our system as a state generator and suggest that conditioning using homodyne measurements is an important tool in the generation of the non-Gaussian states complementarily to photon number detection.

Measurement-induced nonclassical effects in a two-mode interferometer are investigated theoretically using numerical simulations and analytical results. We demonstrate that for certain parameters measurements within the interferometer lead to the occurrence of two-mode squeezing. The results strongly depend on the detection probability, the phase inside the interferometer, and the choice of the input states. The appropriate parameters for maximized squeezing are obtained. We analyze the influence of losses and confirm that the predicted effects are within reach of current experimental techniques.

Multiphoton catalysis is a non-Gaussian operation which has aroused interest recently in quantum information processing. There are two schemes of multiphoton catalysis in the current literature: one uses a beam splitter, while the other uses an optical parametric amplifier. We derive the catalysis operator in the Fock basis for each scheme and find a close relationship between them. We propose to enhance quantum entanglement of the two-mode squeezed vacuum state by performing two-mode multiphoton catalysis independently using two beam splitters. We calculate the optimal entanglement, which is maximized over two independent transmissivities, for the catalyzed states with various pairs of catalysis photon numbers, and find that in each case entanglement is enhanced only when the squeezing parameter is below a certain threshold. We also find that the optimal entanglement is nonzero in the limit of low squeezing, and deduce the analytical forms of the limiting catalyzed state and its entanglement in each case from numerical coefficients.

A state of a quantum-mechanical system is completely described by a density matrix or a phase-space distribution such as the Wigner function. The complete family of squeezed states of light (states that have less uncertainty in one observable than does the vacuum state) have been generated using an optical parametric amplifier, and their density matrices and Wigner functions have been reconstructed from measurements of the quantum statistics of their electric fields.

Quantum teleportation of optical coherent states was demonstrated experimentally using squeezed-state entanglement. The quantum
nature of the achieved teleportation was verified by the experimentally determined fidelity F
exp = 0.58 ± 0.02, which describes the match between input and output states. A fidelity greater than 0.5 is not possible for
coherent states without the use of entanglement. This is the first realization of unconditional quantum teleportation where
every state entering the device is actually teleported.

Recent developments in quantum optics have led to new proposals to generate number states of the electromagnetic field using conditioned measurement techniques or the properties of atom-field interactions in microwave cavities in the micromaser. The number-state field prepared in such a way may be transformed by the action of a displacement operator; for the microwave micromaser state this could be implemented by the action of a classical current that drives the cavity field. We evaluate some properties of such displaced number states, especially their description in phase space. The photon number distribution is shown to display unusual oscillations, which are interpreted as interference in phase space, analogous to Franck-Condon oscillations in molecular spectra. The possibility of detecting these oscillations is discussed, through the photodetection counting statistics of the displaced number states. We show that the displaced-number-state quantum features are relatively robust when dissipation of the field energy is included.

We provide a simple analytic relation that connects the density operator rho^ of the electromagnetic field with the tomographic homodyne probabilities for generic quantum efficiency eta of detectors. The problem of experimentally ``sampling'' a general matrix element is addressed in the statistically rigorous sense of the central-limit theorem. We show that experimental sampling is possible also for nonunit efficiency, provided that eta satisfies a lower bound related to the ``resolutions'' of vectors ||psi> and ||Phi> in the quadrature representations. For coherent and number states the bound is eta>~1/2. On the basis of computer-simulated experiments we show the feasibility of detecting delicate quantum probability oscillations, which otherwise would be smeared out by inefficient detection.

We have measured probability distributions of quadrature-field amplitude for both vacuum and quadrature-squeezed states of a mode of the electromagnetic field. From these measurements we demonstrate the technique of optical homodyne tomography to determine the Wigner distribution and the density matrix of the mode. This provides a complete quantum mechanical characterization of the measured mode.

We report the first measurement of the joint photon-number probability distribution for a two-mode quantum state created by a nondegenerate optical parametric amplifier. The measured distributions exhibit up to 1.9 dB of quantum correlation between the signal and idler photon numbers, whereas the marginal distributions are thermal as expected for parametric fluorescence.

We have reconstructed the quantum state of optical pulses containing single photons using the method of phase-randomized pulsed optical homodyne tomography. The single-photon Fock state 1> was prepared using conditional measurements on photon pairs born in the process of parametric down-conversion. A probability distribution of the phase-averaged electric field amplitudes with a strongly non-Gaussian shape is obtained with the total detection efficiency of (55+/-1)%. The angle-averaged Wigner function reconstructed from this distribution shows a strong dip reaching classically impossible negative values around the origin of the phase space.

It is well known that spontaneous parametric down-conversion can be used to probabilistically prepare single-photon states. We have performed an experiment in which arbitrary superpositions of zero- and one-photon states can be prepared by appropriate postselection. The optical phase, which is meaningful only for superpositions of photon number, is related to the relative phase between the zero- and one-photon states. Whereas the light from spontaneous parametric down-conversion has an undefined phase, we show that this technique collapses one beam to a state of well-defined optical phase when a measurement succeeds on the other beam.

We demonstrate that local transformations on a composite quantum system can be enhanced in the presence of certain entangled states. These extra states act much like catalysts in a chemical reaction: they allow otherwise impossible local transformations to be realised, without being consumed in any way. In particular, we show that this effect can considerably improve the efficiency of entanglement concentration procedures for finite states.

We show that any single-mode quantum state can be generated from the vacuum by alternate application of the coherent displacement operator and the creation operator. We propose an experimental implementation of the scheme for traveling optical fields, which is based on field mixings and conditional measurements in a beam splitter array, and calculate the probability of state generation. Comment: 1 Table and 2 Postscript figures, using Latex; modifications and changes in Figure 2, Table 1 and Eqs. 11-13,17,18,21

The construction of a conditional quantum control-NOT (CNOT) gate from linear optical elements was described using linear-optical quantum computing method derived by Knill, Laflamme and Milburn (KLM). The phase of the nonlinear sign-shift (NS) gate, a basic nondeterminstic gate was discussed. KLM showed that the teleportation step was could be made deterministic by using the appropriate entangled resource. Calculations showed that operation was not critically dependent on experimental parameters.

A beam splitter is a simple, readily available device which can act to entangle the output optical fields. We show that a necessary condition for the fields at the output of the beam splitter to be entangled is that the pure input states exhibit nonclassical behavior. We generalize this proof for arbitrary (pure or impure) Gaussian input states. Specifically, nonclassicality of the input Gaussian fields is a necessary condition for entanglement of the field modes with the help of the beam splitter. We conjecture that this is a general property of the beam splitter: Nonclassicality of the inputs is a necessary condition for entangling fields in the beam splitter. Comment: 8 pages, 4 figures

We demonstrate, theoretically and experimentally, that statistical mixtures of the vacuum state |0〉 and the single-photon Fock state |1〉 are nonclassical according to the Vogel criterion [W. Vogel, Phys. Rev. Lett. 84, 1849 (2000)], regardless of the vacuum fraction. The ensembles are synthesized via conditional measurements on biphotons generated by means of parametric down conversion, and their quadrature statistics are measured using balanced homodyne detection. A comparative review of various quantum state nonclassicality criteria is presented.

Displaced Fock states of the electromagnetic field have been synthesized by overlapping the pulsed optical single-photon Fock state |1〉 with coherent states on a high-reflection beam splitter and completely characterized by means of quantum homodyne tomography. The reconstruction reveals nonclassical properties of displaced Fock states, such as negativity of the Wigner function and photon number oscillations. To our knowledge, this is the first time complete tomographic reconstruction has been performed on a highly nonclassical optical state.

Quantum teleportation — the transmission and reconstruction over arbitrary distances of the state of a quantum system — is demonstrated experimentally. During teleportation, an initial photon which carries the polarization that is to be transferred and one of a pair of entangled photons are subjected to a measurement such that the second photon of the entangled pair acquires the polarization of the initial photon. This latter photon can be arbitrarily far away from the initial one. Quantum teleportation will be a critical ingredient for quantum computation networks.

State preparation via conditional output measurement on a beam splitter is studied, assuming the signal mode is mixed with a mode prepared in a
Fock state and photon numbers are measured in one of the output channels. It is shown that the mode in the other output channel
is prepared in either a photon-subtracted or a photon-added Jacobi polynomial state, depending upon the difference between
the number of photons in the input Fock state and the number of photons in the output Fock state onto which it is projected.
The properties of the conditional output states are studied for coherent and squeezed input states, and the probabilities
of generating the states are calculated. Relations to other states, such as near-photon-number states and squeezed-state-excitations,
are given and proposals are made for generating them by combining the scheme with others. Finally, effects of realistic photocounting
and Fock-state preparation are discussed.

A detailed theoretical analysis of the spatiotemporal mode of a single photon prepared via conditional measurements on a photon pair generated in the process of parametric down-conversion is presented. The maximum
efficiency of coupling the photon into a transform-limited classical optical mode is calculated and ways for its optimization
are determined. An experimentally feasible technique of generating the optimally matching classical mode is proposed. The
theory is applied to a recent experiment on pulsed homodyne tomography of the single-photon Fock state [A.I. Lvovsky et al., Phys. Rev. Lett. 87, 050402 (2001)].

The Einstein-Podolsky-Rosen paradox is demonstrated experimentally for dynamical variables having a continuous spectrum. As opposed to previous work with discrete spin or polarization variables, the continuous optical amplitudes of a signal beam are inferred in turn from those of a spatially separated but strongly correlated idler beam generated by nondegenerate parametric amplification. The uncertainty product for the variances of these inferences is observed to be 0.70±0.01, which is below the limit of unity required for the demonstration of the paradox.

We show how losses in photodetection and in quantum-state measurements can be numerically compensated after the measurements have been performed. When the overall efficiency exceeds 1/2, our recipe works for all quantum states. For smaller efficiencies, however, the convergence of the compensation procedure depends on the quantum state under investigation.

A Comment on the Letter by Lucien Hardy, Phys. Rev. Lett. 73, 2279 (1994). The authors of the Letter offer a Reply.

We show how to infer the quantum state of a wave packet from position probability distributions measured during the packet{close_quote}s motion in an arbitrary potential. We assume a nonrelativistic one-dimensional or radial wave packet. Temporal Fourier transformation and spatial sampling with respect to a newly found set of functions project the density-matrix elements out of the probability distributions. The sampling functions are derivatives of products of regular and irregular wave functions. We note that the ability to infer quantum states in this way depends on the structure of the Schr{umlt o}dinger equation. {copyright} {ital 1996 The American Physical Society.}

Quantum computers promise to increase greatly the efficiency of solving problems such as factoring large integers, combinatorial optimization and quantum physics simulation. One of the greatest challenges now is to implement the basic quantum-computational elements in a physical system and to demonstrate that they can be reliably and scalably controlled. One of the earliest proposals for quantum computation is based on implementing a quantum bit with two optical modes containing one photon. The proposal is appealing because of the ease with which photon interference can be observed. Until now, it suffered from the requirement for non-linear couplings between optical modes containing few photons. Here we show that efficient quantum computation is possible using only beam splitters, phase shifters, single photon sources and photo-detectors. Our methods exploit feedback from photo-detectors and are robust against errors from photon loss and detector inefficiency. The basic elements are accessible to experimental investigation with current technology.

We show how entanglement can be used, without being consumed, to accomplish unitary operations that could not be performed without it. When applied to infinitesimal transformations, our method makes equivalent, in the sense of Hamiltonian simulation, a whole class of otherwise inequivalent two-qubit interactions. The new catalysis effect also implies the asymptotic equivalence of all such interactions.

A pulsed, balanced homodyne detector has been developed for precise measurement of the electric field quadratures of pulsed optical quantum states. A high level of common mode suppression (>85 dB) and low electronic noise (730 electrons per pulse) provide a signal-to-noise ratio of 14 dB for measurement of the quantum noise of individual pulses. Measurements at repetition rates as high as 1 MHz are possible. As a test, quantum tomography of the coherent state was performed, and the Wigner function and the density matrix were reconstructed with 99.5% fidelity. The detection system can be used for ultrarsensitive balanced detection in cw mode, e.g., for weak absorption measurements.

We present the experimental observation of polarization entanglement for three spatially separated photons. Such states of more than two entangled particles, known as GHZ states, play a crucial role in fundamental tests of quantum mechanics versus local realism and in many quantum information and quantum computation schemes. Our experimental arrangement is such that we start with two pairs of entangled photons and register one photon in a way that any information as to which pair it belongs to is erased. The registered events at the detectors for the remaining three photons then exhibit the desired GHZ correlations. Comment: Revtex, 4 pages, uses floats, epsfig

When a one-photon state is mixed with a (separate) weak coherent state at a beamsplitter the probability for detecting one photon in each beamsplitter output approaches zero due to destructive interference. We demonstrate this non-classical interference effect using pulse-gated single photons and weak mode-locked laser pulses.

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