We propose a long-distance continuous-variable quantum key distribution (CVQKD) with four-state protocol using non-Gaussian state-discrimination detection. A photon subtraction operation, which is deployed at the transmitter, is used for splitting the signal required for generating the non-Gaussian operation to lengthen the maximum transmission distance of CVQKD. Whereby an improved state-discrimination detector, which can be deemed as an optimized quantum measurement that allows the discrimination of nonorthogonal coherent states beating the standard quantum limit, is applied at the receiver to codetermine the measurement result with conventional coherent detector. By tactfully exploiting multiplexing technique, the resulting signals can be simultaneously transmitted through an untrusted quantum channel, and subsequently sent to the state-discrimination detector and coherent detector respectively. Security analysis shows that the proposed scheme can lengthen the maximum transmission distance up to hundreds of kilometers. Furthermore, by taking finite-size effect and composable security into account we obtain the tightest bound of the secure distance, which is more practical than that obtained in the asymptotic limit.
We address the limit of the Gaussian operations and classical communication in the problem of quantum state discrimination. We show that the optimal Gaussian strategy for the discrimination of the binary phase shift keyed (BPSK) coherent signal is a simple homodyne detection. We also propose practical near-optimal quantum receivers that beat the BPSK homodyne limit in all areas of the signal power. Our scheme is simple and does not require realtime electrical feedback.
Consider two bosonic modes which are prepared in one of two possible Gaussian states with the same local energy: either a tensor-product thermal state (with zero correlations) or a separable Gaussian state with maximal correlations (with both classical and quantum correlations, the latter being quantified by quantum discord). For the discrimination of these states, we compare the optimal joint coherent measurement with the best local measurement based on single-mode Gaussian detections. We show how the coherent measurement always strictly outperforms the local detection strategy for both singl…
We determine the optimal method of discriminating and comparing quantum states from a certain class of multimode Gaussian states and their mixtures when arbitrary global Gaussian operations and general Gaussian measurements are allowed. We consider the so-called constant-$\hat{p}$ displaced states which include mixtures of multimode coherent states arbitrarily displaced along a common axis. We first show that no global or local Gaussian transformations or generalized Gaussian measurements can lead to a better discrimination method than simple homodyne measurements applied to each mode separate…
The non-linearity of a conditional photon-counting measurement can be used to `de-Gaussify' a Gaussian state of light. Here we present and experimentally demonstrate a technique for photon number resolution using only homodyne detection. We then apply this technique to inform a conditional measurement; unambiguously reconstructing the statistics of the non-Gaussian one and two photon subtracted squeezed vacuum states. Although our photon number measurement relies on ensemble averages and cannot be used to prepare non-Gaussian states of light, its high efficiency, photon number resolving capabi…
We experimentally demonstrate a new measurement scheme for the discrimination of two coherent states. The measurement scheme is based on a displacement operation followed by a photon number resolving detector, and we show that it outperforms the standard homodyne detector which we, in addition, proof to be optimal within all Gaussian operations including conditional dynamics. We also show that the non-Gaussian detector is superior to the homodyne detector in a continuous variable quantum key distribution scheme.
By combining the Minkowski inequality and the quantum Chernoff bound, we derive easy-to-compute upper bounds for the error probability affecting the optimal discrimination of Gaussian states. In particular, these bounds are useful when the Gaussian states are unitarily inequivalent, i.e., they differ in their symplectic invariants.
Inspired by the notion that environmental noise is in principle observable, whilst fundamental noise due to spontaneous localisation would not be, we study the estimation of the diffusion parameter induced by wave function collapse models under continuous monitoring of the environment. We take into account finite measurement efficiencies and, in order to quantify the advantage granted by monitoring, we analyse the quantum Fisher information associated with such a diffusion parameter, identify optimal measurements in limiting cases, and assess the performance of such measurements in more realis…
We address extended systems interacting with classical fluctuating environments and analyze the use of quantum probes to discriminate local noise, described by independent fluctuating fields, from common noise, corresponding to the interaction with a common one. In particular, we consider a bipartite system made of two non interacting harmonic oscillators and assess discrimination strategies based on homodyne detection, comparing their performances with the ultimate bounds on the error probabilities of quantum-limited measurements. We analyze in details the use of Gaussian probes, with emphasi…
We investigate the performance of a Kennedy receiver, which is known as a beneficial tool in optical coherent communications, to the quantum state discrimination of the two superpositions of vacuum and single photon states corresponding to the $\hatσ_x$ eigenstates in the single-rail encoding of photonic qubits. We experimentally characterize the Kennedy receiver in vacuum-single photon two-dimensional space using quantum detector tomography and evaluate the achievable discrimination error probability from the reconstructed measurement operators. We furthermore derive the minimum error rate ob…
We investigate the energy-constrained (EC) diamond norm distance between unitary channels acting on possibly infinite-dimensional quantum systems, and establish a number of results. Firstly, we prove that optimal EC discrimination between two unitary channels does not require the use of any entanglement. Extending a result by Acín, we also show that a finite number of parallel queries suffices to achieve zero error discrimination even in this EC setting. Secondly, we employ EC diamond norms to study a novel type of quantum speed limits, which apply to pairs of quantum dynamical semigroups. We …
Projected Dynamic Programming for Sequential Quantum State Discrimination
College of Liberal Studies, Seoul National University（首尔大学通识学院）；Team QST（QST团队）；Department of Mathematics, College of Science, Korea University（高丽大学理学院数学系）；Institute of Computer Technology, College of Engineering, Seoul National University（首尔大学工程学院计算机技术研究所）
Quantum metrology utilizes nonclassical resources, such as entanglement or squeezed light, to realize sensors whose performance exceeds that afforded by classical-state systems. Environmental loss and noise, however, easily destroy nonclassical resources, and thus nullify the performance advantages of most quantum-enhanced sensors. Quantum illumination (QI) is different. It is a robust entanglement-enhanced sensing scheme whose 6 dB performance advantage over a coherent-state sensor of the same average transmitted photon number survives the initial entanglement's eradication by loss and noise.…
We propose an innovative strategy to discriminate between two coherent states affected by either uniform or gaussian phase noise. The strategy is based on a homodyne-like detection scheme with photon-number-resolving detectors in the regime of low-intensity local oscillator. The experimental implementation of the detection scheme involves two hybrid photodetectors, whose outputs are used in post processing to calculate the shot-by-shot photon-number difference. The performance of this strategy is quantified in terms of the error probability in discriminating the noisy coherent signals as a fun…
We demonstrate an optical receiver that achieves the quantum Chernoff bound for discriminating coherent states from thermal states in the multi-copy scenario. In contrast, we find that repeated use of the receiver approaching the Helstrom bound for single-copy measurement is sub-optimal in this multi-copy case. Furthermore, for a large class of multi-copy discrimination tasks between a pure and a mixed state, we prove that any Helstrom-bound achieving single-copy receiver is suboptimal by a factor of at least two in error-probability exponent compared to the multi-copy quantum Chernoff bound. …
We investigate quantum measurement strategies capable of discriminating two coherent states probabilistically with significantly smaller error probabilities than can be obtained using non- probabilistic state discrimination. We apply a postselection strategy to the measurement data of a homodyne detector as well as a photon number resolving detector in order to lower the error probability. We compare the two different receivers with an optimal intermediate measurement scheme where the error rate is minimized for a fixed rate of inconclusive results. The photon number resolving (PNR) receiver i…
Quantum-elevated Chiral Discrimination for Bio-molecules
Tsung-Dao Lee Institute, Shanghai Jiao Tong University（上海交大李政道研究所）；School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院）；Shanghai Branch, Hefei National Laboratory（合肥国家实验室上海分部）；Shanghai Research Center for Quantum Sciences（上海量子科学研究中心）；Collaborative Innovation Center of Extreme Optics, Shanxi University（山西大学极端光学协同创新中心）
We investigate the discrimination of two candidates of an unknown parameter in quantum systems with continuous weak measurement, inspired by the application of hypothesis testing in distinguish-ing two Hamiltonians [Kiilerich and Mølmer, Phys. Rev. A, 98, 022103 (2018)]. Based on the measurement output and stochastic master equation, temporal evolutions of posterior probabilities of two hypotheses are given by Bayes' formula. The Bayes criterion is presented by the likelihood ratio conditioned on the outcome of measurements. Different from the calculation method based on maximum a posteriori c…
We analyze the capability of discriminating the statistical nature of a thermal bath by exploiting the interaction with an additional environment. We first shows that, at difference with the standard scenario where the additional environment is not present, the modified evolution induced by the mere presence of the extra bath allows to improve the discrimination task. We then also consider the possibility of continuously monitoring the additional environment and we discuss in detail how to obtain improved performances in the discrimination by considering different kinds of interaction, i.e. di…
The characterisation of Quantum Channel Discrimination (QCD) offers critical insight for future quantum technologies in quantum metrology, sensing and communications. The task of multi-channel discrimination creates a scenario in which the discrimination of multiple quantum channels can be equated to the idea of pattern recognition, highly relevant to the tasks of quantum reading, illumination and more. Whilst the optimal quantum strategy for many scenarios is an entangled idler-assisted protocol, the extension to a multi-hypothesis setting invites the exploration of discrimination strategies …
We present a class of non-Gaussian two-mode continuous variable states for which the separability criterion for Gaussian states can be employed to detect whether they are separable or not. These states reduce to the two-mode Gaussian states as a special case.
Boosting State Discrimination in Quantum Brownian Motion Channel via Memory-Induced Coherence Preservation
Federal University of Piauí（皮奥伊联邦大学）；Federal University of Paraná（巴拉那联邦大学）；Universidade Federal do ABC（ABC联邦大学）；Southern University of Science and Technology（南方科技大学）；University of Coimbra（科英布拉大学）
We study a class of mixed non-Gaussian entangled states that, whilst closely related to Gaussian entangled states, none-the-less exhibit distinct properties previously only associated with more exotic, pure non-Gaussian states.
We present a general analysis of the state obtained by subjecting the output from a continuous-wave (cw) Gaussian field to non-Gaussian measurements. The generic multimode state of cw Gaussian fields is characterized by an infinite dimensional covariance matrix involving the noise correlations of the source. Our theory extracts the information relevant for detection within specific temporal output modes from these correlation functions . The formalism is applied to schemes for production of non-classical light states from a squeezed beam of light.
We characterise Gaussian quantum channels that are Gaussian incompatibility breaking, that is, transform every set of Gaussian measurements into a set obtainable from a joint Gaussian observable via Gaussian postprocessing. Such channels represent local noise which renders measurements useless for Gaussian EPR-steering, providing the appropriate generalisation of entanglement breaking channels for this scenario. Understanding the structure of Gaussian incompatibility breaking channels contributes to the resource theory of noisy continuous variable quantum information protocols.
In this paper, we give a simple proof of the fact that the optimal collective attacks against continous-variable quantum key distribution with a Gaussian modulation are Gaussian attacks. Our proof makes use of symmetry properties of the quantum key distribution protocol in phase-space.
Noise-robust discrimination of incoherent point sources with spatial-mode demultiplexing
School of Sciences and Zhejiang Key Laboratory of Quantum State Control and Optical Field Manipulation, Hangzhou Dianzi University（杭州电子科技大学理学院与浙江省量子态调控与光场操纵重点实验室）
Quantum state separation is a probabilistic map that transforms a given set of pure states into another set of more distinguishable ones. Here we investigate such a map acting onto uniparametric families of symmetric linearly dependent or independent quantum states. We obtained analytical solutions for the success probability of the maps--which is shown to be optimal--as well as explicit constructions in terms of positive operator valued measures. Our results can be used for state discrimination strategies interpolating continuously between minimum-error and unambiguous (or maximum-confidence)…
We describe a continuous variable error correction protocol that can correct the Gaussian noise induced by linear loss on Gaussian states. The protocol can be implemented using linear optics and photon counting. We explore the theoretical bounds of the protocol as well as the expected performance given current knowledge and technology.
We investigate an optimization problem of finding quantum sequential measurements, which forms a wide class of state discrimination problems with the restriction that only sequential measurements are allowed. Sequential measurements from Alice to Bob on a bipartite system are considered. Using the fact that the optimization problem can be formulated as a problem with only Alice's measurement and is convex programming, we derive its dual problem and necessary and sufficient conditions for an optimal solution. In the problem we address, the output of Alice's measurement can be infinite or contin…
We analyze the asymptotic security of the family of Gaussian modulated Quantum Key Distribution protocols for Continuous Variables systems. We prove that the Gaussian unitary attack is optimal for all the considered bounds on the key rate when the first and second momenta of the canonical variables involved are known by the honest parties.
We investigate the security of continuous-variable (CV) quantum key distribution (QKD) using coherent states and reverse reconciliation against Gaussian individual attacks based on an optimal Gaussian $1 \to 2$ cloning machine. We provide an implementation of the optimal Gaussian individual attack. we also find a Bell-measurement attack which works without delayed choice of measurements and has better performance than the cloning attack.
We investigate non-Gaussian states of light as ancillary inputs for generating nonlinear transformations required for quantum computing with continuous variables. We consider a recent proposal for preparing a cubic phase state, find the exact form of the prepared state and perform a detailed comparison to the ideal cubic phase state. We thereby identify the main challenges to preparing an ideal cubic phase state and describe the gates implemented with the non-ideal prepared state. We also find the general form of operations that can be implemented with ancilla Fock states, together with Gaussi…
Schroedinger operators with certain Gaussian random potentials in multi-dimensional Euclidean space possess almost surely an absolutely continuous integrated density of states and no absolutely continuous spectrum at sufficiently low energies.
The required set of operations for universal continuous-variable quantum computation can be divided into two primary categories: Gaussian and non-Gaussian operations. Furthermore, any Gaussian operation can be decomposed as a sequence of phase-space displacements and symplectic transformations. Although Gaussian operations are ubiquitous in quantum optics, their experimental realizations generally are approximations of the ideal Gaussian unitaries. In this work, we study different performance criteria to analyze how well these experimental approximations simulate the ideal Gaussian unitaries. …
The Gaussian state description of continuous variables is adapted to describe the quantum interaction between macroscopic atomic samples and continuous-wave light beams. The formalism is very efficient: a non-linear differential equation for the covariance matrix of the atomic system explicitly accounts for both the unitary evolution, the dissipation and noise due to the atom-light interaction, and the back-action due to homodyne optical detection on the beam after its interaction with the atoms. Applications to atomic spin squeezing and estimation of unknown classical parameters are presented…
The asymptotic state discrimination problem with simple hypotheses is considered for a cubic lattice of bosons. A complete solution is provided for the problems of the Chernoff and the Hoeffding bounds and Stein's lemma in the case when both hypotheses are gauge-invariant Gaussian states with translation-invariant quasi-free parts.
Measures of entanglement, fidelity and purity are basic yardsticks in quantum information processing. We propose how to implement these measures using linear devices and homodyne detectors for continuous variable Gaussian states. In particular, the test of entanglement becomes simple with some prior knowledge which is relevant to current experiments.
We suggest an experimentally realizable scheme to test entanglement of a mixed Gaussian continuous variable state. We find that the entanglement condition is simplified for the family of Gaussian states which are relevant to experimental realization. The entanglement condition is then shown to be directly related to joint homodyne measurements. We show how robust the proposed test of entanglement is against imperfect detection efficiency.
Efficiently Computable Strategies and Limits for Bosonic Channel Discrimination
School of Science, STEM College, RMIT University（斯威本科技大学理学院STEM学院）；Scuola Normale Superiore（比萨高等师范学院）；Korteweg–de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学科特韦格-德弗里斯数学研究所）；Mathematical Quantum Information RIKEN Hakubi Research Team, RIKEN Pioneering Research Institute (PRI) and RIKEN Center for Quantum Computing (RQC)（理科院系量子信息RIKEN Hakubi研究团队，理化学研究所前沿研究院（PRI）和理化学研究所量子计算中心（RQC））；School of Electrical and Computer Engineering, Cornell University（康奈尔大学电气与计算机工程学院）
We describe an entanglement purification protocol to generate maximally entangled states with high efficiencies from two-mode squeezed states or from mixed Gaussian continuous entangled states. The protocol relies on a local quantum non-demolition measurement of the total excitation number of several continuous variable entangled pairs. We propose an optical scheme to do this kind of measurement using cavity enhanced cross--Kerr interactions.
We propose a non-Gaussian continuous variable (CV) gate which is able to conditionally produce superposition of two "copies" of an arbitrary input state well separated in the coordinate and momentum plane - a Schrödinger cate state. The gate uses cubic phase state of an ancillary oscillator as a non-Gaussian resource, an entangling Gaussian gate, and homodyne measurement which provides nonunique information about the target system canonical variables, which is a key feature of the scheme. We show that this nonuniqueness manifests problems which may arise by extension of the Heisenberg picture …
We use quantum entanglement witnesses derived from Gaussian operators to study the separable criteria of continuous variable states. We transform the validity of a Gaussian witness to a Bosonic Gaussian channel problem. It follows that the maximal means of two-mode and some four-mode Gaussian operators over product pure states are achieved by vacuum (or coherent states and squeezed states) according to the properties of Bosonic Gaussian channels. Then we have necessary and sufficient criteria of separability not only for Gaussian quantum states, but also for non-Gaussian states prepared by pho…
Entanglement distillation is a fundamental building block in long-distance quantum communication. Though known to be useless on their own for distilling Gaussian entangled states, local Gaussian operations may still help to improve non-Gaussian entanglement distillation schemes. Here we show that by applying local squeezing operations, both the performance and the efficiency of existing distillation protocols can be enhanced. We derive the optimal enhancement through local Gaussian unitaries, which can be obtained even in the most natural scenario when Gaussian mixed entangled states are share…
Two Kerr-squeezed optical beams can be combined in a beamsplitter to produce non-Gaussian continuous-variable entangled states. We characterize the non-Gaussian nature of the output by calculating the third-order cumulant of quadrature variables, and predict the level of entanglement that could be generated by evaluating the Duan-Simon and Reid Einstein-Podolsky-Rosen criteria. These states have the advantage over Gaussian states and non-Gaussian measurement schemes in that the well known, efficient and proven technology of homodyne detection may be used for their characterisation. A physical …
We study nonclassical correlations beyond entanglement in a family of two-mode non-Gaussian states which represent the continuous-variable counterpart of two-qubit Werner states. We evaluate quantum discord and other quantumness measures obtaining exact analytical results in special instances, and upper and lower bounds in the general case. Non-Gaussian measurements such as photon counting are in general necessary to solve the optimization in the definition of quantum discord, whereas Gaussian measurements are strictly suboptimal for the considered states. The gap between Gaussian and optimal …
We consider two classes of non-Gaussian entangled states generated from the product of number states with the action of the beamsplitter or the two-mode squeezer. It is shown that, for many of these states, the covariance matrix is compatible with the covariance matrix of separable Gaussian states and their separability cannot be verified by the measurements of the first and second moments of canonical variables. We identify a couple of continuous-variable entanglement criteria with higher order moments to verify these non-Gaussian entanglement.
