This paper introduces coherent quantum channel discrimination as a coherent version of conventional quantum channel discrimination. Coherent channel discrimination is phrased here as a quantum interactive proof system between a verifier and a prover, wherein the goal of the prover is to distinguish two channels called in superposition in order to distill a Bell state at the end. The key measure considered here is the success probability of distilling a Bell state, and I prove that this success probability does not increase under the action of a quantum superchannel, thus establishing this measure as a fundamental measure of channel distinguishability. Also, I establish some bounds on this success probability in terms of the success probability of conventional channel discrimination. Finally, I provide an explicit semi-definite program that can compute the success probability.
It is well known that for the discrimination of classical and quantum channels in the finite, non-asymptotic regime, adaptive strategies can give an advantage over non-adaptive strategies. However, Hayashi [IEEE Trans. Inf. Theory 55(8), 3807 (2009)] showed that in the asymptotic regime, the exponential error rate for the discrimination of classical channels is not improved in the adaptive setting. We extend this result in several ways. First, we establish the strong Stein's lemma for classical-quantum channels by showing that asymptotically the exponential error rate for classical-quantum cha…
Quantum channel discrimination is a fundamental problem in quantum information science. In this study, we consider general quantum channel discrimination problems, and derive the lower bounds of the error probability. Our lower bounds are based on the triangle inequalities of the Bures angle and the trace distance. As a consequence of the lower bound based on the Bures angle, we prove the optimality of Grover's search if the number of marked elements is fixed to some integer $\ell$. This result generalizes Zalka's result for $\ell=1$. We also present several numerical results in which our lowe…
The discrimination of non-orthogonal quantum states with reduced or without errors is a fundamental task in quantum measurement theory. In this work, we investigate a quantum measurement strategy capable of discriminating two coherent states probabilistically with significantly smaller error probabilities than can be obtained using non-probabilistic state discrimination. We find that appropriate postselection of the measurement data of a photon number resolving detector can be used to discriminate two coherent states with small error probability. We compare our new receiver to an optimal inter…
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.
The optimal discrimination of non-orthogonal quantum states with minimum error probability is a fundamental task in quantum measurement theory as well as an important primitive in optical communication. In this work, we propose and experimentally realize a new and simple quantum measurement strategy capable of discriminating two coherent states with smaller error probabilities than can be obtained using the standard measurement devices; the Kennedy receiver and the homodyne receiver.
In this work we propose the technique for phase-coded weak coherent states protocols utilizing two signal states and one decoy state which is found as linear combination of signal states (Schrödinger Cat states); the latter allows to overcome the USD attack. For instance, Schrödinger Cat states can be considered as even coherent states. Moreover we consider decoy states implementation based on squeezed vacuum states which might not disables USD completely yet produces discrimination probabilities low enough to distribute keys in channel with particular losses. Thus we can detect Eve simply by …
The optimal exponential error rate for adaptive discrimination of two channels is discussed. In this problem, adaptive choice of input signal is allowed. This problem is discussed in various settings. It is proved that adaptive choice does not improve the exponential error rate in these settings. These results are applied to quantum state discrimination.
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.
A general scenario for an $N$-sequential conclusive state discrimination introduced recently in Loubenets and Namkung [arXiv:2102.04747] can provide a multipartite quantum communication realizable in the presence of a noise. In the present article, we propose a new experimental scheme for the implementation of a sequential conclusive discrimination between binary coherent states via indirect measurements within the Jaynes-Cummings interaction model. We find that if the mean photon number is less than 1.6, then, for our two-sequential state discrimination scheme, the optimal success probability…
We propose and experimentally demonstrate a near-optimal discrimination scheme for the quadrature phase shift keying protocol (QPSK). We show in theory that the performance of our hybrid scheme is superior to the standard scheme - heterodyne detection - for all signal amplitudes and underpin the predictions with our experimental results. Furthermore, our scheme provides the hitherto best performance in the domain of highly attenuated signals. The discrimination is composed of a quadrature measurement, a conditional displacement and a threshold detector.
It is well known that a minimum error quantum measurement for arbitrary binary optical coherent states can be realized by a receiver that comprises interfering with a coherent reference light, photon counting, and feedback control. We show that, for ternary optical coherent states, a minimum error measurement cannot always be realized by such a receiver. The problem of finding an upper bound on the maximum success probability of such a receiver can be formulated as a convex programming. We derive its dual problem and numerically find the upper bound. At least for ternary phase-shift keyed cohe…
School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）数据科学学院）；Institute for Quantum Information, RWTH Aachen University（亚琛工业大学量子信息研究所）
The so-called quasi-Bell entangled coherent states in a thermal environment are studied. In the analysis, we assume thermal noise affects only one of the two modes of each state. First the matrix representation of the density operators of the quasi-Bell entangled coherent states in a thermal environment is derived. Secondly we investigate the entanglement property of one of the quasi-Bell entangled coherent states with thermal noise. At that time a lower bound of the entanglement of formation for the state is computed. Thirdly the minimax discrimination problem for two cases of the binary set …
We demonstrate a sub-shot-noise-limit discrimination of on-off keyed coherent signals by an optimal displacement quantum receiver in which a superconducting transition edge sensor is installed. Use of a transition edge sensor and a fiber beam splitter realizes high total detection efficiency and high interference visibility of the receiver and the observed average error surpasses the shot-noise-limit in a wider range of the signal power. Our technique opens up a new technology for the sub-shot-noise-limit detection of coherent signals in optical communication channels.
We consider quantum-memory assisted protocols for discriminating quantum channels. We show that for optimal discrimination of memory channels, memory assisted protocols are needed. This leads to a new notion of distance for channels with memory. For optimal discrimination and estimation of sets of unitary channels memory-assisted protocols are not required.
We quantify the usefulness of a bipartite quantum state in the ancilla-assisted channel discrimination of arbitrary quantum channels, formally defining a worst-case-scenario channel discrimination power for bipartite quantum states. We show that such a quantifier is deeply connected with the operator Schmidt decomposition of the state. We compute the channel discrimination power exactly for pure states, and provide upper and lower bounds for general mixed states. We show that highly entangled states can outperform any state that passes the realignment criterion for separability. Furthermore, w…
The discrimination of two nonorthogonal states is a fundamental element for secure and efficient communication. Quantum measurements of nonorthogonal coherent states can enhance information transfer beyond the limits of conventional technologies. We demonstrate a strategy for binary state discrimination based on optimized single-shot measurements with photon number resolving (PNR) detection with finite number resolution. This strategy enables a high degree of robustness to noise and imperfections while being scalable to high rates and in principle allows for surpassing the quantum noise limit …
We prove that every entangled state is useful as a resource for the problem of minimum-error channel discrimination. More specifically, given a single copy of an arbitrary bipartite entangled state, it holds that there is an instance of a quantum channel discrimination task for which this state allows for a correct discrimination with strictly higher probability than every separable state.
The use of linearly independent signal states in realistic implementations of quantum key distribution (QKD) enables an eavesdropper to perform unambiguous state discrimination. We explore quantitatively the limits for secure QKD imposed by this fact taking into account that the receiver can monitor to some extend the photon number statistics of the signals even with todays standard detection schemes. We compare our attack to the beamsplitting attack and show that security against beamsplitting attack does not necessarily imply security against the attack considered here.
We study the discrimination of weak coherent states of light with significant overlaps by nondestructive measurements on the light states through measuring atomic states that are entangled to the coherent states via dipole coupling. In this way, the problem of measuring and discriminating coherent light states is shifted to finding the appropriate atom-light interaction and atomic measurements. We show that this scheme allows us to attain a probability of error extremely close to the Helstrom bound, the ultimate quantum limit for discriminating binary quantum states, through the simple Jaynes-…
We provide a simple example that illustrates the advantage of adaptive over non-adaptive strategies for quantum channel discrimination. In particular, we give a pair of entanglement-breaking channels that can be perfectly discriminated by means of an adaptive strategy that requires just two channel evaluations, but for which no non-adaptive strategy can give a perfect discrimination using any finite number of channel evaluations.
We show that every entangled state provides an advantage in bi- and multi-channel discrimination that singles out its degree of entanglement, quantified in terms of its Schmidt number and of the corresponding robustness measures.
The discrimination of coherent states is a key task in optical communication and quantum key distribution protocols. In this work, we use a photon-number-resolving detector, the transition-edge sensor, to discriminate binary-phase-shifted coherent states at a telecom wavelength. Owing to its dynamic range and high efficiency, we achieve a bit error probability that unconditionally exceeds the standard quantum limit (SQL) by up to 7.7 dB. The improvement to the SQL persists for signals containing up to approximately seven photons on average and is achieved in a single shot (i.e. without measure…
Generalized quantum measurements identifying non-orthogonal states without ambiguity often play an indispensable role in various quantum applications. For such unambiguous state discrimination scenario, we have a finite probability of obtaining inconclusive results and minimizing the probability of inconclusive results is of particular importance. In this paper, we experimentally demonstrate an adaptive generalized measurement that can unambiguously discriminate the quaternary phase-shift-keying coherent states with a near-optimal performance. Our scheme is composed of displacement operations,…
We consider the sequential quantum channel discrimination problem using adaptive and non-adaptive strategies. In this setting the number of uses of the underlying quantum channel is not fixed but a random variable that is either bounded in expectation or with high probability. We show that both types of error probabilities decrease to zero exponentially fast and, when using adaptive strategies, the rates are characterized by the measured relative entropy between two quantum channels, yielding a strictly larger region than that achievable by non-adaptive strategies. Allowing for quantum memory,…
We solve the problem of discriminating with minimum error probability two given Pauli channels. We show that, differently from the case of discrimination between unitary transformations, the use of entanglement with an ancillary system can strictly improve the discrimination, and any maximally entangled state allows to achieve the optimal discrimination. We also provide a simple necessary and sufficient condition in terms of the structure of the channels for which the ultimate minimum error probability can be achieved without entanglement assistance. When such a condition is satisfied, the opt…
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…
A collection of quantum channels is called incompatible if they cannot be obtained as marginals from a single channel. No-cloning theorem is the most prominent instance of incompatibility of quantum channels. We show that every collection of incompatible channels can be more useful than compatible ones as preprocessings in a state discrimination task with multiple ensembles of states. This is done by showing that the robustness of channel incompatibility which is a measure for incompatibility of channels exactly quantifies the maximum advantage in the state discrimination. We also show that in…
We study the power of measurements implementable with local quantum operations and classical communication (or LOCC measurements for short) in the setting of quantum channel discrimination. More precisely, we consider discrimination procedures that attempt to identify an unknown channel, chosen uniformly from two known alternatives, that take the following form: (i) the input to the unknown channel is prepared in a possibly entangled state with an ancillary system, (ii) the unknown channel is applied to the input system, and (iii) an LOCC measurement is performed on the output and ancillary sy…
We introduce and study norms in the space of hermitian operators, obtained from base norms in positively generated subspaces. These norms are closely related to discrimination of so-called generalized quantum channels, including quantum states, channels and networks. We further introduce generalized quantum decision problems and show that the maximal average payoff of decision procedures is again given by these norms. We also study optimality of decision procedures, in particular, we obtain a necessary and sufficient condition under which an optimal 1-tester for discrimination of quantum chann…
We discuss several methods for unambiguous state discrimination of N symmetric coherent states using linear optics and photodetectors. One type of measurements is shown to be optimal in the limit of small photon numbers for any N. For the special case of N=4 this measurement can be fruitfully used by the receiving end (Bob) in an implementation of the BB84 quantum key distribution protocol using faint laser pulses. In particular, if Bob detects only a single photon the procedure is equivalent to the standard measurement that he would have to perform in a single-photon implementation of BB84, i…
Query Complexity of Classical and Quantum Channel Discrimination
School of Electrical；Computer Engineering, Cornell University, Ithaca, New York 14850, USA；Department of Mathematics \& Illinois Quantum Information Science；Technology (IQUIST) Center, University of Illinois Urbana-Champaign, Urbana, IL 61801, USA
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 …
For minimum-error channel discrimination tasks that involve only unitary channels, we show that sequential strategies may outperform the parallel ones. Additionally, we show that general strategies that involve indefinite causal order are also advantageous for this task. However, for the task of discriminating a uniformly distributed set of unitary channels that forms a group, we show that parallel strategies are, indeed, optimal, even when compared to general strategies. We also show that strategies based on the quantum switch cannot outperform sequential strategies in the discrimination of u…
We consider the problem of optimally discriminating two Pauli channels in the minimax strategy, maximizing the smallest of the probabilities of correct identification of the channel. We find the optimal input state at the channel and show the conditions under which using entanglement strictly enhances distinguishability. We finally compare the minimax strategy with the Bayesian one.
A no-signalling channel transforming quantum systems in Alice's and Bob's laboratories is compatible with two different causal structures: (A < B) Alice's output causally precedes Bob's input and (B< A) Bob's output causally precedes Alice's input. I show that a quantum superposition of circuits operating within these two causal structures enables the perfect discrimination between no-signalling channels that can not be perfectly distinguished by any ordinary circuit.
We present a method to separate coherent and incoherent contributions of cathodoluminescence (CL) by using a time-resolved coincidence detection scheme. For a proof-of-concept experiment, we generate CL by irradiating an optical multimode fiber with relativistic electrons in a transmission electron microscope. A temporal analysis of the CL reveals a large peak in coincidence counts for small time delays, also known as photon bunching. Additional measurements allow us to attribute the bunching peak to the temporal correlations of coherent CL (Cherenkov radiation) created by individual electrons…
A scheme for the detection of low-intensity optical coherent signals was studied which uses a probabilistic amplifier operated in the non-heralded version, as the underlying non-linear operation to improve the detection efficiency. This approach allows us to improve the statistics by keeping track of all possible outcomes of the amplification stage (including failures). When compared with an optimized Kennedy receiver, the resulting discrimination success probability we obtain presents a gain up to ~1.85% and it approaches the Helstrom bound appreciably faster than the Dolinar receiver, when e…
In this work, we consider optimal state discrimination for a quantum system that interacts with an environment, i.e., states evolve under a quantum channel. We show the conditions on a quantum channel and an ensemble of states such that a measurement for optimal state discrimination is preserved. In particular, we show that when an ensemble of states with equal {\it a priori} probabilities is given, an optimal measurement can be preserved over any quantum channel by applying local operations and classical communication, that is, by manipulating the quantum states before and after the channel a…
We present an instance of a task of minimum-error discrimination of two qubit-qubit quantum channels for which a sequential strategy outperforms any parallel strategy. We then establish two new classes of strategies for channel discrimination that involve indefinite causal order and show that there exists a strict hierarchy among the performance of all four strategies. Our proof technique employs a general method of computer-assisted proofs. We also provide a systematic method for finding pairs of channels that showcase this phenomenon, demonstrating that the hierarchy between the strategies i…
Many quantum mechanical experiments can be viewed as multi-round interactive protocols between known quantum circuits and an unknown quantum process. Fully quantum "coherent" access to the unknown process is known to provide an advantage in many discrimination tasks compared to when only incoherent access is permitted, but it is unclear if this advantage persists when the process is noisy. Here, we show that a quantum advantage can be maintained when distinguishing between two noisy single qubit rotation channels. Numerical and analytical calculations reveal a distinct transition between optim…
