No-Signaling Principle Can Determine Optimal Quantum State Discrimination

In this paper we consider the general problem of designing the optimal quantum detection for minimum-error quantum state discrimination. Since it is hard to get an explicit solution and we must resort to numerical methods for the general case. We show the problem can be cast into a semidefinite programming(SDP) problem, we use the theory of SDP to solve it. Based on the idea, we propose a method for minimum-error discrimination by formulating the design of optimal quantum detection as a standard SDP problem; furthermore, based on the optimality conditions for SDP, we derive a set of optimality conditions for quantum detection. It is exemplified that this method is simple to implement and can be used to design optimum detection operators for minimum-error discrimination efficiently.

For the optimal success probability under minimum-error discrimination between $r\geq2$ arbitrary quantum states prepared with any a priori probabilities, we find new general analytical lower and upper bounds and specify the relations between these new general bounds and the general bounds known in the literature. We also present the example where the new general analytical bounds, lower and upper, on the optimal success probability are tighter than most of the general analytical bounds known in the literature. The new upper bound on the optimal success probability explicitly generalizes to $r>2$ the form of the Helstrom bound. For $r=2$, each of our new bounds, lower and upper, reduces to the Helstrom bound.

Iterative algorithm for minimum-error quantum state discrimination: Convergence for pure-state ensembles

Correlations that violate a Bell inequality are said to be nonlocal; i.e., they do not admit a local and deterministic explanation. Great effort has been devoted to study how the amount of nonlocality (as measured by a Bell inequality violation) serves to quantify the amount of randomness present in observed correlations. In this work we reverse this research program and ask what do the randomness certification capabilities of a theory tell us about the nonlocality of that theory. We find that, contrary to initial intuition, maximal randomness certification cannot occur in maximally nonlocal theories. We go on and show that quantum theory, in contrast, permits certification of maximal randomness in all dichotomic scenarios. We hence pose the question of whether quantum theory is optimal for randomness; i.e., is it the most nonlocal theory that allows maximal randomness certification? We answer this question in the negative by identifying a larger-than-quantum set of correlations capable of this feat. Not only are these results relevant to understanding quantum mechanics' fundamental features, but also put fundamental restrictions on device-independent protocols based on the no-signaling principle.

The discovery of postquantum nonlocality, i.e., the existence of nonlocal correlations that are stronger than any quantum correlations but nevertheless consistent with the no-signaling principle, has deepened our understanding of the foundations of quantum theory. In this work, we investigate whether the phenomenon of Einstein-Podolsky-Rosen steering, a different form of quantum nonlocality, can also be generalized beyond quantum theory. While post-quantum steering does not exist in the bipartite case, we prove its existence in the case of three observers. Importantly, we show that postquantum steering is a genuinely new phenomenon, fundamentally different from postquantum nonlocality. Our results provide new insight into the nonlocal correlations of multipartite quantum systems.

Quantum measurement and control science provides a toolkit for implementing quantum information protocols, overcoming noisy operation and minimizing the use of costly quantum resources. The importance of finding optimal measurement and control strategies is revealed in the task of quantum state discrimination, a characteristic feature of quantum mechanics and a primitive for quantum information science and technology. When a quantum system is prepared in one of two known nonorthogonal quantum states, no measurement can deterministically tell which state the system is in. Two strategies can be applied to discriminate non-orthogonal states efficiently. Unambiguous state discrimination (USD) is strategy that provides a guess which is either correct, or inconclusive [1]. Alternatively, minimum error discrimination (MED) strategy will always provide a conclusive answer, which can sometimes be incorrect [2]. Perfect nonorthogonal state discrimination is still impossible even when multiple copies of the system are available, although different strategies increase the chance of having conclusive (for USD), or correct (for MED) result. Multiple-copy measurement strategies can be either collective, when a single measurement involves all the copies of the system, or local, when each copy of the system is measured separately. The latter can be further divided into fixed, where the measurement applied to each copy is the same, or adaptive, where the measurement on each subsequent copy depends on the outcomes of the previous measurements [3,4]. A multiple-copy MED strategy is defined by its goal of minimizing the average error for fixed resources, i.e. the number of copies of the system. An alternative approach we consider, is to minimize the average resources required, while keeping errors below a given bound [5]. This approach is central for fault-tolerant quantum computing and has been applied to a number of quantum control strategies [6,7]. We call the corresponding state discrimination task the guaranteed bounded error discrimination (GBED) task. Intuitively, one may assume that the multiple-copy strategies optimal for MED would be also optimal for GBED. We experimentally apply two known local non-adaptive strategies, previously considered for the MED task, to the two-state GBED problem [5]. We then derive and experimentally demonstrate a new local strategy, designed specifically for the GBED task that outperforms other strategies. Moreover, it performs usually better than, and in the regime of small error, scales better than, the theoretical performance of the optimal adaptive strategy for the MED task. The discovered reversal in the performance of schemes when swapping the task definition from performance-maximization to resource-minimization, is similar to that previously observed in state purification [6], suggesting that this phenomenon is a generic one. [1] I. D. Ivanovic, Phys. Lett. A 123, 257 (1987). [2] C. W. Helstrom, Quantum detection and estimation theory (1976). [3] A. Acín, et al., Phys. Rev. A 71, 032338 (2005). [4] B. L. Higgins, et al., Phys. Rev. Lett. 103, 220503 (2009). [5] S. Slussarenko, et al., PRL 118, 030502 (2017). [6] H. M. Wiseman and J. F. Ralph. New J. Phys. 8, 90 (2006). [7] J. Combes, et al., Phys. Rev. Lett. 100, 160503 (2008).

We study cheating strategies against a practical four-state quantum\nbit-commitment protocol and its two-state variant when the underlying quantum\nchannels are noisy and the cheating party is constrained to using single-qubit\nmeasurements only. We show that simply inferring the transmitted photons'\nstates by using the Breidbart basis, optimal for ambiguous (minimum-error)\nstate discrimination, does not directly produce an optimal cheating strategy\nfor this bit-commitment protocol. We introduce a new strategy, based on certain\npost-measurement processes, and show it to have better chances at cheating than\nthe direct approach. We also study to what extent sending forged geographical\ncoordinates helps a dishonest party in breaking the binding security\nrequirement. Finally, we investigate the impact of imperfect single-photon\nsources in the protocols. Our study shows that, in terms of the resources used,\nthe four-state protocol is advantageous over the two-state version. The\nanalysis performed can be straightforwardly generalised to any finite-qubit\nmeasurement, with the same qualitative results.\n

We study the quantum state discrimination problem under the minimum error (ME) strategy for a set of N pure equidistant states. These states are characterized by the property that the inner product between any pair of states is given by a unique complex number S. We provide the explicit form of the states and analyze their main structural properties. The optimal success probability for ME discrimination is evaluated as a function of the number of states, as well as the modulus and phase of the inner product S. Furthermore, we propose an experimental scheme for implementing the ME discrimination of equidistant states. We also investigate the quantum coherence consumed in the implementation of the minimum error discrimination of the equidistant states, which has an established operational interpretation as cryptographic randomness gain. As an application, we propose a quantum communication protocol in which Alice prepares and sends one of the equidistant states, while Bob applies the minimum error discrimination to extract the classical information encoded in the state. Finally, we discuss the optimal conditions under which the protocol achieves an optimal balance of classical correlations and quantum coherence, thereby ensuring effective information transfer and cryptographic security.

Using the equality form of the necessary and sufficient conditions introduced in Jafarizadeh (Phys Rev A 84:012102 (9 pp), 2011), minimum error discrimination between states of the two sets of equiprobable similarity transformed quantum qudit states is investigated. In the case that the unitary operators describing the similarity transformations are generating sets of two irreducible representations and the states fulfill a certain constraint, the optimal set of measurements and the corresponding maximum success probability of discrimination are determined in closed form. In the cases that they are generating sets of reducible representations, there exist no closed-form formula in general, but the procedure can be applied properly in each case provided that the states obey some constraints. Finally, we give the maximum success probability of discrimination and optimal measurement operators for some important examples of mixed quantum states, such as generalized Bloch sphere m-qubit states, qubit states and their three special cases.

Standard deterministic techniques for quantum state reconstruction, as for example optical homodyne tomography, photon chopping, unbalanced homodyning etc., are based on the deterministic inversion of measured data. Since the frequencies obtained in realistic experiments always differ from probabilities predicted by quantum theory due to fluctuations, imperfections and realistic restrictions, the algorithm of inversion cannot guarantee the positive definiteness of the reconstructed density matrix. Hence the estimation of the noises may appear as doubtful.Quantum states may be successfully reconstructed within quantum and information theories using the maximum likelihood estimation. The question of deterministic schemes: “What quantum state is determined by that measurement?” is replaced by the formulation consistent with quantum theory: “What quantum state(s) seems to be most likely for that measurement?” Nonlinear equation for reconstructed state is formulated. An exact solution may be approached by subsequent iterations. Reconstruction is formulated as a problem of proper normalization of incompatible (nonorthogonal) measurements. The results obtained by this novel method may differ significantly from the standard predictions. Data are fitted better keeping the constraint of positive definiteness of reconstructed density matrix. However, this interpretation may enlarge uncertainty in prediction of quantum state in comparison with deterministic schemes, since, in general, there is a whole family of states which fit the measured data equally well. The novel technique is nonlinear and the reinterpretation of existing reconstruction schemes represents an advanced program.KeywordsQuantum StateDensity MatrixQuantum TheoryRelative EntropyQuantum TomographyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we propose a novel multi-user access in wireless optical communication based on the quantum detection of the coherent state. In this case, the coherent states are used as the signal carrier and a technique of quantum detection is applied to distinguish between signals from different users. To accomplish this task, two main quantum measurement methods are introduced; one is minimum error discrimination (MED), and the other is unambiguous state discrimination (USD). The theoretical derivation implies that the two methods can both distinguish between the signals from different users efficiently when the average photon number is large enough. Typically, the numerical result shows that in the two-user case, the channel capacity will approach the theoretical maximum limit when the average photon number is greater than 2.5 for MED and 5 for USD in the absence of noise. The MED gains more channel capacity than the USD at the same average photon number. However, the USD wins the error-correction scene with its free-error capability. Furthermore, the detection error probability and channel capacity for the USD with the thermal noise are examined. The result shows that increasing the signal average photon number can continue the USD’s advantage of error-free detection even if in the presence of thermal noise. In addition, compared with non-orthogonal multiple access (NOMA), the bit error rate (BER) against signal-to-noise rate (SNR) performance of USD has been improved.

Blind quantum computation is a new secure quantum computing protocol which enables Alice who does not have sufficient quantum technology to delegate her quantum computation to Bob who has a fully-fledged quantum computer in such a way that Bob cannot learn anything about Alice's input, output, and algorithm. In previous protocols, Alice needs to have a device which generates quantum states, such as single-photon states. Here we propose another type of blind computing protocol where Alice does only measurements, such as the polarization measurements with a threshold detector. In several experimental setups, such as optical systems, the measurement of a state is much easier than the generation of a single-qubit state. Therefore our protocols ease Alice's burden. Furthermore, the security of our protocol is based on the no-signaling principle, which is more fundamental than quantum physics. Finally, our protocols are device independent in the sense that Alice does not need to trust her measurement device in order to guarantee the security.

Feynman insisted 'no one understands quantum theory'. Yet, experimentalists tell us quantum theory is the most successful theory in history. Quantum theory cannot be understood as a classical mechanical theory since it arose through the 'interpolation' of two highly successful but complementary classical mechanics: Newtonian particle mechanics and Maxwellian wave mechanics. The two-slit experiment illustrates that what is experienced depends on choice of experimental set-up. Quantum theory is properly understood within the more general framework of engineering thermodynamics. In Part One, I point to four essential characteristics of quantum theory that cannot be understood in any framework defined by the classical mechanical presuppositions of symmetry and conservation. These four characteristics are the participatory, the complementary, the indeterminate and the new non-commutative geometry. In Part Two, articulating engineering thermodynamics, I note there are two histories and two formulations of thermodynamics: Carnot's engineering thermodynamics and the 'rational mechanical' tradition of Clausius-Boltzmann. These four essential characteristics of quantum theory are also characteristics of engineering thermodynamics. In Part Three, I trace the precursors of Lazare Carnot's engineering thermodynamics to earlier insights of Huygens, d'Alembert, Leibniz and the Bernoullis. Leibniz brought these forth in his meta-paradigm shift from Statics to Dynamics. This article is part of the theme issue 'Thermodynamics 2.0: Bridging the natural and social sciences (Part 2)'.

Standard formulations of quantum theory are based on complex numbers: Quantum states can be in superpositions, with weights given by complex probability amplitudes. Motivated by quantum theory promising a range of practical advantages over classical for a multitude of tasks, we investigate how the presence of complex amplitudes in quantum theory can yield operational advantages over counterpart real formulations. We identify a straightforward communication task for which complex quantum theory exhibits a provably lower communication cost than not just any classical approach, but also any approach based on real quantum theory. We certify the necessity of complex quantum theory for optimal approaches to the task through geometric properties of quantum state ensembles that witness the presence of basis-independent complexity. This substantiates a strict operational advantage of complex quantum theory. We discuss the relevance of this finding for quantum advantages in stochastic simulation.

Nonlocality without entanglement(NLWE) is a nonlocal phenomenon that occurs in quantum state discrimination of multipartite separable states. In the discrimination of orthogonal separable states, the term NLWE is used when the quantum states cannot be discriminated perfectly by local operations and classical communication. In this case, the occurrence of NLWE is independent of nonzero prior probabilities of quantum states being prepared. Recently, it has been found that the occurrence of NLWE can depend on nonzero prior probabilities in minimum-error discrimination of nonorthogonal separable states. Here, we show that even in optimal unambiguous discrimination, the occurrence of NLWE can depend on nonzero prior probabilities. We further show that NLWE can occur regardless of nonzero prior probabilities, even if only one state can be locally discriminated without error. Our results provide new insights into classifying sets of multipartite quantum states in terms of quantum state discrimination.