As a modern approach for the foundation of quantum theory, existing studies of General Probabilistic Theories gave various models of states and measurements that are quite different from quantum theory. In this paper, to seek a more realistic situation, we investigate models approximately close to quantum theory. We define larger measurement classes that are smoothly connected with the class of POVMs via a parameter, and investigate the performance of perfect discrimination. As a result, we give a sufficient condition of perfect discrimination, which shows a significant improvement beyond the class of POVMs.
General Probabilistic Theories provide the most general mathematical framework for the theory of probability in an operationally natural manner, and generalize classical and quantum theories. In this article, we study state-discrimination problems in general probabilistic theories using a Bayesian strategy. After re-formulation of the theories with mathematical rigor, we first prove that an optimal observable to discriminate any (finite) number of states always exists in the most general setting. Next, we revisit our recently proposed geometric approach for the problem and show that, for two-s…
The indistinguishability of non-orthogonal pure states lies at the heart of quantum information processing. Although the indistinguishability reflects the impossibility of measuring complementary physical quantities by a single measurement, we demonstrate that the distinguishability can be perfectly retrieved simply with the help of posterior classical partial information. We demonstrate this by showing an ensemble of non-orthogonal pure states such that a state randomly sampled from the ensemble can be perfectly identified by a single measurement with help of the post-processing of the measur…
We consider optimal state discrimination in a general convex operational framework, so-called generalized probabilistic theories (GPTs), and present a general method of optimal discrimination by applying the complementarity problem from convex optimization. The method exploits the convex geometry of states but not other detailed conditions or relations of states and effects. We also show that properties in optimal quantum state discrimination are shared in GPTs in general: i) no measurement sometimes gives optimal discrimination, and ii) optimal measurement is not unique.
This thesis deals with General Probabilistic Theories (GPTs) and Entanglement Structures (ESs). An ES is a possible structure of a quantum composite system in GPTs, which is not uniquely determined as the Standard Entanglement Structure (SES). It is an important problem to find reasonable postulates that determine various ESs as the SES. In order to solve this problem, this thesis explores the diversity of ESs. The topics of this thesis are roughly divided into two parts. First, this thesis considers state discrimination in ESs. As a main result, this thesis gives equivalent conditions for a g…
Discrimination of unitary operations is a fundamental quantum information processing task. Assisted with linear optical elements, we experimentally demonstrate perfect discrimination between single-bit unitary operations using two methods--sequential scheme and parallel scheme. The complexity and resource consumed in these two schemes are analyzed and compared.
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.
Recently, the problem of discriminating multipartite unitary operations by local operations and classical communication (LOCC) has attracted significant attention. The latest work in the literature on this problem showed that two multipartite unitary operations can always be perfectly distinguished by LOCC when a finite number of runs are allowable. However, in these schemes, local entanglement (an entangled state holden by one party) was required, which seems to imply that local entanglement is necessary for perfect discrimination between unitary operations by LOCC. In this article, we show t…
We investigate a state discrimination problem in operationally the most general framework to use a probability, including both classical, quantum theories, and more. In this wide framework, introducing closely related family of ensembles (which we call a {\it Helstrom family of ensembles}) with the problem, we provide a geometrical method to find an optimal measurement for state discrimination by means of Bayesian strategy. We illustrate our method in 2-level quantum systems and in a probabilistic model with square-state space to reproduce e.g., the optimal success probabilities for binary sta…
We address perfect discrimination of two separable states. When available states are restricted to separable states, we can theoretically consider a larger class of measurements than the class of measurements allowed in quantum theory. The framework composed of the class of separable states and the above extended class of measurements is a typical example of general probabilistic theories. In this framework, we give a necessary and sufficient condition to discriminate two separable pure states perfectly. In particular, we derive measurements explicitly to discriminate two separable pure states…
We revisit the problem of discriminating orthogonal quantum states within the local quantum operation and classical communication (LOCC) paradigm. Our particular focus is on the asymptotic situation where the parties have infinite resources and the protocol may become arbitrarily long. Our main result is a necessary condition for perfect asymptotic LOCC discrimination. As an application, we prove that for complete product bases, unlimited resources are of no advantage. On the other hand, we identify an example, for which it still remains undecided whether unlimited resources are superior.
We analyze entanglement swapping (ES) of partially entangled pure states with arbitrary Schmidt rank from the perspective of quantum state discrimination. It is shown that the standard deterministic ES protocol is related with an optimal minimum-error strategy. In this case the amount of entanglement of the states resulting from swapping is, in general, lower than the maximum achievable for the quantum channels involved. In this regard, we show that the ES protocol can be probabilistically improved resorting to optimal maximum-confidence (MC) discrimination strategy. Additionally, we show that…
Distinguishing physical processes is one of the fundamental problems in quantum physics. Although distinguishability of quantum preparations and quantum channels have been studied considerably, distinguishability of quantum measurements remains largely unexplored. We investigate the problem of single-shot discrimination of quantum measurements using two strategies, one based on single quantum systems and the other one based on entangled quantum systems. First, we formally define both scenarios. We then construct sets of measurements (including non-projective) in arbitrary finite dimensions tha…
A state discrimination problem in an operational probabilistic theory (OPT) is investigated in diagrammatic terms. It is well-known that, in the case of quantum theory, if a state set has a certain symmetry, then there exists a minimum-error measurement having the same type of symmetry. However, to our knowledge, it is not yet clear whether this property also holds in a more general OPT. We show that it also holds in OPTs, i.e., for a symmetric state set, there exists a minimum-error measurement that has the same type of symmetry. It is also shown that this result can be utilized to optimize o…
It is shown that while entanglement remains a significant factor in discriminating a set of mutually orthogonal entangled states perfectly by local operations and classical communication (LOCC), entanglement content is not. In particular, for a class of entangled multi-qubit states, the maximum number of perfectly LOCC distinguishable orthogonal states is shown to be independent of the average entanglement of the states, and the spatial configuration with respect to which LOCC operations may be carried out. It is also pointed out that the make-up of an ensemble, that is, whether it consists on…
It is shown that local distinguishability of orthogonal mixed states can be completely characterized by local distinguishability of their supports irrespective of entanglement and mixedness of the states. This leads to two kinds of upper bounds on the number of perfect LOCC distinguishable orthogonal mixed states. The first one depends only on pure-state entanglement within the supports of the states, and therefore may be easy to compute in many instances. The second bound is optimal in the sense that it optimizes the bounding quantities, not necessarily function of entanglement alone, over al…
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…
Despite well-established no-go theorems on a perfect linear optical Bell state analyzer, we find a numerical trend that appears to approach a near-perfect measurement if we incorporate eight or more un-entangled ancilla photons into our device. Following this trend, we begin a promising inductive approach to building an ideal optical Bell measurement device. In the process, we determine that any Bell state analyzer that (even occasionally) bunches all photons into only two of the output modes cannot perform an ideal measurement and we find a set of conditions on our linear optical circuit that…
We show that a unitary operation (quantum circuit) secretely chosen from a finite set of unitary operations can be determined with certainty by sequentially applying only a finite amount of runs of the unknown circuit. No entanglement or joint quantum operations is required in our scheme. We further show that our scheme is optimal in the sense that the number of the runs is minimal when discriminating only two unitary operations.
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 present an in-depth study of the problem of multiple-shot discrimination of von Neumann measurements in finite-dimensional Hilbert spaces. Specifically, we consider two scenarios: minimum error and unambiguous discrimination. In the case of minimum error discrimination, we focus on discrimination of measurements with the assistance of entanglement. We provide an alternative proof of the fact that all pairs of distinct von Neumann measurements can be distinguished perfectly (i.e. with the unit success probability) using only a finite number of queries. Moreover, we analytically find the mini…
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…
In this paper, we address the problem of discriminating two given quantum operations. Firstly, based on the Bloch representation of single qubit systems, we give the exact minimum error probability of discriminating two single qubit quantum operations by unentangled input states. In particular, for the Pauli channels discussed in [Phys. Rev. A {\bf 71}, 062340 (2005)], we use a more intuitional and visual method to deal with their discrimination problem. Secondly, we consider the condition for perfect discrimination of two quantum operations. Specially, we get that two generalized Pauli channe…
We study discrimination of m quantum measurements in the scenario when the unknown measurement with n outcomes can be used only once. We show that ancilla-assisted discrimination procedures provide a nontrivial advantage over simple (ancilla-free) schemes for perfect distinguishability and we prove that inevitably m <= n. We derive necessary and sufficient conditions of perfect distinguishability of general binary measurements. We show that the optimization of the discrimination of projective qubit measurements and their mixtures with white noise is equivalent to the discrimination of specific…
We provide a feasible necessary and sufficient condition for when an unknown quantum operation (quantum device) secretely selected from a set of known quantum operations can be identified perfectly within a finite number of queries, and thus complete the characterization of the perfect distinguishability of quantum operations. We further design an optimal protocol which can achieve the perfect discrimination between two quantum operations by a minimal number of queries. Interestingly, employing the techniques from the theory of $q$-numerical range we find that an optimal perfect discrimination…
The intrinsic idea of superdense coding is to find as many gates as possible such that they can be perfectly discriminated. In this paper, we consider a new scheme of discrimination of quantum gates, called ancilla-assisted discrimination, in which a set of quantum gates on a $d-$dimensional system are perfectly discriminated with assistance from an $r-$dimensional ancilla system. The main contribution of the present paper is two-fold: (1) The number of quantum gates that can be discriminated in this scheme is evaluated. We prove that any $rd+1$ quantum gates cannot be perfectly discriminated …
Recently, the nonlocal optimal probabilistic cloning (NLOPC) of two non-orthogonal qubit states has been proposed [Phys. Rev. A 86, 052332 (2012)] by means of an experimental setup based on a pair of twin photons in a maximally entangled state. Here we study the performance of the NLOPC protocol when implemented via a partially entangled state. We show that the errors introduced by the use of partial entanglement can be undone by applying a quantum state discrimination process. Since quantum state discrimination is a probabilistic process the correction succeeds with a certain probability but …
We address the problem of unambiguous discrimination and identification among quantum observables. We set a general framework and investigate in details the case of qubit observables. In particular, we show that perfect discrimination with two shots is possible only for sharp qubit observables (e.g. Stern-Gerlach apparatuses) associated with mutually orthogonal directions. We also show that for sharp qubit observables associated to nonorthogonal directions unambiguous discrimination with an inconclusive result is always possible.
In this work we relate the well-known no-go theorem that two non-orthogonal (mixed) quantum states cannot be perfectly discriminated, to the general principle in physics, the no-signalling condition. In fact, we derive the minimum error in discrimination between two quantum states, using the no-signalling condition.
We solve the long-standing problem of making n perfect clones from m copies of one of two known pure states with minimum failure probability in the general case where the known states have arbitrary a priori probabilities. The solution emerges from a geometric formulation of the problem. This formulation also reveals a deeper connection between cloning and state discrimination. The convergence of cloning to state discrimination as the number of clones goes to infinity exhibits a phenomenon analogous to a second order symmetry breaking phase transition.
In this work we study the problem of single-shot discrimination of von Neumann measurements, which we associate with measure-and-prepare channels. There are two possible approaches to this problem. The first one is simple and does not utilize entanglement. We focus only on the discrimination of classical probability distributions, which are outputs of the channels. We find necessary and sufficient criterion for perfect discrimination in this case. A more advanced approach requires the usage of entanglement. We quantify the distance between two measurements in terms of the diamond norm (called …
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…
In this work, we investigate the question, which objects one can discriminate perfectly by 'interaction-free' measurements. To this end, we interpret the Elitzur-Vaidman bomb-tester experiment as a quantum channel discrimination problem and generalize the notion of 'interaction-free' measurement to arbitrary quantum channels. Our main result is a necessary and sufficient criterion for when it is possible or impossible to discriminate quantum channels in an 'interaction-free' manner (i.e., such that the discrimination error probability and the 'interaction' probability can be made arbitrarily s…
We prove that the states secretly chosen from a mixed state set can be perfectly discriminated if and only if these states are orthogonal. The sufficient and necessary condition when nonorthogonal quantum mixed states can be unambiguously discriminated is also presented. Furthermore, we derive a series of lower bounds on the inconclusive probability of unambiguous discrimination of states from a mixed state set with \textit{a prior} probabilities.
We establish an operational characterization of general convex resource theories -- describing the resource content of not only states, but also measurements and channels, both within quantum mechanics and in general probabilistic theories (GPTs) -- in the context of state and channel discrimination. We find that discrimination tasks provide a unified operational description for quantification and manipulation of resources by showing that the family of robustness measures can be understood as the maximum advantage provided by any physical resource in several different discrimination tasks, as …
Strategies to optimally discriminate between quantum states are critical in quantum technologies. We present an experimental demonstration of minimum error discrimination between entangled states, encoded in the polarization of pairs of photons. Although the optimal measurement involves projecting onto entangled states, we use a result of Walgate et al. to design an optical implementation employing only local polarization measurements and feed-forward, which performs at the Helstrom bound. Our scheme can achieve perfect discrimination of orthogonal states and minimum error discrimination of no…
The main goal of this work is to provide an insight into the problem of discrimination of positive operator valued measures with rank-one effects. It is our intention to study multiple shot discrimination of such measurements, that is the case when we are able to use to unknown measurement a given number of times. Furthermore, we are interested in comparing two possible discrimination schemes: the parallel and adaptive ones. To this end we construct a pair of symmetric, information complete positive operator valued measures which can be perfectly discriminated in a two-shot adaptive scheme. On…
One of the most fundamental results in quantum information theory is that no measurement can perfectly discriminate between non-orthogonal quantum states. In this work, we investigate quantum advantages for discrimination tasks over noncontextual theories by considering a maximum confidence measurement that unifies different strategies of quantum state discrimination, including minimum-error and unambiguous discrimination. We first show that maximum confidence discrimination, as well as unambiguous discrimination, contains contextual advantages. We then consider a semi-device independent scena…
We assess the resources needed to identify a reversible quantum gate among a finite set of alternatives, including in our analysis both deterministic and probabilistic strategies. Among the probabilistic strategies we consider unambiguous gate discrimination, where errors are not tolerated but inconclusive outcomes are allowed, and we prove that parallel strategies are sufficient to unambiguously identify the unknown gate with minimum number of queries. This result is used to provide upper and lower bounds on the query complexity and on the minimum ancilla dimension. In addition, we introduce …
Quantum state filtering is a variant of the unambiguous state discrimination problem: the states are grouped in sets and we want to determine to which particular set a given input state belongs.The simplest case, when the N given states are divided into two subsets and the first set consists of one state only while the second consists of all of the remaining states, is termed quantum state filtering. We derived previously the optimal strategy for the case of N non-orthogonal states, {|ψ_{1} >, ..., |ψ_{N} >}, for distinguishing |ψ_1 > from the set {|ψ_2 >, ..., |ψ_N >} and the corresponding op…
We provide a bound on the minimum error when discriminating among quantum states, using the no-signaling principle. The bound is general in that it depends on neither dimensions nor specific structures of given quantum states to be discriminated among. We show that the bound is tight for the minimum-error state discrimination between symmetric (both pure and mixed) qubit states. Moreover, the bound can be applied to a set of quantum states for which the minimum-error state discrimination is not known yet. Finally, our results strengthen the quantitative connection between two no-go theorems, t…
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 s…
The problem of quantum state filtering consists of determining whether an unknown quantum state, which is chosen from a known set of states, is either a particular, specified state, or not. We consider this problem for the case that the filtering procedure is required to be unambiguous, which necessitates admitting inconclusive answers when the given states are not orthogonal. As an application, we propose an efficient, probabilistic procedure for distinguishing between sets of Boolean functions, which is an extension of the Deutsch-Jozsa algorithm.
For distinguishing quantum states sampled from a fixed ensemble, the gap in bipartite and single-party distinguishability can be interpreted as a nonlocality of the ensemble. In this paper, we consider bipartite state discrimination in a composite system consisting of $N$ subsystems, where each subsystem is shared between two parties and the state of each subsystem is randomly sampled from a particular ensemble comprising the Bell states. We show that the success probability of perfectly identifying the state converges to $1$ as $N\rightarrow\infty$ if the entropy of the probability distributi…
The problem of discriminating the state of a quantum system among a number of hypothetical states is usually addressed under the assumption that one has perfect knowledge of the possible states of the system. In this thesis, I analyze the role of the prior information available in facing such problems, and consider scenarios where the information regarding the possible states is incomplete. In front of a complete ignorance of the possible states' identity, I discuss a quantum "programmable" discrimination machine for qubit states that accepts this information as input programs using a quantum …
This thesis investigates the connection between quantum theory, thermodynamics and information theory. Theories with structure similar to that of quantum theory are considered, mathematically described by the framework of "Generalized Probabilistic Theories". For these theories, a thought experiment by von Neumann is adapted to obtain a natural thermodynamic entropy definition, following a proposal by J. Barrett. Mathematical properties of this entropy are compared to physical consequences of the thought experiment. The validity of the second law of thermodynamics is investigated. In that cont…
We investigate the ability of a quantum measurement device to discriminate two states or, generically, two hypothesis. In full generality, the measurement can be performed a number $n$ of times, and arbitrary pre-processing of the states and post-processing of the obtained data is allowed. Even if the two hypothesis correspond to orthogonal states, perfect discrimination is not always possible. There is thus an intrinsic error associated to the measurement device, which we aim to quantify, that limits its discrimination power. We minimize various error probabilities (averaged or constrained) o…
