We present the optimal measurement strategy for distinguishing between three quantum states exhibiting a mirror symmetry. The three states live in a two-dimensional Hilbert space, and are thus overcomplete. By mirror symmetry we understand that the transformation {|+> -> |+>, |-> -> -|->} leaves the set of states invariant. The obtained measurement strategy minimizes the error probability. An experimental realization for polarized photons, realizable with current technology, is suggested.
We consider two different optimized measurement strategies for the discrimination of nonorthogonal quantum states. The first is conclusive discrimination with a minimum probability of inferring an erroneous result, and the second is unambiguous, i. e. error-free, discrimination with a minimum probability of getting an inconclusive outcome, where the measurement fails to give a definite answer. For distinguishing between two mixed quantum states, we investigate the relation between the minimum error probability achievable in conclusive discrimination, and the minimum failure probability that ca…
We propose a numerical algorithm for finding optimal measurements for quantum-state discrimination. The theory of the semidefinite programming provides a simple check of the optimality of the numerically obtained results.
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…
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…
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.
Distinguishing different quantum states is a fundamental task having practical applications for information processing. Despite the efforts devoted so far, however, strategies for optimal discrimination are known only for specific examples. We here consider the problem of minimum-error quantum state discrimination where the average error is attempted to be minimized. We show the general structure of minimum-error state discrimination as well as useful properties to derive analytic solutions. Based on the general structure, we present a geometric formulation of the problem, which can be applied…
We consider the optimal discrimination of bipartite quantum states and provide an upper bound for the maximum success probability of optimal local discrimination. We also provide a necessary and sufficient condition for a measurement to realize the upper bound. We further establish a necessary and sufficient condition for this upper bound to be saturated. Finally, we illustrate our results using an example.
In this work, we provide a complete analysis to minimum-error discrimination of mixed four qubit states with arbitrary prior probabilities. For the complete anaysis, the most important work to do is to find the necessary and sufficient conditions for the existence of null measurement operator. From the geometric structure of qubit states, we obtain the analytic condition for deciding the existence of a null operator in minimum-error measurement for mixed four qubit states, which also gives the necessary and sufficient conditions for every optimal POVM to have non-zero elements. Using the condi…
We derive a general lower bound on the minimum-error probability for {\it ambiguous discrimination} between arbitrary $m$ mixed quantum states with given prior probabilities. When $m=2$, this bound is precisely the well-known Helstrom limit. Also, we give a general lower bound on the minimum-error probability for discriminating quantum operations. Then we further analyze how this lower bound is attainable for ambiguous discrimination of mixed quantum states by presenting necessary and sufficient conditions related to it. Furthermore, with a restricted condition, we work out a upper bound on th…
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…
Using the necessary and sufficient conditions, minimum error discrimination among two sets of similarity transformed equiprobable quantum qudit states is investigated. In the case that the unitary operators are generating sets of two irreducible representations, the optimal set of measurements and the corresponding maximum success probability of discrimination are determined in closed form. In the case of reducible representations, there exists no closed-form formula in general, but the procedure can be applied in each case accordingly. Finally, we give the maximum success probability of optim…
We provide a solution of finding optimal measurement strategy for distinguishing between symmetric mixed quantum states. It is assumed that the matrix elements of at least one of the symmetric quantum states are all real and nonnegative in the basis of the eigenstates of the symmetry operator.
Using the known necessary and sufficient conditions for minimum error discrimination (MED), first it is shown that a Helstrom family of ensembles is equivalent to these conditions and then by a convex combination of the initial states (the states which we try to discriminate them) and the corresponding conjugate states, a more suitable and convenient form for the MED conditions is extracted, so that optimal set of measurements and corresponding optimal success probability of discrimination can be determined. Then, using the introduced identity, MED between N similarity transformed equiprobable…
The minimum-error probability of ambiguous discrimination for two quantum states is the well-known {\it Helstrom limit} presented in 1976. Since then, it has been thought of as an intractable problem to obtain the minimum-error probability for ambiguously discriminating arbitrary $m$ quantum states. In this paper, we obtain a new lower bound on the minimum-error probability for ambiguous discrimination and compare this bound with six other bounds in the literature. Moreover, we show that the bound between ambiguous and unambiguous discrimination does not extend to ensembles of more than two st…
The problem of discriminating with minimum error between two mixed quantum states is reviewed, with emphasize on the detection operators necessary for performing the measurement. An analytical result is derived for the minimum probability of errors in deciding whether the state of a quantum system is either a given pure state or a uniform statistical mixture of any number of mutually orthogonal states. The result is applied to two-qubit states, and the minimum error probabilities achievable by collective and local measurements on the qubits are compared.
We study an optimized measurement that discriminates two mixed quantum states with maximum confidence for each conclusive result, thereby keeping the overall probability of inconclusive results as small as possible. When the rank of the detection operators associated with the two different conclusive outcomes does not exceed unity we obtain a general solution. As an application, we consider the discrimination of two mixed qubit states. Moreover, for the case of higher-rank detection operators we give a solution for particular states. The relation of the optimized measurement to other discrimin…
Error probability is a popular and well-studied optimization criterion in discriminating non-orthogonal quantum states. It captures the threat from an adversary who can only query the actual state once. However, when the adversary is able to use a brute-force strategy to query the state, discrimination measurement with minimum error probability does not necessarily minimize the number of queries to get the actual state. In light of this, we take Massey's guesswork as the underlying optimization criterion and study the problem of minimum guesswork discrimination. We show that this problem can b…
The necessary and sufficient conditions for minimization of the generalized rate error for discriminating among $N$ pure qubit states are reformulated in terms of Bloch vectors representing the states. For the direct optimization problem an algorithmic solution to these conditions is indicated. A solution to the inverse optimization problem is given. General results are widely illustrated by particular cases of equiprobable states and $N=2,3,4$ pure qubit states given with different prior probabilities.
We investigate the discrimination between photon polarizations when measured by moving detectors. Both unambiguous and minimum-error discriminations are considered, and we analyze the the optimal successful (correct) probability as a function of the apparatus' velocity. The Holevo bound for polarization discrimination is also discussed and explicit calculation shows that the Holevo bound and the optimal successful (correct) probability for unambiguous (minimum-error) discrimination simultaneously increase or decrease.
In this paper we study the Minimum Error Discrimination problem (MED) for ensembles of linearly independent (LI) states. We define a bijective map from the set of those ensembles to itself and we show that the Pretty Good Measurement (PGM) and the optimal measurement for the MED are related by the map. In particular, the fixed points of the map are those ensembles for which the PGM is the optimal measurement. Also, we simplify the optimality conditions for the measurement of an ensemble of LI states.
We study several variations of the question of minimum-error discrimination of thermal states. Besides of providing the optimal values for the probability of error, we also characterize the optimal measurements. For the case of a fixed Hamiltonian, we show that for a general discrimination problem the optimal measurement is the measurement in the energy basis of the Hamiltonian. We identify a critical temperature, determining whether the given temperature is best distinguishable from thermal state of very high or very low temperatures. Further, we investigate the decision problem of whether th…
We investigate the performance of discrimination strategy in the comparison task of known quantum states. In the discrimination strategy, one infers whether or not two quantum systems are in the same state on the basis of the outcomes of separate discrimination measurements on each system. In some cases with more than two possible states, the optimal strategy in minimum-error comparison is that one should infer the two systems are in different states without any measurement, implying that the discrimination strategy performs worse than the trivial "no-measurement" strategy. We present a suffic…
Using the convex optimization method and Helstrom family of ensembles introduced in Ref. [1], we have discussed optimal ambiguous discrimination in qubit systems. We have analyzed the problem of the optimal discrimination of N known quantum states and have obtained maximum success probability and optimal measurement for N known quantum states with equiprobable prior probabilities and equidistant from center of the Bloch ball, not all of which are on the one half of the Bloch ball and all of the conjugate states are pure. An exact solution has also been given for arbitrary three known quantum s…
In this paper we present the solution to the problem of optimally discriminating among quantum states, i.e., identifying the states with maximum probability of success when a certain fixed rate of inconclusive answers is allowed. By varying the inconclusive rate, the scheme optimally interpolates between Unambiguous and Minimum Error discrimination, the two standard approaches to quantum state discrimination. We introduce a very general method that enables us to obtain the solution in a wide range of cases and give a complete characterization of the minimum discrimination error as a function o…
A measurement strategy is developed for a new kind of hypothesis testing. It assigns, with minimum probability of error, the state of a quantum system to one or the other of two complementary subsets of a set of N given non-orthogonal quantum states occurring with given a priori probabilities. A general analytical solution is obtained for N states that are restricted to a two-dimensional subspace of the Hilbert space of the system. The result for the special case of three arbitrary but linearly dependent states is applied to a variety of sets of three states that are symmetric and equally prob…
We formulate minimum-error and unambiguous discrimination problems for quantum processes in the language of process positive operator valued measures (PPOVM). In this framework we present the known solution for minimum-error discrimination of unitary channels. We derive a "fidelity-like" lower bound on the failure probability of the unambiguous discrimination of arbitrary quantum processes. This bound is saturated (in a certain range of apriori probabilities) in the case of unambiguous discrimination of unitary channels. Surprisingly, the optimal solution for both tasks is based on the optimiz…
We introduce the concept of boundariness capturing the most efficient way of expressing a given element of a convex set as a probability mixture of its boundary elements. In other words, this number measures (without the need of any explicit topology) how far the given element is from the boundary. It is shown that one of the elements from the boundary can be always chosen to be an extremal element. We focus on evaluation of this quantity for quantum sets of states, channels and observables. We show that boundariness is intimately related to (semi)norms that provide an operational interpretati…
A quantum analogue of the famous Blackwell Theorem in classical statistics has recently been proposed. Given two quantum channels A and B, a set of payoff functions have been proven to have values for B at least as high as they are for A if and only if there exists a quantum garbling channel E such that A=EB. When such a channel E exists, we can globally compare A and B in terms of their `noisiness'. We show that this method of channel noise comparison is equivalent to one obtained by considering the degradation of the distinguishability of states. Here, the channel A is said to be at least as…
Finding quantitative aspects of quantum phenomena which cannot be explained by any classical model has foundational importance for understanding the boundary between classical and quantum theory. It also has practical significance for identifying information processing tasks for which those phenomena provide a quantum advantage. Using the framework of generalized noncontextuality as our notion of classicality, we find one such nonclassical feature within the phenomenology of quantum minimum error state discrimination. Namely, we identify quantitative limits on the success probability for minim…
Inspired by the work done by Belavkin [Belavkin V. P., Stochastics, 1, 315 (1975)], and independently by Mochon, [Phys. Rev. A 73, 032328, (2006)], we formulate the problem of minimum error discrimination of any ensemble of $n$ linearly independent pure states by stripping the problem of its rotational covariance and retaining only the rotationally invariant aspect of the problem. This is done by embedding the optimal conditions in a matrix equality as well as matrix inequality. Employing the implicit function theorem in these conditions we get a set of first-order coupled ordinary non-linear …
We study an optimum measurement for quantum state discrimination, which maximizes the probability of correct results when the probability of inconclusive results is fixed at a given value. The measurement describes minimum-error discrimination if this value is zero, while under certain conditions it corresponds to optimized maximum-confidence discrimination, or to optimum unambiguous discrimination, respectively, when the fixed value reaches a definite minimum. Using operator conditions that determine the optimum measurement, we derive analytical solutions for the discrimination of two mixed q…
Laser light is widely used for communication and sensing applications, so the optimal discrimination of coherent states--the quantum states of light emitted by a laser--has immense practical importance. However, quantum mechanics imposes a fundamental limit on how well different coher- ent states can be distinguished, even with perfect detectors, and limits such discrimination to have a finite minimum probability of error. While conventional optical receivers lead to error rates well above this fundamental limit, Dolinar found an explicit receiver design involving optical feedback and photon c…
This paper deals with the quantum optimal discrimination among mixed quantum states enjoying geometrical uniform symmetry with respect to a reference density operator $ρ_0$. It is well-known that the minimal error probability is given by the positive operator-valued measure (POVM) obtained as a solution of a convex optimization problem, namely a set of operators satisfying geometrical symmetry, with respect to a reference operator $Π_0$, and maximizing $\textrm{Tr}(ρ_0 Π_0)$. In this paper, by resolving the dual problem, we show that the same result is obtained by minimizing the trace of a sem…
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 optimization conditions for minimum error discrimination of linearly independent pure states comprise of two kinds: stationary conditions over the space of rank one projective measurements and the global maximization conditions. A discrete number of projective measurments will solve th former of which a unique one will solve the latter. In the case of three real linearly independent pure states we show that the stationary conditions translate to a system of simultaneous polynomial (non linear) equations in three variabes thus explaining why it's so difficult to obtain a closed-form solutio…
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…
State discrimination with the aim to minimize the error probability is a well studied problem. Instead, here the binary decision problem for operators with a given prior is investigated. A black box containing the unknown operator is probed by selected wave functions. The output is analyzed with conventional methods developed for state discrimination. An error probability bound for all binary operator choices is provided, and it is shown how probe entanglement enhances the result.
Quantum hypothesis testing is an important tool for quantum information processing. Two main strategies have been widely adopted: in a minimum error discrimination strategy, the average error probability is minimized; while in an unambiguous discrimination strategy, an inconclusive decision (abstention) is allowed to vanish any possibility of errors when a conclusive result is obtained. In both scenarios, the testing between quantum states are relatively well-understood, for example, the ultimate limits of the performance are established decades ago; however, the testing between quantum channe…
The discrimination between two unknown states can be performed by a universal programmable discriminator, where the copies of the two possible states are stored in two program systems respectively and the copies of data, which we want to confirm, are provided in the data system. In the present paper, we propose a group-theretic approach to the multi-copy programmable state discrimination problem. By equivalence of unknown pure states to known mixed states and with the representation theory of U(n) group, we construct the Jordan basis to derive the analytical results for both the optimal unambi…
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…
In the task of discriminating between nonorthogonal quantum states from multiple copies, the key parameters are the error probability and the resources (number of copies) used. Previous studies have considered the task of minimizing the average error probability for fixed resources. Here we introduce a new state discrimination task: minimizing the average resources for a fixed admissible error probability. We show that this new task is not performed optimally by previously known strategies, and derive and experimentally test a detection scheme that performs better.
We consider the problem of minimum-error quantum state discrimination for single-qubit mixed states. We present a method which uses the Helstrom conditions constructively and analytically; this algebraic approach is complementary to existing geometric methods, and solves the problem for any number of arbitrary signal states with arbitrary prior probabilities.
The minimum error discrimination problem for ensembles of linearly independent pure states are known to have an interesting structure; for such a given ensemble the optimal POVM is given by the pretty good measurment of another ensemble which can be related to the former ensemble by a bijective mapping $\mathscr{R}$ on the "space of ensembles". In this paper we generalize this result to ensembles of general linearly independent states (not necessarily pure) and also give an analytic expression for the inverse of the map, i.e., for $\mathscr{R}^{-1}$. In the process of proving this we also simp…
Quantum state discrimination is a central problem in quantum measurement theory, with applications spanning from quantum communication to computation. Typical measurement paradigms for state discrimination involve a minimum probability of error or unambiguous discrimination with a minimum probability of inconclusive results. Alternatively, an optimal inconclusive measurement, a non-projective measurement, achieves minimal error for a given inconclusive probability. This more general measurement encompasses the standard measurement paradigms for state discrimination and provides a much more pow…
In quantum state discrimination, one aims to identify unknown states from a given ensemble by performing measurements. Different strategies such as minimum-error discrimination or unambiguous state identification find different optimal measurements. Maximum-confidence measurements (MCMs) maximise the confidence with which inputs can be identified given the measurement outcomes. This unifies a range of discrimination strategies including minimum-error and unambiguous state identification, which can be understood as limiting cases of MCM. In this work, we investigate MCMs for general ensembles o…
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…
Quantum mechanics forbids perfect discrimination among nonorthogonal states through a single shot measurement. To optimize this task, many strategies were devised that later became fundamental tools for quantum information processing. Here, we address the pioneering minimum-error (ME) measurement and give the first experimental demonstration of its application for discriminating nonorthogonal states in high dimensions. Our scheme is designed to distinguish symmetric pure states encoded in the transverse spatial modes of an optical field; the optimal measurement is performed by a projection ont…
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…
