We present a generic study of unambiguous discrimination between two mixed quantum states. We derive operational optimality conditions and show that the optimal measurements can be classified according to their rank. In Hilbert space dimensions smaller or equal to five this leads to the complete optimal solution. We demonstrate our method with a physical example, namely the unambiguous comparison of n quantum states, and find the optimal success probability.
In this paper, we introduce a mixed form of ambiguous and unambiguous quantum state discriminations, and show that the mixed form has higher success probability than the unambiguous quantum state discriminations.
We propose an optimal discrimination scheme for a case of four linearly independent nonorthogonal symmetric quantum states, based on linear optics only. The probability of discrimination is in agreement with the optimal probability for unambiguous discrimination among N symmetric states [Phys. Lett. A \textbf{250}, 223 (1998)]. The experimental setup can be extended for the case of discrimination among $2^M$ nonorthogonal symmetric quantum states.
We present an experimental implementation of optimum measurements for quantum state discrimination. Optimum maximum-confidence discrimination and optimum unambiguous discrimination of two mixed single-photon polarization states were performed. For the latter the states of rank two in a four-dimensional Hilbert space are prepared using both path and polarization encoding. Linear optics and single photons from a true single-photon source based on a semiconductor quantum dot are utilized.
We address the problem of unambiguous discrimination among a given set of quantum operations. The necessary and sufficient condition for them to be unambiguously distinguishable is derived in the cases of single use and multiple uses respectively. For the latter case we explicitly construct the input states and corresponding measurements that accomplish the task. It is found that the introduction of entanglement can improve the discrimination.
In this paper, we consider the problem of unambiguous discrimination between a set of mixed quantum states. We first divide the density matrix of each mixed state into two parts by the fact that it comes from ensemble of pure quantum states. The first part will not contribute anything to the discrimination, the second part has support space linearly independent to each other. Then the problem we consider can be reduced to a problem in which the strategy of set discrimination can be used in designing measurements to discriminate mixed states unambiguously. We find a necessary and sufficient con…
Quantum mechanics forbids deterministic discrimination among non-orthogonal states. Nonetheless, the capability to distinguish nonorthogonal states unambiguously is an important primitive in quantum information processing. In this work, we experimentally implement generalized measurements in an optical system and demonstrate the first optimal unambiguous discrimination between three nonorthogonal states, with a success rate of 55%, to be compared with the 25% maximum achievable using projective measurements. Furthermore we present the first realization of unambiguous discrimination between a p…
We present a general scheme to realize the POVMs for the unambiguous discrimination of quantum states. For any set of pure states it enables us to set up a feasible linear optical circuit to perform their optimal discrimination, if they are prepared as single-photon states. An example of unknown states discrimination is discussed as the illustration of the general scheme.
We consider the unambiguous discrimination between two unknown qudit states in $n$-dimensional ($n\geqslant2$) Hilbert space. By equivalence of unknown pure states to known mixed states and with the Jordan-basis method, we demonstrate that the optimal success probability of the discrimination between two unknown states is independent of the dimension $n$. We also give a scheme for a physical implementation of the programmable state discriminator that unambiguously discriminate between two unknown states with optimal probability of success.
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.
A set of quantum states can be unambiguously discriminated if and only if they are linearly independent. However, for a linearly dependent set, if C copies of the state are available, then the resulting C particle states may form a linearly independent set, and be amenable to unambiguous discrimination. We obtain necessary and sufficient conditions for the possibility of unambiguous discrimination between N states given that C copies are available and that the single copies span a D dimensional space. These conditions are found to be identical for qubits. We then examine in detail the linearly…
Roa et al. showed that quantum state discrimination between two nonorthogonal quantum states does not require quantum entanglement but quantum dissonance only. We find that quantum coherence can also be utilized for unambiguous quantum state discrimination. We present a protocol and quantify the required coherence for this task. We discuss the optimal unambiguous quantum state discrimination strategy in some cases. In particular, our work illustrates an avenue to find the optimal strategy for discriminating two nonorthogonal quantum states by measuring quantum coherence.
We present reduction theorems for the problem of optimal unambiguous state discrimination (USD) of two general density matrices. We show that this problem can be reduced to that of two density matrices that have the same rank $n$ and are described in a Hilbert space of dimensions $2n$. We also show how to use the reduction theorems to discriminate unambiguously between N mixed states (N \ge 2).
We address the problem of unambiguous discrimination among oracle operators. The general theory of unambiguous discrimination among unitary operators is extended with this application in mind. We prove that entanglement with an ancilla cannot assist any discrimination strategy for commuting unitary operators. We also obtain a simple, practical test for the unambiguous distinguishability of an arbitrary set of unitary operators on a given system. Using this result, we prove that the unambiguous distinguishability criterion is the same for both standard and minimal oracle operators. We then show…
We present the conditions under which probabilistic error-free discrimination of mixed states is possible, and provide upper and lower bounds on the maximum probability of success for the case of two mixed states. We solve certain special cases exactly, and demonstrate how the problems of state filtering and state comparison can be recast as problems of mixed state unambiguous discrimination.
The paper studies unambiguous discrimination of Fermionic states through local operations and classical communication (LOCC). In the task of unambiguous discrimination, no error is tolerated but an inconclusive result is allowed. We show that contrary to the quantum case, it is not always possible to distinguish two Fermionic states through LOCC unambiguously with the same success probability as if global measurements were allowed. Furthermore, we prove that we can overcome such a limit through an ancillary system made of two Fermionic modes, independently of the dimension of the system, prepa…
We show how to optimally unambiguously discriminate between two subspaces of a Hilbert space. In particular we suppose that we are given a quantum system in either the state ψ_{1}, where ψ_{1} can be any state in the subspace S_{1}, or ψ_{2}, where ψ_{2} can be any state in the subspace S_{2}, and our task is to determine in which of the subspaces the state of our quantum system lies. We do not want to make a mistake, which means that our procedure will sometimes fail if the subspaces are not orthogonal. This is a special case of the unambiguous discrimination of mixed states. We present the P…
Unambiguous non-orthogonal state discrimination has fundamental importance in quantum information and quantum cryptography. The discrimination is carried out by POVM generalized measurements. For this process, we find a tradeoff between the needed energy resources and the evolution time, and express it in terms of action-like cost inequality. We find the realization that minimizes this action-like cost and show that in this case the cost is determined by the maximal population transfer from the system to the ancilla needed for the POVM implementation.
Unambiguous discrimination and exact cloning reduce the square-overlap between quantum states, exemplifying the more general type of procedure we term state separation. We obtain the maximum probability with which two equiprobable quantum states can be separated by an arbitrary degree, and find that the established bounds on the success probabilities for discrimination and cloning are special cases of this general bound. The latter also gives the maximum probability of successfully producing N exact copies of a quantum system whose state is chosen secretly from a known pair, given M initial re…
We discuss the problem of designing an unambiguous programmable discriminator for mixed quantum states. We prove that there does not exist such a universal unambiguous programmable discriminator for mixed quantum states that has two program registers and one data register. However, we find that we can use the idea of programmable discriminator to unambiguously discriminate mixed quantum states. The research shows that by using such an idea, when the success probability for discrimination reaches the upper bound, the success probability is better than what we do not use the idea to do, except f…
The theory of generalised measurements is used to examine the problem of discriminating unambiguously between non-orthogonal pure quantum states. Measurements of this type never give erroneous results, although, in general, there will be a non-zero probability of a result being inconclusive. It is shown that only linearly-independent states can be unambiguously discriminated. In addition to examining the general properties of such measurements, we discuss their application to entanglement concentration.
We present a criterion, based on three commutator relations, that allows to decide whether two self-adjoint matrices with non-overlapping support are simultaneously unitarily similar to quasidiagonal matrices, i.e., whether they can be simultaneously brought into a diagonal structure with 2x2-dimensional blocks. Application of this criterion to unambiguous state discrimination provides a systematic test whether the given problem is reducible to a solvable structure. As an example, we discuss unambiguous state comparison.
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…
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 obtain a necessary and sufficient condition for a finite set of states of a finite dimensional multiparticle quantum system to be amenable to unambiguous discrimination using local operations and classical communication. This condition is valid for states which may be be mixed, entangled or both. When the support of the set of states is the entire multiparticle Hilbert space, this condition is found to have an intriguing connection with the theory of entanglement witnesses.
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…
We analyze the optimal unambiguous discrimination of two arbitrary mixed quantum states. We show that the optimal measurement is unique and we present this optimal measurement for the case where the rank of the density operator of one of the states is at most 2 ("solution in 4 dimensions"). The solution is illustrated by some examples. The optimality conditions proved by Eldar et al. [Phys. Rev. A 69, 062318 (2004)] are simplified to an operational form. As an application we present optimality conditions for the measurement, when only one of the two states is detected. The current status of op…
Recently the problem of Unambiguous State Discrimination (USD) of mixed quantum states has attracted much attention. So far, bounds on the optimum success probability have been derived [1]. For two mixed states they are given in terms of the fidelity. Here we give tighter bounds as well as necessary and sufficient conditions for two mixed states to reach these bounds. Moreover we construct the corresponding optimal measurement strategies. With this result, we provide analytical solutions for unambiguous discrimination of a class of generic mixed states. This goes beyond known results which are…
We present the first full demonstration of unambiguous state discrimination between non-orthogonal quantum states. Using a novel free space interferometer we have realised the optimum quantum measurement scheme for two non-orthogonal states of light, known as the Ivanovic-Dieks-Peres (IDP) measurement. We have for the first time gained access to all three possible outcomes of this measurement. All aspects of this generalised measurement scheme, including its superiority over a standard von Neumann measurement, have been demonstrated within 1.5% of the IDP predictions.
In this paper, we propose a method to discriminate two extremely similar quantum states via a weak measurement. For the two states with equal prior probabilities, the optimum discrimination probability given by Ivanovic-Dieks-Peres limit can be achieved by our protocol with an appropriate choice of the interaction strength. However, compared with the conventional method for state discrimination, our approach shows the advantage of error-tolerance by achieving a better ratio of the success probability to the probability of error.
We describe how to device-independently generate a set of quantum states for which unambiguous state discrimination is not possible. First, we derive a formula that a certain non-signaling black box must satisfy. Then, we describe how to generate a set of quantum states. Devices for generating states and possible measurements on the states can be put into a black box. Because this black box is non-signaling, it satisfies the formula. Using the formula, we prove that unambiguous state discrimination is not possible for the generated states. Because we did not consider internal mechanisms but on…
We propose a general description on the unambiguous discrimination of mixed states according to the system-environment coupling, and present a procedure to reduce this to a standard semidefinite programming problem. In the two states case, we introduce the canonical vectors and partly simplify the problem to the case of discrimination between pairs of canonical vectors. By considering the positivity of the two by two matrices, we obtain a series of new upper bounds of the total success probability which depends on both the prior probabilities and specific state structures.
Complete solutions are given in a closed analytic form for unambiguous discrimination of three general pure states with real mutual inner products. For this purpose, we first establish some general results on unambiguous discrimination of n linearly independent pure states. The uniqueness of solution is proved. The condition under which the problem is reduced to an (n-1)-state problem is clarified. After giving the solution for three pure states with real mutual inner products, we examine some difficulties in extending our method to the case of complex inner products. There is a class of set o…
Given $n$ linearly independent pure states and their prior probabilities, we study the problem of optimum unambiguous discrimination of these states. We derive the properties of the optimum solution and the equations that must be satisfied by the optimum measurement strategy which achieves the maximum average success probability, and also give the detailed steps to obtain the optimum solution and the optimum measurement strategy. The general method and results we obtain are also illustrated both numerically and geometrically. We derive a simple analytical formula of the maximum average success…
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.
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 quantum formalism permits one to discriminate sometimes between any set of linearly-independent pure states with certainty. We obtain the maximum probability with which a set of equally-likely, symmetric, linearly-independent states can be discriminated. The form of this bound is examined for symmetric coherent states of a harmonic oscillator or field mode.
It is known that unambiguous discrimination among non-orthogonal but linearly independent quantum states is possible with a certain probability of success. Here, we consider a variant of that problem. Instead of discriminating among all of the different states, we shall only discriminate between two subsets of them. In particular, for the case of three non-orthogonal states, we show that the optimal strategy to distinguish between a set containing one of the states from the set containing the other two has a higher success rate than if we wish to discriminate among all three states. Somewhat s…
We consider the problem of unambiguous (error-free) discrimination of N linearly independent pure quantum states with prior probabilities, where the goal is to find a measurement that maximizes the average probability of success. We derive an upper bound on the optimal average probability of success using a result on optimal local conversion between two bipartite pure states. We prove that an optimal measurement in general saturates our bound. In the exceptional cases we show that the bound is tight, but not always optimal.
We investigate generalized measurements, based on positive-operator-valued measures, and von Neumann measurements for the unambiguous discrimination of two mixed quantum states that occur with given prior probabilities. In particular, we derive the conditions under which the failure probability of the measurement can reach its absolute lower bound, proportional to the fidelity of the states. The optimum measurement strategy yielding the fidelity bound of the failure probability is explicitly determined for a number of cases. One example involves two density operators of rank d that jointly spa…
We discuss the problem of designing unambiguous programmable discriminators for any $n$ unknown quantum states in an $m$-dimensional Hilbert space. The discriminator is a fixed measurement which has two kinds of input registers: the program registers and the data register. The program registers consist of the $n$ states, while the data register is prepared among them. The task of the discriminator is to tell us which state stored in the program registers is equivalent to that in the data register. First, we give a necessary and sufficient condition for judging an unambiguous programmable discr…
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…
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 investigate the discrimination of pure-mixed (quantum filtering) and mixed-mixed states and compare their optimal success probability with the one for discriminating other pairs of pure states superposed by the vectors included in the mixed states. We prove that under the equal-fidelity condition, the pure-pure state discrimination scheme is superior to the pure-mixed (mixed-mixed) one. With respect to quantum filtering, the coherence exists only in one pure state and is detrimental to the state discrimination for lower dimensional systems; while it is the opposite for the mixed-mixed case …
We propose an oversimplified scheme to unambiguously discriminate nonorthogonal quantum field states inside high-Q cavities. Our scheme, which is based on positive operator-valued mea- sures (POVM) technique, uses a single three-level atom interacting resonantly with a single mode of a cavity-field and selective atomic state detectors. While the single three-level atom takes the role of the ancilla, the single cavity mode field represents the system we want to obtain information. The efficiency of our proposal is analyzed considering the nowadays achievements in the context of cavity QED.
Unambiguously distinguishing between nonorthogonal but linearly independent quantum states is a challenging problem in quantum information processing. In this work, an exact analytic solution to an optimum measurement problem involving an arbitrary number of pure linearly independent quantum states is presented. To this end, the relevant semi-definite programming task is reduced to a linear programming one with a feasible region of polygon type which can be solved via simplex method. The strength of the method is illustrated through some explicit examples. Also using the close connection betwe…
We discuss the problem of designing unambiguous programmable discriminators for any n unknown quantum states in an m-dimensional Hilbert space. The discriminator is a fixed measurement that has two kinds of input registers: the program registers and the data register. The quantum state in the data register is what users want to identify, which is confirmed to be among the n states in program registers. The task of the discriminator is to tell the users which state stored in the program registers is equivalent to that in the data register. First, we give a necessary and sufficient condition for…
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 study the measurement for the unambiguous discrimination of two mixed quantum states that are described by density operators $ρ_1$ and $ρ_2$ of rank d, the supports of which jointly span a 2d-dimensional Hilbert space. Based on two conditions for the optimum measurement operators, and on a canonical representation for the density operators of the states, two equations are derived that allow the explicit construction of the optimum measurement, provided that the expression for the fidelity of the states has a specific simple form. For this case the problem is mathematically equivalent to dis…
An approach to quantum random number generation based on unambiguous quantum state discrimination (USD) is developed. We consider a prepare-and-measure protocol, where two non-orthogonal quantum states can be prepared, and a measurement device aims at unambiguously discriminating between them. Because the states are non-orthogonal, this necessarily leads to a minimal rate of inconclusive events whose occurrence must be genuinely random and which provide the randomness source that we exploit. Our protocol is semi-device-independent in the sense that the output entropy can be lower bounded based…
Deterministic discrimination of nonorthogonal states is forbidden by quantum measurement theory. However, if we do not want to succeed all the time, i.e. allow for inconclusive outcomes to occur, then unambiguous discrimination becomes possible with a certain probability of success. A variant of the problem is set discrimination: the states are grouped in sets and we want to determine to which particular set a given pure input state belongs. We consider here the simplest case, termed quantum state filtering, when the $N$ given non-orthogonal states, $\{|ψ_{1} >,..., |ψ_{N} > \}$, are divided i…
In this thesis we study the problem of unambiguously discriminating two mixed quantum states. We first present reduction theorems for optimal unambiguous discrimination of two generic density matrices. We show that this problem can be reduced to that of two density matrices that have the same rank $r$ in a 2$r$-dimensional Hilbert space. These reduction theorems also allow us to reduce USD problems to simpler ones for which the solution might be known. As an application, we consider the unambiguous comparison of $n$ linearly independent pure states with a simple symmetry. Moreover, lower bound…
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.
