The goal of comparison is to reveal the difference of compared objects as fast and reliably as possible. In this paper we formulate and investigate the unambiguous comparison of unknown quantum measurements represented by non-degenerate sharp POVMs. We distinguish between measurement devices with apriori labeled and unlabeled outcomes. In both cases we can unambiguously conclude only that the measurements are different. For the labeled case it is sufficient to use each unknown measurement only once and the average conditional success probability decreases with the Hilbert space dimension as 1/d. If the outcomes of the apparatuses are not labeled, then the problem is more complicated. We analyze the case of two-dimensional Hilbert space. In this case single shot comparison is impossible and each measurement device must be used (at least) twice. The optimal test state in the two-shots scenario gives the average conditional success probability 3/4. Interestingly, the optimal experiment detects unambiguously the difference with nonvanishing probability for any pair of observables.
We consider N quantum systems initially prepared in pure states and address the problem of unambiguously comparing them. One may ask whether or not all $N$ systems are in the same state. Alternatively, one may ask whether or not the states of all N systems are different. We investigate the possibility of unambiguously obtaining this kind of information. It is found that some unambiguous comparison tasks are possible only when certain linear independence conditions are satisfied. We also obtain measurement strategies for certain comparison tasks which are optimal under a broad range of circumst…
We address the problem of unambiguous comparison of a pair of unknown qudit unitary channels. Using the framework of process positive operator valued measures (PPOVM) we characterize all solutions and identify the optimal ones. We prove that the entanglement is the key ingredient in designing the optimal experiment for comparison of unitary channels. Without entanglement the optimality can not be achieved. The proposed scheme is also experimentally feasible.
We present a solution of the problem of the optimal unambiguous comparison of two ensembles of unknown quantum states (psi_1)^k and (psi_2)^l. We consider two cases: 1) The two unknown states psi_1 and psi_2 are arbitrary states of qudits. 2) Alternatively, they are coherent states of a harmonic oscillator. For the case of coherent states we propose a simple experimental realization of the optimal "comparison" machine composed of a finite number of beam-splitters and a single photodetector.
We propose a scheme for unambiguous state comparison (USC) of two unknown squeezed vacuum states of an electromagnetic field. Our setup is based on linear optical elements and photon-number detectors, and achieves optimal USC in an ideal case of unit quantum efficiency. In realistic conditions, i.e., for non-unit quantum efficiency of photodetectors, we evaluate the probability of getting an ambiguous result as well as the reliability of the scheme, thus showing its robustness in comparison to previous proposals.
In this article, we study the problem of comparing mixed quantum states: given $n$ unknown mixed quantum states, can one determine whether they are identical or not with an unambiguous quantum measurement? We first study universal comparison of mixed quantum states, and prove that this task is generally impossible to accomplish. Then, we focus on unambiguous comparison of $n$ mixed quantum states arbitrarily chosen from a set of $k$ mixed quantum states. The condition for the existence of an unambiguous measurement operator which can produce a conclusive result when the unknown states are actu…
We discuss the unambiguous measurement of quantum nonorthogonal states in connection with the quantum cryptography. We show that checking a ratio of null one to signal is essential in detecting a certain kind of eavesdropping in the case of two nonorthogonal states quantum cryptography. We prove that it is not needed in the case of the four states quantum cryptography.
In this article, we study an opposite problem of universal quantum state comparison, that is unambiguous determining whether multiple unknown quantum states from a Hilbert space are orthogonal or not. We show that no unambiguous quantum measurement can accomplish this task with a non-zero probability. Moreover, we extend the problem to a more general case, that is to compare how orthogonal multiple unknown quantum states are with a threshold, and it turns out that given any threshold this extended task is also impossible by any unambiguous quantum measurement except for a trivial case. It will…
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 propose and theoretically investigate an unambiguous Bell measurement of atomic qubits assisted by multiphoton states. The atoms interact resonantly with the electromagnetic field inside two spatially separated optical cavities in a Ramsey-type interaction sequence. The qubit states are postselected by measuring the photonic states inside the resonators. We show that if one is able to project the photonic field onto two coherent states on opposite sites of phase space, an unambiguous Bell measurement can be implemented. Thus our proposal may provide a core element for future components of q…
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.
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,…
Based on our previous publication [U. Herzog and J. A. Bergou, Phys.Rev. A 71, 050301(R) (2005)] we investigate the optimum measurement for the unambiguous discrimination of two mixed quantum states that occur with given prior probabilities. Unambiguous discrimination of nonorthogonal states is possible in a probabilistic way, at the expense of a nonzero probability of inconclusive results, where the measurement fails. Along with a discussion of the general problem, we give an example illustrating our method of solution. We also provide general inequalities for the minimum achievable failure p…
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 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 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.
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…
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…
Entangled multi-photon states have the potential to provide improved measurement accuracy, but are sensitive to photon loss. It is possible to calculate ideal loss-resistant states that maximize the Fisher information, but it is unclear how these could be experimentally generated. Here we propose a set of states that can be obtained by processing the output from parametric down-conversion. Although these states are not optimal, they provide performance very close to that of optimal states for a range of parameters. Moreover, we show how to use sequences of such states in order to obtain an una…
We investigate unambiguous discrimination between given quantum states with a sequential measurement, which is restricted to local measurements and one-way classical communication. If the given states are binary or those each of whose individual systems is two-dimensional, then it is in some cases known whether a sequential measurement achieves a globally optimal unambiguous measurement. In contrast, for more than two states each of whose individual systems is more than two-dimensional, the problem becomes extremely complicated. This paper focuses on symmetric ternary pure states each of whose…
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…
In the present paper I formulate a framework that accommodates many unambiguous discrimination problems. I show that the prior information about any type of constituent (state, channel, or observable) allows us to reformulate the discrimination among finite number of alternatives as the discrimination among finite number of average constituents. Using this framework I solve several unambiguous tasks. I present a solution to optimal unambiguous comparison of two ensembles of unknown quantum states. I consider two cases: 1) The two unknown states are arbitrary pure states of qudits. 2) Alternati…
We consider the problem of designing an optimal quantum detector that distinguishes unambiguously between a collection of mixed quantum states. Using arguments of duality in vector space optimization, we derive necessary and sufficient conditions for an optimal measurement that maximizes the probability of correct detection. We show that the previous optimal measurements that were derived for certain special cases satisfy these optimality conditions. We then consider state sets with strong symmetry properties, and show that the optimal measurement operators for distinguishing between these sta…
We address the problem of unambiguously identifying the state of a probe qudit with the state of one of d reference qudits. The reference states are assumed pure and linearly independent but we have no knowledge of them. The state of the probe qudit is assumed to coincide equally likely with either one of the d unknown reference states. We derive the optimum measurement strategy that maximizes the success probability of unambiguous identification and find that the optimum strategy is a generalized measurement. We give both the measurement operators and the optimum success probability explicitl…
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…
We consider unambiguous identification of coherent states of electromagnetic field. In particular, we study possible generalizations of an optical setup proposed in M. Sedlák {\it et al.}, Phys. Rev. A {\bf 76}, 022326 (2007). We show how the unambiguous identification of coherent states can be performed in a general case when multiple copies of unknown and the reference states are available. We also investigate whether reference states after the measurement can be "recovered" and further used for subsequent unambiguous identification tasks. We show that in spite of the fact that the recovered…
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.
We consider an infinite class of unambiguous quantum state discrimination problems on multipartite systems, described by Hilbert space $\cal{H}$, of any number of parties. Restricting consideration to measurements that act only on $\cal{H}$, we find the optimal global measurement for each element of this class, achieving the maximum possible success probability of $1/2$ in all cases. This measurement turns out to be both separable and unique, and by our recently discovered necessary condition for local quantum operations and classical communication (LOCC), it is easily shown to be impossible b…
We derive, and experimentally demonstrate, an interferometric scheme for unambiguous phase estimation with precision scaling at the Heisenberg limit that does not require adaptive measurements. That is, with no prior knowledge of the phase, we can obtain an estimate of the phase with a standard deviation that is only a small constant factor larger than the minimum physically allowed value. Our scheme resolves the phase ambiguity that exists when multiple passes through a phase shift, or NOON states, are used to obtain improved phase resolution. Like a recently introduced adaptive technique [Hi…
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…
Quantum state elimination measurements tell us what states a quantum system does not have. This is different from state discrimination, where one tries to determine what the state of a quantum system is, rather than what it is not. Apart from being of fundamental interest, quantum state elimination may find uses in quantum communication and quantum cryptography. We consider unambiguous quantum state elimination for two or more qubits, where each qubit can be in one of two possible states. Optimal measurements for eliminating one and two states out of four two-qubit states are given. We also pr…
We investigate the scrambling of information in a hierarchical star-topology system using out-of-time-ordered correlation (OTOC) functions. The system consists of a central qubit directly interacting with a set of satellite qubits, which in turn interact with a second layer of satellite qubits. This particular topology not only allows convenient preparation and filtering of multiple quantum coherences between the central qubit and the first layer but also to engineer scrambling in a controlled manner. Hence, it provides us with an opportunity to experimentally study scrambling of information l…
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.
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 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…
In this paper, we consider the generalized measurement where one particular quantum signal is unambiguously extracted from a set of non-commutative quantum signals and the other signals are filtered out. Simple expressions for the maximum detection probability and its POVM are derived. We applyl such unambiguous quantum state filtering to evaluation of the sensing of decoherence channels. The bounds of the precision limit for a given quantum state of probes and possible device implementations are discussed.
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 unambiguous comparison of quantum states in a scenario that is more general than the one that was originally suggested by Barnett et al. First, we find the optimal solution for the comparison of two states taken from a set of two pure states with arbitrary a priori probabilities. We show that the optimal coherent measurement is always superior to the optimal incoherent measurement. Second, we develop a strategy for the comparison of two states from a set of N pure states, and find an optimal solution for some parameter range when N=3. In both cases we use the reduction metho…
We initially consider a quantum system consisting of two qubits, which can be in one of two nonorthogonal states, Ψ_0 or Ψ_1. We distribute the qubits to two parties, Alice and Bob. They each measure their qubit and then compare their measurement results to determine which state they were sent. This procedure is error-free, which implies that it must sometimes fail. In addition, no quantum memory is required; it is not necessary to store one of the qubits until the result of the measurement on the other is known. We consider the cases in which, should failure occur, both parties receive a fail…
A universal programmable discriminator can perform the discrimination between two unknown states, and the optimal solution can be approached via the discrimination between the two averages over the uniformly distributed unknown input pure states, which has been widely discussed in previous works. In this paper, we consider the success probabilities of the optimal universal programmable unambiguous discriminators when applied to the pure input states. More precisely, the analytic results of the success probabilities are derived with the expressions of the optimal measurement operators for the u…
Unambiguous unitary maps and unambiguous unitary quantum channels are introduced and some of their properties are derived. These properties ensure certain simple form for the measurements involved in realizing an unambiguous unitary quantum channel. Error correction and unambiguous error correction with nonzero probability are discussed in terms of unambiguous unitary quantum channels. We not only re-derive the well-known condition for a set of errors to be correctable with certainty, but also obtain a necessary and sufficient condition for the errors caused by a noisy channel to be correctabl…
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…
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…
In this paper we consider the problem of unambiguous discrimination between a set of linearly independent pure quantum states. We show that the design of the optimal measurement that minimizes the probability of an inconclusive result can be formulated as a semidefinite programming problem. Based on this formulation, we develop a set of necessary and sufficient conditions for an optimal quantum measurement. We show that the optimal measurement can be computed very efficiently in polynomial time by exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed…
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…
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.
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.
The problem of unambiguously distinguishing among nonorthogonal but linearly independent quantum states can be solved by mapping the set of nonorthogonal quantum states onto a set of orthogonal ones, which can then be distinguished without error. Such nonunitary transformations can be performed conditionally on quantum systems; a unitary transformation is carried out on a larger system of which the system of interest is a subsytem, a measurement is performed, and if the proper result is obtained, the desired nonunitary transformation will have been performed on the subsystem. We show how to co…
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 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…
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…
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…
We consider an unambiguous identification of an unknown coherent state with one of two unknown coherent reference states. Specifically, we consider two modes of an electromagnetic field prepared in unknown coherent states alpha_1 and alpha_2, respectively. The third mode is prepared either in the state alpha_1 or in the state alpha_2. The task is to identify (unambiguously) which of the two modes are in the same state. We present a scheme consisting of three beamsplitters capable to perform this task. Although we don't prove the optimality, we show that the performance of the proposed setup is…
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…
