We show that the quantum measurement known as the pretty good measurement can be used to identify an unknown quantum state picked from any set of $n$ mixed states that have pairwise fidelities upper-bounded by a constant below 1, given $O(\log n)$ copies of the unknown state, with high success probability in the worst case. If the unknown state is promised to be pure, there is an explicit measurement strategy which solves this worst-case quantum state discrimination problem with $\widetilde{O}(\|G\|)$ copies, where $G$ is the Gram matrix of the states.
Paper withdrawn. Results are similar to V. P. Belavkin, "Optimum distinction of non-orthogonal quantum signals," Radio Engineering and Electronic Physics, vol. 20, June 1975, p. 39-47.
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.
Port-based teleportation (PBT) is a protocol in which Alice teleports an unknown quantum state to Bob using measurements on a shared entangled multipartite state called the port state and forward classical communication. In this paper, we give an explicit proof that the so-called pretty good measurement, or square-root measurement, is optimal for the PBT protocol with independent copies of maximally entangled states as the port state. We then show that the very same measurement remains optimal even when the port state is optimized to yield the best possible PBT protocol. Hence, there is one pa…
Pretty Good Measurement for Radiomics: A Quantum-Inspired Multi-Class Classifier for Lung Cancer Subtyping and Prostate Cancer Risk Stratification
Università degli Studi di Cagliari（卡利亚里大学）；Institute of Bioimaging and Complex Biological Systems - National Research Council (IBSBC - CNR)（生物成像与复杂生物系统研究所 - 国家研究委员会）；National Laboratory of South, National Institute for Nuclear Physics(LNS-INFN)（南部国家实验室，核物理研究所）
We present quantum-inspired algorithms for classification tasks inspired by the problem of quantum state discrimination. By construction, these algorithms can perform multiclass classification, prevent overfitting, and generate probability outputs. While they could be implemented on a quantum computer, we focus here on classical implementations of such algorithms. The training of these classifiers involves Semi-Definite Programming. We also present a relaxation of these classifiers that utilizes Linear Programming (but that can no longer be interpreted as a quantum measurement). Additionally, …
We examine conditions for a pair of strongly cospectral vertices to have pretty good quantum state transfer in terms of minimal polynomials, and provide cases where pretty good state transfer can be ruled out. We also provide new examples of simple, unweighted graphs exhibiting pretty good state transfer. Finally, we consider modifying paths by adding symmetric weighted edges, and apply these results to this case.
We construct the exact solution for a family of one-half spin chains explicitly. The spin chains Hamiltonian corresponds to an isotropic Heisenberg Hamiltonian, with staggered exchange couplings that take only two different values. We work out the exact solutions in the one-excitation subspace. Regarding the problem of quantum state transfer, we use the solution and some theorems concerning the approximation of irrational numbers, to show the appearance of conclusive pretty good transmission for chains with particular lengths. We present numerical evidence that pretty good transmission is achi…
The Petz recovery channel plays an important role in quantum information science as an operation that approximately reverses the effect of a quantum channel. The pretty good measurement is a special case of the Petz recovery channel, and it allows for near-optimal state discrimination. A hurdle to the experimental realization of these vaunted theoretical tools is the lack of a systematic and efficient method to implement them. This paper sets out to rectify this lack: using the recently developed tools of quantum singular value transformation and oblivious amplitude amplification, we provide a…
We construct infinite families of graphs in which pretty good state transfer can be induced by adding a potential to the nodes of the graph (i.e. adding a number to a diagonal entry of the adjacency matrix). Indeed, we show that given any graph with a pair of cospectral nodes, a simple modification of the graph, along with a suitable potential, yields pretty good state transfer (i.e. asymptotically perfect state transfer) between the nodes. This generalizes previous work, concerning graphs with an involution, to asymmetric graphs.
The polar decomposition of a matrix is a key element in the quantum linear algebra toolbox. We show that the problem of quantum polar decomposition, recently studied in Lloyd et al. [LBP+20], has a simple and concise implementation via the quantum singular value transform (QSVT). We focus on the applications to pretty-good measurements, a close-to-optimal measurement to distinguish quantum states, and the quantum Procrustes problem, the task of learning an optimal unitary mapping between given `input' and `output' quantum states. By transforming the state-preparation unitaries into a block-enc…
We compare several instances of pure-state Belavkin weighted square-root measurements from the standpoint of minimum-error discrimination of quantum states. The quadratically weighted measurement is proven superior to the so-called "pretty good measurement" (PGM) in a number of respects: (1) Holevo's quadratic weighting unconditionally outperforms the PGM in the case of two-state ensembles, with equality only in trivial cases. (2) A converse of a theorem of Holevo is proven, showing that a weighted measurement is asymptotically optimal only if it is quadratically weighted. Counterexamples for …
We address the limit of the Gaussian operations and classical communication in the problem of quantum state discrimination. We show that the optimal Gaussian strategy for the discrimination of the binary phase shift keyed (BPSK) coherent signal is a simple homodyne detection. We also propose practical near-optimal quantum receivers that beat the BPSK homodyne limit in all areas of the signal power. Our scheme is simple and does not require realtime electrical feedback.
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 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 find optimality conditions for testers in discrimination of quantum channels. These conditions are obtained using semidefinite programming and are similar to optimality conditions for POVMs obtained by Holevo for ensembles of quantum states. We get a simple condition for existence of an optimal tester with any given input state with maximal Schmidt rank, in particular with a maximally entangled input state and we show the pitfalls of using input states with not maximal Schmidt rank. In case when maximally entangled state is not the optimal input state an error estimate is obtained. The resu…
The optimal discrimination of non-orthogonal quantum states with minimum error probability is a fundamental task in quantum measurement theory as well as an important primitive in optical communication. In this work, we propose and experimentally realize a new and simple quantum measurement strategy capable of discriminating two coherent states with smaller error probabilities than can be obtained using the standard measurement devices; the Kennedy receiver and the homodyne receiver.
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…
Dimension witnesses allow one to test the dimension of an unknown physical system in a device-independent manner, that is, without placing assumptions about the functioning of the devices used in the experiment. Here we present simple and general dimension witnesses for quantum systems of arbitrary Hilbert space dimension. Our approach is deeply connected to the problem of quantum state discrimination, hence establishing a strong link between these two research topics. Finally, our dimension witnesses can distinguish between classical and quantum systems of the same dimension, making them pote…
We address the problem of energy-error tradeoff in the discrimination between two linear passive quantum optical devices with a single use. We provide an analytical derivation of the optimal strategy for beamsplitters and an iterative algorithm converging to the optimum in the general case. We then compare the optimal strategy with a simpler strategy using coherent input states and homodyne detection. It turns out that the former requires much less energy in order to achieve the same performances.
We describe a simple multivariate technique of likelihood ratios for improved discrimination of signal and background in multi-dimensional quantum target detection. The technique combines two independent variables, time difference and summed energy, of a photon pair from the spontaneous parametric down-conversion source into an optimal discriminant. The discriminant performance was studied in experimental data and in Monte-Carlo modelling with clear improvement shown compared to previous techniques. As novel detectors become available, we expect this type of multivariate analysis to become inc…
The discrimination of quantum processes, including quantum states, channels, and superchannels, is a fundamental topic in quantum information theory. It is often of interest to analyze the optimal performance that can be achieved when discrimination strategies are restricted to a given subset of all strategies allowed by quantum mechanics. In this paper, we present a general formulation of the task of finding the maximum success probability for discriminating quantum processes as a convex optimization problem whose Lagrange dual problem exhibits zero duality gap. The proposed formulation can b…
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 …
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…
We analyse the problem of finding sets of quantum states that can be deterministically discriminated. From a geometric point of view this problem is equivalent to that of embedding a simplex of points whose distances are maximal with respect to the Bures distance (or trace distance). We derive upper and lower bounds for the trace distance and for the fidelity between two quantum states, which imply bounds for the Bures distance between the unitary orbits of both states. We thus show that when analysing minimal and maximal distances between states of fixed spectra it is sufficient to consider d…
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 a special form of state discrimination in which after the measurement we are given additional information that may help us identify the state. This task plays a central role in the analysis of quantum cryptographic protocols in the noisy-storage model, where the identity of the state corresponds to a certain bit string, and the additional information is typically a choice of encoding that is initially unknown to the cheating party. We first provide simple optimality conditions for measurements for any such problem, and show upper and lower bounds on the success probability. For a c…
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 a scheme in which sequential state-discrimination measurements are performed on qudits to determine the quantum state in which they were initially prepared. The qudits belong to a set of nonorthogonal quantum states and hence cannot be distinguished with certainty. Unambiguous state discrimination allows error-free measurements at the expense of occasionally failing to give a conclusive answer about the state of the qudit. Qudits have the potential to carry more information per transmission than qubits. We considered the situation in which Alice sends one of N qudits, where the dime…
A fundamental problem in quantum information is to explore the roles of different quantum correlations in a quantum information procedure. Recent work [Phys. Rev. Lett., 107 (2011) 080401] shows that the protocol for assisted optimal state discrimination (AOSD) may be implemented successfully without entanglement, but with another correlation, quantum dissonance. However, both the original work and the extension to discrimination of $d$ states [Phys. Rev. A, 85 (2012) 022328] have only proved that entanglement can be absent in the case with equal a \emph{priori} probabilities. By improving the…
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…
We study the discrimination of weak coherent states of light with significant overlaps by nondestructive measurements on the light states through measuring atomic states that are entangled to the coherent states via dipole coupling. In this way, the problem of measuring and discriminating coherent light states is shifted to finding the appropriate atom-light interaction and atomic measurements. We show that this scheme allows us to attain a probability of error extremely close to the Helstrom bound, the ultimate quantum limit for discriminating binary quantum states, through the simple Jaynes-…
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 examine the results of the paper "Precision metrology using weak measurements", [Zhang, Datta, and Walmsley, arXiv:1310.5302] from a quantum state discrimination point of view. The Heisenberg scaling of the photon number for the precision of the interaction parameter between coherent light and a spin one-half particle (or pseudo-spin) has a simple interpretation in terms of the interaction rotating the quantum state to an orthogonal one. In order to achieve this scaling, the information must be extracted from the spin rather than from the coherent state of light, limiting the applications o…
We determine the optimal method of discriminating and comparing quantum states from a certain class of multimode Gaussian states and their mixtures when arbitrary global Gaussian operations and general Gaussian measurements are allowed. We consider the so-called constant-$\hat{p}$ displaced states which include mixtures of multimode coherent states arbitrarily displaced along a common axis. We first show that no global or local Gaussian transformations or generalized Gaussian measurements can lead to a better discrimination method than simple homodyne measurements applied to each mode separate…
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…
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…
Motivated by the recent discovery of a quantum Chernoff theorem for asymptotic state discrimination, we investigate the distinguishability of two bipartite mixed states under the constraint of local operations and classical communication (LOCC), in the limit of many copies. While for two pure states a result of Walgate et al. shows that LOCC is just as powerful as global measurements, data hiding states (DiVincenzo et al.) show that locality can impose severe restrictions on the distinguishability of even orthogonal states. Here we determine the optimal error probability and measurement to dis…
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 present a theoretical study of minimum error probability discrimination, using quantum- optical probe states, of M optical phase shifts situated symmetrically on the unit circle. We assume ideal lossless conditions and full freedom for implementing quantum measurements and for probe state selection, subject only to a constraint on the average energy, i.e., photon number. In particular, the probe state is allowed to have any number of signal and ancillary modes, and to be pure or mixed. Our results are based on a simple criterion that partitions the set of pure probe states into equivalence …
We propose an interferometric method for statistically discriminating between nonorthogonal states in high dimensional Hilbert spaces for use in quantum information processing. The method is illustrated for the case of photon orbital angular momentum (OAM) states. These states belong to pairs of bases that are mutually unbiased on a sequence of two-dimensional subspaces of the full Hilbert space, but the vectors within the same basis are not necessarily orthogonal to each other. Over multiple trials, this method allows distinguishing OAM eigenstates from superpositions of multiple such eigenst…
We consider the problem of discriminating finite-dimensional quantum processes, also called quantum supermaps, that can consist of multiple time steps. Obtaining the ultimate performance for discriminating quantum processes is of fundamental importance, but is challenging mainly due to the necessity of considering all discrimination strategies allowed by quantum mechanics, including entanglement-assisted strategies and adaptive strategies. In the case in which the processes to be discriminated have internal memories, the ultimate performance would generally be more difficult to analyze. In thi…
