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Quantum state discrimination and its applications

Joonwoo Bae, Leong-Chuan Kwek

arXiv:1707.02571v2quant-ph

TL;DR

Quantum state discrimination concerns how distinguishable quantum states can be and how classical information can be extracted from them, but optimal strategies remain unsolved in many cases. This review surveys discrimination methods and selected applications, reporting recent progress while emphasizing the continuing difficulty of general optimal discrimination.

  • Problem

    Optimal quantum state discrimination remains unsolved in many general cases, limiting the characterization of discrimination strategies and related applications.

  • Method

    The review synthesizes quantum state discrimination methods, recent progress, and selected applications connecting quantum information processing with quantum foundations.

  • Results

    Recent progress includes a general approach based on Karush-Kuhn-Tucker conditions and a reduction to one parameter for equal a priori probabilities.

  • Takeaways & Limitations

    Quantum state discrimination provides a basic tool linking quantum information theory and the foundations of quantum mechanics.

  • Takeaways & Limitations

    General optimal discrimination remains difficult, and the review focuses on minimum-error discrimination while referring to other reviews for several alternative methods and experimental progress.

Abstract

from arXiv · show

Quantum state discrimination underlies various applications in quantum information processing tasks. It essentially describes the distinguishability of quantum systems in different states, and the general process of extracting classical information from quantum systems. It is also useful in quantum information applications, such as the characterisation of mutual information in cryptographic protocols, or as a technique to derive fundamental theorems in quantum foundations. It has deep connections to physical principles such as relativistic causality. Quantum state discrimination traces a long history of several decades, starting with the early attempts to formalise information processing of physical systems such as optical communication with photons. Nevertheless, in most cases, optimal strategies of quantum state discrimination remain unsolved, and related applications are valid in some limited cases only. The present review aims to provide an overview on quantum state discrimination, covering some recent progress, and addressing applications in some selected topics. This review serves to strengthen the link between results in quantum state discrimination and quantum information applications, by showing the ways in which the fundamental results are exploited in applications and vice versa.

1 Introduction

Quantum state discrimination is central to quantum information and foundations because non-orthogonal states cannot generally be distinguished perfectly. This review surveys discrimination methods, recent progress, and selected applications linking these areas.

  • Non-orthogonal quantum states cannot be discriminated perfectly, connecting state discrimination to no-cloning and the impossibility of instantaneous communication.
  • The optimal strategy depends on the figure of merit, such as minimizing average errors or maximizing confidence in detection events.
  • Quantum state discrimination is used in quantum information processing and foundations, including no-go theorems, dimension witnesses, and operational interpretations of conditional mutual entropy.
  • Optimal discrimination is generally difficult beyond two-state discrimination, although recent progress includes an analytic solution for randomly assigned qubit states.
  • The review provides a comprehensive introduction to quantum state discrimination, recent progress, and selected applications across quantum information science.

2 Discrimination of Quantum States

Quantum state discrimination models how one party prepares a state and another measures it to infer which state was sent. Because arbitrary quantum states need not be orthogonal, the measurement strategy must be chosen according to the application's objective.

  • Quantum state discrimination is framed as a game in which Alice prepares a state and Bob applies a measurement to identify it.
  • Bob combines measurement outcomes with prior knowledge of the agreed state set to learn about Alice’s preparation.
  • Measurements cannot always reveal full information because arbitrary quantum states need not be orthogonal and are generally not perfectly distinguishable.
  • Since perfect discrimination is unavailable, schemes optimize a figure of merit and may allow inconclusive outcomes.
  • The review covers minimum-error discrimination, unambiguous state discrimination, and maximum confidence methods.

2.1 A basic setting

The basic setting consists of state preparation followed by a quantum measurement whose outcomes provide information about the prepared state. Different figures of merit produce different optimal measurement strategies, and the review focuses especially on minimum-error discrimination.

  • Before measurement, ignorance about the prepared state is represented by a mixed state formed from the possible states and their prior probabilities.
  • Alice prepares one state ρ_i from an agreed set with prior probability q_i, while Bob measures the received system.
  • Bob’s measurement uses a POVM consisting of positive operators M_k that sum to the identity and correspond to detection outcomes.
  • For a prepared state ρ_i, outcome k occurs with probability p(k|i) = tr[M_kρ_i].
  • The review emphasizes minimum-error discrimination while referring to other reviews for unambiguous discrimination, maximum confidence measurement, and experimental progress.

2.2 Minimum-error discrimination

Minimum-error discrimination optimizes measurements to maximize correct guesses, with the guessing probability determined exactly for two states and tractably in several symmetric or low-dimensional cases. General optimal measurements may be non-unique, and multiple-state discrimination remains difficult outside special settings.

  • Definition and optimization: Minimum-error discrimination chooses POVMs so each detection outcome is assigned to a state while minimizing average error, with p_error = 1 − p_guess.The correct-guess contribution for state ρ_i is q_i p(i|i) = tr[M_i q_i ρ_i].
  • Two-state discrimination: For two states, the optimal guessing probability is obtained from the spectral decomposition of X = q_1ρ_1 − q_2ρ_2, yielding the Helstrom bound.The optimal measurement uses the positive and negative spectral projectors of X.
  • Multiple-state discrimination: For multiple states, optimal discrimination is known in limited cases, including arbitrary qubit states with equal priors and geometrically uniform states.Symmetry facilitates computation of both the guessing probability and the optimal measurement.
  • Multiple-state discrimination: The square-root measurement is optimal for some symmetric ensembles but is not a general optimal strategy for minimum-error discrimination.Mirror-symmetric states provide an example where the square-root measurement neither is optimal nor attains the maximum guessing probability.
  • A general approach: A general approach characterizes optimal discrimination through an operator K and complementary states using conditions derived from convex optimization.K is uniquely determined for a state set even when optimal POVMs are not unique, and obtaining K achieves the optimal discrimination.
  • Qubit state discrimination: For equal priors, the guessing probability reduces to finding a single parameter r, while geometric constructions express it through distances or polytope ratios.For three qubit states, examples include P_guess = (1 + sin θ)/3 and p_guess = 2/3 in the stated regimes.

2.3 Selected topics in applications of two-state discrimination

Two-state discrimination quantifies state-preparation quality, distinguishes measurement classes, and gives operational norms that expose gaps between operations such as SEP and LOCC.

  • Applications: Two-state discrimination is completely analyzed and applied to state preparation, measurement classification, and operationally meaningful norms.These applications form the focus of the section.
  • State preparation: State distinguishability quantifies differences between desired and actually prepared states, including in quantum key distribution.The actual state can be identified by tomography, and distinguishability measures deviation from the desired state.
  • Distinguishing measurements: Separable operations can perfectly distinguish some state sets that LOCC protocols cannot, establishing the strict inclusion LOCC ⊂ SEP.The distinction is demonstrated using quantum state discrimination.
  • Distinguishing measurements: Characterising LOCC capabilities remains an open problem connected to the study of entangled states and the boundary between entangled and separable states.This makes the measurement-classification application relevant to broader quantum-information questions.
  • Distinguishing measurements: Restricting measurements to classes such as LOCC or separable operations induces corresponding norms that describe gaps between operational capabilities.The general trace norm is recovered when all measurements are allowed.

2.4 Unambiguous state discrimination

Unambiguous discrimination guarantees correct conclusions whenever a conclusive detection occurs, while assigning ambiguous outcomes to an additional port; its feasibility depends on the states.

  • Definition: Unambiguous discrimination uses conclusive outcomes associated exclusively with individual states and an additional outcome collecting ambiguous results.The POVM elements are positive and complete, with the extra element completing the measurement when needed.
  • Conditions: For pure states, unambiguous discrimination can be designed when the states are linearly independent; for two mixed states, their supports must not completely overlap.The required measurement may be found numerically using semidefinite programming.
  • Two-state case: For two pure states with equal prior probabilities, the ambiguous-result probability is determined by their overlap and is known as the Ivanovic-Dieks-Peres limit.The supplied passage identifies the overlap dependence but does not provide the formula.
  • Extensions and applications: Closed-form unambiguous discrimination is available for linearly independent three pure states, and the strategy also applies to state comparison, state filtering, and entanglement concentration.The cited approach extends to arbitrary numbers of linearly independent pure states.
  • Operational behavior: A detection event on the ith conclusive port identifies ρi with certainty, whereas the additional port makes no state assignment.This is the operational distinction from minimum-error discrimination.

2.5 Maximum confidence discrimination

Maximum confidence discrimination optimizes the correctness of each detected outcome, interpolating between unambiguous and minimum-error strategies and recovering the latter on average.

  • Definition: Maximum confidence discrimination maximizes the conditional probability that each detection event corresponds to the indicated state.An additional output port collects ambiguous answers when needed.
  • Example: For linearly dependent geometrically uniform three-qubit states, unambiguous discrimination is unavailable, so maximum confidence uses conclusive POVM elements plus an inconclusive element.The POVM coefficients depend on θ.
  • Example: The maximum-confidence conditional probability is 3 for all i = 1, 2, 3, whereas minimum-error discrimination gives (1 + sin 2θ)/3.These are the two conditional-probability expressions reported for the example.
  • Optimization: Its optimal POVM is constructed by choosing operators with maximal overlap with the relevant states and then enforcing measurement completeness.The coefficients are determined under the completeness constraint.
  • Relation to minimum-error discrimination: Maximizing conditional probabilities averaged over all states is equivalent to obtaining the guessing probability, thereby recovering minimum-error discrimination on average.The equality follows from maximizing average conditional probabilities over all measurements.

2.6 Relation between different strategies of state discrimination

Minimum-error and unambiguous discrimination can be generalized by trading error probability against inconclusive outcomes, while maximum-confidence strategies connect to minimum-error discrimination on projected supports.

  • Generalization: Maximum-confidence discrimination generalizes unambiguous and minimum-error discrimination, coinciding with them in their respective applicable regimes.It therefore interpolates between the two strategies.
  • Fixed inconclusive rate: Fixing the inconclusive-outcome rate Q turns generalized minimum-error discrimination into minimizing the error rate perror.Q = 0 recovers minimum-error discrimination, while sufficiently large Q can recover unambiguous discrimination for linearly independent states.
  • Error-constrained unambiguous discrimination: Allowing error in unambiguous discrimination instead minimizes Q subject to perror ≤ pc, with pc = 0 recovering unambiguous discrimination.This provides the converse generalization of the fixed-Q formulation.
  • Trade-off: The two generalizations are related by a trade-off between error rate perror and inconclusive rate Q, with analytic relations obtained for some settings.General results include optimality conditions, but arbitrary state sets are not fully analyzed.
  • Reduction to minimum-error discrimination: For fixed Q, projecting states onto the support of the conclusive measurement reduces the guessing-probability problem to minimum-error discrimination of normalized projected states.The resulting optimal strategies share similarities with maximum-confidence measurement.

2.7 Maximising mutual information

Maximising mutual information characterises the information accessible through measurements on quantum states. Although the optimisation has an upper bound, optimal measurements can be difficult to determine; convexity provides a route to optimisation.

  • Accessible information is the mutual information maximised over Bob’s measurements for a given quantum-state ensemble.It quantifies the information accessible through measurement.
  • The mutual-information optimisation over Bob’s POVMs is a convex optimisation problem.The text states that mutual information is convex in Bob’s measurements.
  • Convex optimisation techniques can solve the problem and yield a unique solution.
  • For geometrically uniform qubit states, symmetry and group-theoretical properties provide optimal POVMs.The resulting POVM can coincide with a minimum-error optimal measurement, even when that measurement is not unique generally.

2.8 State discrimination in the asymptotic limit

In the asymptotic setting, independently prepared copies enable analysis of how discrimination errors converge with the number of copies. Collective, repeated, and adaptive measurements offer different implementation strategies, while their asymptotic relationship remains unresolved.

  • For i.i.d. preparations, n copies produce tensor-power states ρ1^⊗n or ρ2^⊗n, and the quantum Chernoff bound characterises error convergence.
  • Two-state minimum-error discrimination is completely analysed by the Helstrom bound, although computing the trace distance for tensor-power states is not straightforward.
  • The quantum Chernoff exponent has an explicit form analogous to the classical critical exponent.
  • For multiple states, the error convergence is bounded using pairwise quantum Chernoff quantities, but the exact multi-state relation remains conjectural.The lower bound contains a factor 1/3 whose removal remains open.
  • Measurement strategies: Collective measurements act jointly on all copies, whereas repeated and adaptive measurements operate on individual copies.Adaptive measurements update later settings using previous outcomes.
  • Measurement strategies: Collective measurements are currently impractical because they require quantum memory and complicated interactions among copies.
  • Measurement strategies: Whether individual measurements match collective measurements, including asymptotically, remains an open question.

3 State Discrimination as a Tool

Quantum state discrimination serves as a tool linking quantum information processing with quantum foundations and applications. The review discusses exclusion, unitary discrimination, dimension witnesses, and connections to asymptotic cloning.

  • Minimum-error discrimination can certify the minimum Hilbert-space dimension of a quantum system, supporting device-independent communication.
  • Quantum state exclusion: Quantum state exclusion seeks measurements that minimise or ideally eliminate detection events associated with designated states.
  • Quantum state exclusion: State exclusion can be formulated as a semidefinite program with primal and dual optimisation problems.
  • Unitary transformations: Unlike state discrimination, ancillary systems can enhance unitary distinguishability, and finite repetitions can make any two unitaries perfectly distinguishable.
  • Unitary transformations: For two non-commuting unitaries, ancillary systems can yield guessing probability p_guess = 1 even when the corresponding states are not perfectly distinguishable.
  • Dimension witnesses: Dimension-witness methods exploit the analytically solved two-state discrimination problem, while generalisation to multiple states remains open.
  • Asymptotic cloning: State discrimination converts quantum-state information into classical information and can therefore be viewed as asymptotic 1 →∞ cloning.
  • Asymptotic cloning: State discrimination or estimation is equivalent to optimal asymptotic quantum cloning for states with continuous spectrum, while general equivalence remains open.

4 Characterizations in quantum communication

Quantum state discrimination characterises operational quantities in quantum communication, including min-entropy and constraints imposed by no-signaling. These connections relate optimal guessing to privacy, communication limits, and relativistic causality.

  • Quantum min-entropy has an operational meaning equal to the guessing probability of minimum-error state discrimination.
  • Alice’s measurement on an entangled state prepares a classical-quantum ensemble of conditional states for Bob.
  • Bob’s optimal measurement on those conditional states is formulated as a convex optimisation and is equivalent to minimum-error discrimination.
  • In cryptographic applications, minimum-error discrimination identifies the optimal measurement for maximising Bob’s information and quantifies the operational task associated with min-entropy.
  • No-signaling prevents Bob from learning Alice’s unannounced measurement choice with arbitrary precision.
  • A guessing probability above random guessing sufficient to infer Alice’s measurement would enable faster-than-light communication, so no-signaling bounds it.
  • The no-signaling upper bound is tight: optimal state discrimination attains equality for any ensemble {q_i, ρ_i}.
  • No-signaling constraints can be used to discover optimality conditions for state discrimination.

5 Conclusion

Quantum state discrimination is a basic tool for quantum information theory and the foundations of quantum mechanics, but general optimal-discrimination theorems remain unsolved despite progress in special cases. The review provides a comprehensive introduction and selected applications while excluding continuous-variable discrimination.

  • Quantum state discrimination serves as a basic tool for quantum information theory and the foundation of quantum mechanics.
  • General theorems for optimal state discrimination remain unsolved, while recent progress has addressed some special cases.
  • Technical difficulty in general scenarios may place strict limitations on developing some quantum information tasks.
  • The review provides a comprehensive introduction to quantum state discrimination and selected applications.
  • Discrimination of continuous-variable states is outside the review's coverage, although related experimental and cryptographic applications exist.
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