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arXiv:2405.03872 · Physical Review A

Nonprojective Bell-state measurements

Amanda Wei, Gabriele Cobucci, Armin Tavakoli

Published Oct 7, 2024• 3 citations
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Abstract

The Bell-state measurement (BSM) is the projection of two qubits onto four orthogonal maximally entangled states. Here we first propose how to appropriately define more general BSMs, which have more than four possible outcomes, and then study whether they exist in quantum theory. We observe that nonprojective BSMs can be defined in a systematic way in terms of equiangular tight frames of maximally entangled states, i.e., a set of maximally entangled states, where every pair is equally, and in a sense maximally, distinguishable. We show that there exists a five-outcome BSM through an explicit construction and find that it admits a simple geometric representation. Then we prove that there exists no larger BSM on two qubits by showing that no six-outcome BSM is possible. We also determine the most distinguishable set of six equiangular maximally entangled states and show that it falls only somewhat short of forming a valid quantum measurement. Finally, we study the nonprojective BSM in the contexts of both local state discrimination and entanglement-assisted quantum communication. Our results put forward natural forms of nonprojective joint measurements and provide insight into the geometry of entangled quantum states. Published by the American Physical Society 2024

DOI
10.1103/physreva.110.042206

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The Bell-state measurement (BSM) is the projection of two qubits onto four orthogonal maximally entangled states. Here we first propose how to appropriately define more general BSMs, which have more than four possible outcomes, and then study whether they exist in quantum theory. We observe that nonprojective BSMs can be defined in a systematic way in terms of equiangular tight frames of maximally entangled states, i.e., a set of maximally entangled states, where every pair is equally, and in a sense maximally, distinguishable. We show that there exists a five-outcome BSM through an explicit construction and find that it admits a simple geometric representation. Then we prove that there exists no larger BSM on two qubits by showing that no six-outcome BSM is possible. We also determine the most distinguishable set of six equiangular maximally entangled states and show that it falls only somewhat short of forming a valid quantum measurement. Finally, we study the nonprojective BSM in the contexts of both local state discrimination and entanglement-assisted quantum communication. Our results put forward natural forms of nonprojective joint measurements and provide insight into the geometry of entangled quantum states. Published by the American Physical Society 2024
In the $4\ensuremath{\bigotimes}4$ quantum system, four qubit lattice states are found to be either locally indistinguishable or distinguishable by using one-way classical communication only, presenting a curious and counterintuitive case in which two-way classical communication has no advantage. Based on these results, we explicitly construct sets of four, five, and even more maximally entangled states that can be locally distinguished only with the help of two-way classical communication, solving partially the open problem described by Nathanson [Phys. Rev. A 88, 062316 (2013)] and highlighting in more detail the difference between local operations with and without two-way classical communication in distinguishing more than three states.
Bell-state measurements are essential ingredients in many protocols for quantum information processing, ranging from quantum teleportation and dense coding to entanglement distribution in quantum networks. Their power relies on the fact that they are measurements in an entangled basis of a two-particle system and that the used Bell-state basis can be generated from a single Bell state by local unitary transformations. How can these measurements be generalized to more particles? We develop a general framework for this state-to-measurement problem: We introduce a hierarchy of classes of measurement bases, distinguished by the local transformations the parties may use for their generation from a single state. This leads to a generalization of the concept of maximally entangleable (or weighted hypergraph) states and the identification of a novel maximally entangled basis of four qubits, being a candidate for data-hiding tasks or distillation protocols. Finally, we prove that not all forms of entanglement can be encoded in an entire measurement basis.
Bell inequalities reveal the fundamentally nonlocal character of quantum mechanics. In this regard, one of the interesting problems is to explore all possible Bell inequalities that demonstrate a gap between local and nonlocal quantum behaviour. This is useful for the geometric characterisation of the set of nonlocal correlations achievable within quantum theory. Moreover, it provides a systematic way to construct Bell inequalities that are tailored to specific quantum information processing tasks. This characterisation is well understood in the simplest (2,2,2)(2,2,2)(2,2,2) scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the (2,2,3)(2,2,3)(2,2,3) scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.
Quantum measurements on a two-level system can have more than two independent outcomes, and in this case, the measurement cannot be projective. Measurements of this general type are essential to an operational approach to quantum theory, but so far, the nonprojective character of a measurement can only be verified experimentally by already assuming a specific quantum model of parts of the experimental setup. Here, we overcome this restriction by using a device-independent approach. In an experiment on pairs of polarization-entangled photonic qubits we violate by more than 8 standard deviations a Bell-like correlation inequality that is valid for all sets of two-outcome measurements in any dimension. We combine this with a device-independent verification that the system is best described by two qubits, which therefore constitutes the first device-independent certification of a nonprojective quantum measurement.
Numerous quantum information protocols make use of maximally entangled two-particle states, or Bell states, in which information is stored in the correlations between the two particles rather than their individual properties. Retrieving information stored in this way means distinguishing between different Bell states, yet the well known no-go theorem establishes that projective linear evolution and local measurement (LELM) detection schemes can only reliably distinguish three of the four qubit Bell states. We establish maximum distinguishability of the qutrit Bell states of bosons via projective LELM measurements; only three of the nine Bell states can be distinguished. Next, we extend to the case of non-projective measurements. We strengthen the no-go theorem by showing that general LELM measurements cannot reliably distinguish all four qubit Bell states. We also establish that at most five qutrit Bell states can be distinguished with generalized LELM measurements.
Among the surprising features of quantum measurements, the problem of distinguishing and antidistinguishing general quantum measurements is fundamentally appealing. Unlike classical systems, quantum theory offers entangled states and peculiar state update rule of the postmeasurement state, which gives rise to four distinct scenarios for distinguishing (and antidistinguishing) quantum measurements: (i) probing single systems and without access to the postmeasurement states, (ii) probing entangled systems and without access to the postmeasurement states, (iii) probing single systems with access to the postmeasurement states, and (iv) probing entangled systems with access to the postmeasurement states. In these scenarios, we consider the probability of distinguishing (and antidistinguishing) quantum measurements sampled from a given set in the single-shot regime. For some scenarios, we provide the analytical expressions of distinguishability (and antidistinguishability) for qubit projective measurements. We show that the distinguishability of any pair of qubit projective measurements in scenario (iii) is always greater than its values in scenario (ii). Interestingly, we find certain pairs of qubit nonprojective measurements for which the optimal distinguishability in scenario (ii) is achieved using a nonmaximally entangled state. It turns out that, for any set of measurements, distinguishability (and antidistinguishability) in scenario (i) is always less than or equal to in any other scenario, while it reaches its highest possible value in scenario (iv). We establish that the relations form a strict hierarchy, and there is no hierarchical relation between scenarios (ii) and (iii). In particular, we introduce different variants of the well-known ``trine'' qubit measurement to construct pairs (and triples) of qubit quantum measurements such that they are perfectly distinguishable (and antidistinguishable) in scenario (ii) but not in scenario (iii) and vice versa. Additionally, we present qubit measurements that are perfectly distinguishable (and antidistinguishable) in scenario (iv) but not in any other scenario.