University of Jyväskylä Team Presents Confidence-Based Protocol for Nonlocality Without Entanglement

A new pathway for certifying nonlocality without entanglement has been revealed through a refined analysis of measurement confidence, achieved by Hanwool Lee and Joonwoo Bae at University of Jyväskylä, in collaboration with KAIST. Lee and Bae show that using global measurements sharply improves the discrimination of separable quantum states compared to separable measurements, offering a new approach to demonstrating nonlocality without entanglement. The method is a semi-device-independent certification, verifying global measurements by assessing confidence in measurement outcomes, and a key feature is that experimental demonstration of this phenomenon is now feasible with current quantum technology, even when accounting for imperfect detection efficiencies.

Demonstrating nonlocality via measurement outcomes independent of quantum entanglement

Quantum information processing encompasses states and measurements exceeding classical capabilities. Entanglement has long been considered a vital resource for achieving quantum advantages. It is now feasible to experimentally demonstrate nonlocality without entanglement (NLWE) using current quantum measurement devices, even with imperfect detection efficiencies, as maximum-confidence measurements depend solely on detected outcomes. Methods for verifying entanglement find applications in quantum information, including entanglement witnesses for secure communication, quantum steering for one-sided quantum key distribution, and Bell nonlocality for device-independent quantum information processing.

Quantum measurements performed in an entangled basis can accomplish information tasks beyond the limits of local operations and classical communication (LOCC), a phenomenon known as NLWE. Recently, NLWE certification presented results in a device-independent manner, closely linked to standard entanglement characterisation methods like entanglement witnesses. Minimum-error discrimination, in particular, has demonstrated NLWE. The distinction between GLOBAL and LOCC is illustrated by the guessing probability for ensembles of states, such as double-trine states, where GLOBAL achieves a strictly higher probability than LOCC. These ensembles showcase NLWE. In unambiguous discrimination, a gap exists between GLOBAL and separable measurements (SEP), expressed in terms of the rate of inconclusive outcomes; this rate can be lower with GLOBAL than with SEP. However, these discrimination strategies are often impractical in real-world scenarios with imperfect measurements, noise, and undetected events, potentially hindering the application of NLWE to information technologies.

When applying fundamental results to practical tasks, accounting for experimental noise is crucial. For example, closing loopholes in Bell nonlocality due to noisy detectors is essential for device-independent quantum information applications. This work establishes a framework for both demonstrating and certifying NLWE in practical scenarios where measurement imperfections may be present, allowing certification even for unknown and untrusted measurements.

To demonstrate NLWE, a maximum-confidence measurement (MCM) strategy is exploited, considering only detected events as measurement outcomes. MCMs also generalise minimum-error and unambiguous discrimination strategies. The certification of NLWE can verify that unknown measurements are in GLOBAL but not in SEP, corresponding to a semi-device-independent (sDI) certification. Considering an ensemble of antiparallel qubit states, the framework investigates measurements in GLOBAL and SEP or LOCC using MCMs. GLOBAL yields a higher confidence in a guessing task than SEP or LOCC, thus demonstrating NLWE. An MCM finds confidence in individual detection events, revealing the distinction between GLOBAL and SEP in a fine-grained guessing task encompassing minimum-error and unambiguous state discrimination.

Consequently, it is possible to certify NLWE using only detection events, even with untrusted or unknown measurement devices. The framework for certifying NLWE applies to measurement devices with imperfections, including undetected outcomes due to photon loss or low detection efficiency. A quantum measurement is generally formulated as a prediction allowing computation of outcome probabilities given an ensemble of states. Conversely, an MCM provides retrodictive capability, identifying the most probable state in an ensemble given a measurement outcome.

For an ensemble of states, S = {qx, ρx}, the MCM strategy optimizes POVM elements to maximise the conditional probability, or confidence, Cx,max:= max M pP |M(x|x) = max Mx≥0 qxtr[ρxMx] try[ρMx]. For instance, Cx,max = 1 if an outcome x implies the preparation of state ρx with certainty, or Cx,max = qx if no measurement is optimal. The optimisation problem can be linearized or formulated as a semidefinite program (SDP). Maximum-confidence discrimination, encompassing minimum-error and unambiguous discrimination, can be realised by rank-one POVM elements, where Cx,max = ∥√ρ−1qxρx√ρ−1∥∞ with an operator norm ∥· ∥∞. Measurements can be constrained to be either global or separable. Let C(G) x,max denote confidence with measurements in global form, and C(S) x,max with measurements in separable form.

Since confidence remains identical under scaling a POVM element, confidence with LOCC equals that with separable measurements; therefore, maximum-confidence measurements do not reveal a distinction between LOCC and separable measurements. The optimality conditions of the SDP can be used to analyse measurements for maximum-confidence discrimination. Reformulating the optimisation problem via the complementarity problem identifies optimal parameters for a POVM element Mx and an ensemble of complementary states {rx, σx}, where rx ≥0, satisfying try[σxMx] = 0 and C(G) x,maxρ −qxρx = rxσx. A POVM element may be constrained to be separable to compute the confidence C(S) x,max.

The optimality conditions demonstrate that, upon finding complementary states, a POVM element can be constructed, a process termed complementarity slackness. SEP is optimal if a product vector |e⟩|f⟩ exists, such that M (S) x = |e⟩⟨e| ⊗|f⟩⟨f|, and try[σxM (S) x ] = 0. In other words, if try[σxM (S) x ] > 0 for all M (S) x ∈SEP, an optimal POVM element is not in SEP but in GLOBAL. Consequently, C(G) x,max > C(S) x,max. A proposition states that in the complementary problem to finding an MCM for an ensemble S, consisting of complementary states {σx}, C(G) x,max = C(S) x,max for an outcome x if and only if a product vector |e⟩|f⟩ exists such that σx|e⟩|f⟩= 0. If no product state satisfies this condition, the ensemble S exhibits a distinction between GLOBAL and SEP in terms of confidence, i.e., C(G) x,max > C(S) x,max.

Applying these general results to an ensemble of two-qubit antiparallel states demonstrates NLWE, i.e., a gap between GLOBAL and SEP, S⊥= {|Ψx⟩}4x=1 where |Ψx⟩= |φx⟩⊗|φ⊥ x ⟩, and {|φx⟩} corresponds to symmetric, informationally complete states |φ1⟩= |0⟩, |φ2⟩= 1/√3|0⟩+ 2/√3|1⟩, and |φ3,4⟩= 1/√3|0⟩+ e±2πi/3 2/√3|1⟩. This ensemble has previously demonstrated NLWE in state estimation; while GLOBAL realizes optimal estimation on antiparallel states, SEP suffices for parallel states, which are related by an anti-unitary operation. Therefore, antiparallel states with GLOBAL can be estimated with higher mutual information than SEP on parallel states. With GLOBAL, C(G) x,max = 1, where a POVM element is given as Mx = ax|φx⟩⟨φx| with |φx⟩= 1/√5(2|φx⟩⊗|φ⊥ x ⟩+ |φ⊥ x ⟩⊗|φx⟩) for some ax ∈(0, 1]. The measurement realizes unambiguous discrimination as ⟨Ψy|φx⟩= 0 for all x = y; the POVM element in the direction of |φx⟩ rules out all states but one |Ψx⟩. Conversely, C(S) x,max 0 and have C(S) x,max = 3/4. Thus, NLWE for antiparallel states in terms of MCMs has been demonstrated.

An MCM for an ensemble of parallel states S∥= {|φx⟩⊗2}4x=1 can be achieved by measurements in SEP. This ensemble can be described on a symmetric subspace implemented with SEP; a POVM element in SEP suffices to find an MC, which is 3/4 for all x. Consequently, an ensemble of parallel states cannot demonstrate distinctions between GLOBAL and SEP. This conclusion aligns with the result that state estimation is more efficient with measurements in GLOBAL, achieving higher state fidelity for antiparallel states than parallel ones. The sDI certification of NLWE is then considered. Global measurements can outperform separable ones when discriminating ensembles of separable states, establishing nonlocality without entanglement based on confidence in a detection event.

Verifying achievable confidence in measurement outcomes can certify global measurements, representing a semi-device-independent certification of this nonlocality. A distinction between global and separable measurements exists when considering the rate of inconclusive outcomes; global measurements can achieve a lower rate. If inconclusive outcomes from unknown measurements are infrequent enough to violate a specific limit, they preclude the possibility of measurements realising unambiguous discrimination.

Specifically, a framework for certifying that unknown measurements belong to global but not separable (SEP) measurements is established. This framework contains three parameters. First, an ensemble of states should be specified S = {qx, ρx}. Second, measurement outcomes are collected and the outcome rates, denoted by {ηx}, are found. For a specified a priori ensemble ρ = Σx qxρx, a positive operator-valued measure (POVM) element Mx exists such that the outcome rate is ηx = try[Mxρ] for each x. The probability of an outcome x given a state ρx, denoted by pM|P (x|x), can be found by considering the list of states prepared and outcomes obtained from measurements, where pM|P (x|x) = try[Mxρx] for a POVM element Mx and a state ρx. Third, confidence can be computed from measurement outcomes and trusted quantum states, called certifiable confidence, as follows: Cx = qx ηx pM|P (x|x). With outcome rates ηx and certifiable confidence Cx achieved experimentally, the certification of nonlocality without entanglement (NLWE) works by finding that certifiable confidence cannot be achieved by measurements in SEP. The semidefinite program (SDP) for certifiable maximum confidence (MC) given an outcome rate is as follows: Cx,max = max qx ηx try[ρxMx] subject to try[ρMx] = ηx and 0 ≤Mx ≤I. Note that compared to an equation maximising confidence by optimising a measurement, the optimisation here focuses on finding the maximum confidence given a rate.

Circumventing entanglement unlocks potential for robust quantum demonstrations

The demonstration of nonlocality without entanglement by David Awschalom and his colleagues offers a potential route around a persistent challenge in quantum physics: the difficulty of creating and maintaining entangled states. While entanglement remains a cornerstone of many quantum technologies, its fragility presents a practical hurdle. This research suggests a path where nonlocality, a key feature of quantum mechanics, can be observed even when entanglement is absent. The ability to definitively show this effect with current technology is significant, despite the limitation that this demonstration relies on examining groups of particles rather than individual ones. By circumventing the need for fragile entangled states, a major obstacle in building quantum devices, the team has simplified experimental setups and broadened the scope of potential applications.

The researchers demonstrated nonlocality without entanglement, achieving this by focusing on maximising confidence in detection events rather than relying on entangled states. This is important because entanglement is often difficult to create and maintain in real-world quantum systems, presenting a significant practical challenge. The study establishes that global measurements perform better than separable ones, allowing for the certification of this nonlocality using existing quantum measurement devices and even with imperfect detection efficiencies. The team’s approach simplifies experimental requirements and offers a means of verifying nonlocality that does not depend on the presence of entanglement.