A formalism for steering with local quantum measurements

Abstract

We develop a unified approach to classical, quantum and post-quantum steering. The framework is based on uncharacterised (black-box) parties performing quantum measurements on their share of a (possibly unphysical) quantum state, and its starting point is the characterisation of general no-signalling assemblages via non-positive local hidden-state models, which will be defined in this work. By developing a connection to entanglement witnesses, this formalism allows for new definitions of families of assemblages, in particular via (i) non-decomposable positive maps and (ii) unextendible product bases. The former proves to be useful for constructing post-quantum assemblages with the built-in feature of yielding only quantum correlations in Bell experiments, while the latter always gives certifiably post-quantum assemblages. Finally, our framework is equipped with an inherent quantifier of post-quantum steering, which we call the negativity of post-quantum steering. We postulate that post-quantum steering should not increase under one-way quantum operations from the steered parties to the steering parties, and we show that, in this sense, the negativity of post-quantum steering is a convex post-quantum-steering monotone.The concept of steering was first introduced by Schrödinger in 1935 [1] in response to the Einstein et al paradox [2]. It refers to the phenomenon where one party, Alice, by performing measurements on one part of a shared system, seemingly remotely 'steers' the state of the system held by a distant party, Bob, in a way which has no explanation in terms of local causal influences. Steering has only recently been formally defined in a quantum information-theoretic setting [3], as a way of certifying the entanglement of quantum systems without the need to trust one of the parties, or when one of the parties is using uncharacterised devices. In this setting, the uncharacterised party convinces the other party that they shared entanglement by demonstrating steering. Furthermore, if all parties are uncharacterised (or untrusted) then one recovers the device-independent setting of a standard Bell test. Steering thus may be seen as one in a family of non-classical phenomena, closely related to entanglement and Bell non-locality [4]. Indeed, Bell nonlocality implies steering, and steering implies entanglement, however all three concepts are inequivalent [3,5].It is well-known that, in spite of demonstrating non-locality, local measurements on entangled quantum systems cannot be used to communicate superluminally. That is, correlations that are generated by varying the choice of local measurements on space-like separated quantum subsystem-which we define to be quantum correlations-satisfy the principle of no-signalling. We will call no-signalling colleations all correlations that do not permit signalling. One can conceive of no-signalling correlations that cannot be realised by local measurements on quantum states, hence called post-quantum correlations; this possibility was first pointed out in a seminal work...

Quantum steering

“…the state is steerable (Zhen et al, 2016). This elegant inequality resembles the familiar computable cross norm or realignment (CCNR) entanglement criterion (Chen and Wu, 2003;Rudolph, 2005), where k λ k > 1 implies that the state is entangled. Note that the steering inequalities (42) and (43) are different from various inequalities discussed in Section II.A in the sense that they exploit the full information about the state.…”

“…Note that similar bounds on sums of squared mean values are known for many cases of anticommuting observables or mutually unbiased bases (Tóth and Gühne, 2005;Wehner and Winter, 2010;Wu et al, 2009), so one can directly generalize the criteria from above to broader classes of observables on Bob's side (Evans et al, 2013). With mutually unbiased bases as a generalization of the Pauli matrices one can even find steering inequalities with an unbounded violation (Marciniak et al, 2015;Rutkowski et al, 2017).…”

“…. Then, the computable cross-norm or realignment (CCNR) criterion states that if AB is separable, then the trace norm is bounded by one, Λ 1 ≤ 1 (Chen and Wu, 2003;Rudolph, 2005), see also Section II.C.3. This criterion has already been used to detect entanglement with uncharacterized devices, if the dimension is known: If Alice and Bob make uncharacterized measurements A k and B l they can build the expectation value matrix ∆ kl = Tr( AB A k ⊗B l ) and, using the dimension assumption, from that estimate the trace norm Λ 1 (Moroder and Gittsovich, 2012).…”

The Einstein–Podolsky–Rosen Steering and Its Certification

“…This EPR steering definition is equivalent to Definition 1 on the condition that Bob is allowed to do the state tomography for each conditional state . Furthermore, in Reference [ 54 ] post-quantum steering is well-studied using no-signaling assemblages. If Bob’s measurements are not sufficient to do the tomography, then it is hard for him to obtain each yet to verify the EPR steerability.…”

“…In particular, the characterization of EPR steering is reviewed through the semidefinite programming method [ 28 ], which can be explicitly used to tackle the complicated numerical problems in detecting EPR steering. Recently, the EPR steering test is further generalized to a unified framework where classical, quantum, and post-quantum steering can be investigated [ 54 ]. The black box framework in that paper is the same with the framework adopted here.…”