Fine-grained Einstein-Podolsky-Rosen–steering inequalities

Abstract

We derive a new steering inequality based on a fine-grained uncertainty relation to capture EPRsteering for bipartite systems. Our steering inequality improves over previously known ones since it can experimentally detect all steerable two-qubit Werner state with only two measurement settings on each side. According to our inequality, pure entangle states are maximally steerable. Moreover, by slightly changing the setting, we can express the amount of violation of our inequality as a function of their violation of the CHSH inequality. Finally, we prove that the amount of violation of our steering inequality is, up to a constant factor, a lower bound on the key rate of a one-sided device independent quantum key distribution protocol secure against individual attacks. To show this result, we first derive a monogamy relation for our steering inequality. The development of quantum information led to distinguish three forms of non-local correlations in quantum physics [1][2][3][4][5][6]. These are entanglement, steering and Bell non-local correlations. Einstein, Podolsky and Rosen (EPR) introduced entangled quantum states in an attempt to show the incompleteness of quantum physics known as the EPR paradox [1]. The same year, Schrödinger re-expressed the EPR paradox as the possibility of steering (more generally, known as EPRsteering), i.e., when Alice and Bob share an entangled state, Alice can affect Bob's state throught her own measurement. More precisely, a state exhibits EPR-steering if it cannot be modeled as Bob holding an unknown yet definite state, a description known as a local hidden state (LHS) model [4]. Bell-type inequalities can be used to rule out local hidden variable (LHV) models. Similarly, steering inequalities are used to rule out the existence of LHS model and thus, demonstrate steerability.According to Wiseman, Jones and Doherty, the three forms of non-local correlations are also tightly related to the experimental settings required to test them [4]. To test entanglement, both parties need to trust that they perform quantum operations and also trust their measurement devices. In the case of EPR-steering, only one party assumes that he applies a quantum measurement and that his device is not controlled by a third party. Finally, Bell non-locality can be be tested without assuming quantum theory and trusting measurement devices. This leads to a hierarchy in which EPR-steering lies between Bell non-locality and entanglement.Experimental demonstration of Bell's non-locality has been achieved by several experiments [7]. To test EPRsteering, Reid proposed a testable formulation for continuous variable systems based on the position-momentum uncertainty relation [5]. Denote (X, P x ) and (Y, P y ) the position and corresponding momentum of two correlated modes. According to the Reid criterion, one needs to infer the uncertainty (measured by the standard deviation) of the quadrature amplitude X θk = cos[θk]X + sin[θk]P x for k ∈ {1, 2} from the measurement outcome of the correlated amplitude Y φk = cos[φ...

Nonlocal advantage of quantum coherence

“…Intuitively, for quantum systems, it may seem that all steerable states can achieve the non-local advantage on quantum coherence. But here we show that for mixed states, steerability captured by different steering criteria [8][9][10] based on uncertainty relations are drastically different from the steerability captured by coherence. In other words, we show that there are steerable states, which cannot achieve the non-local advantage of quantum coherence.…”

“…Therefore, Alice's task is to convince Bob that the prepared state is indeed entangled and they share non-local correlation. On the other hand, Bob thinks that Alice may cheat by preparing the system B in a single quantum system, on the basis of possible strategies [8,9]. Bob agrees with Alice that the prepared state is entangled and they share non-local correlation if and only if the state of Bob cannot be written by local hidden state model (LHS) [3] …”

“…Several steering conditions have been derived on the basis of Eq. (2) and the existence of single system description of a part of the bi-partite systems [8][9][10]. It has also been quantified for two-qubit systems [11].…”

Preservation of a lower bound of quantum secret key rate in the presence of decoherence

“…More specifically, we discuss a way to protect the one-sided device independent quantum key distribution (1s-DIQKD) protocol [28] when the system interacts with the environment modeled by ADC. Comparing the preservation of 1s-DIQKD with the preservation of the fidelity of quantum teleportation, we observe that ADC cannot improve the optimal secret key rate in 1s-DIQKD, which is derived using the steering inequality [29] based on the fine-grained uncertainty relation [30], though it can improve the teleportation fidelity for states having teleportation fidelity just below the quantum region [11,12]. We show that improvement of the key rate becomes possible using the technique of weak measurement and its reversal, which may be used to suppress the effect of the amplitude damping decoherence [14][15][16][17][18][19].…”

Geometric Bell-like inequalities for steering

“…There are only few steering inequalities known. For example, linear steering inequality [3,4], steering inequality due to an "All-Versus-Nothing" argument [5] , steering inequality from uncertainty principle [6], and fine-grained uncertainty relation [7]. In this work we address the problem of relation of Bell inequalities with steering.…”