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Physical Review A

Steering sharing for a two-qubit system via weak measurements

Dan Yao, Changliang Ren

Published May 17, 2021• 16 citations
Publisher

Abstract

We aim to explore the scenario that a single Alice steers multiple Bobs by using weak measurements. The multiple violations of two different Einstein-Podolsky-Rosen steering inequalities were discussed. Based on unbiased inputs, a single Alice can steer two Bobs when the double violation of the Clauser-Horne-Shimony-Holt-like inequality is realized. Meanwhile, three Bobs can be steered by a single Alice when the triple violation of the three-settings linear steering inequality is observed. More importantly, we theoretically inferred the analytical formula of the maximal violation for the corresponding inequality, which applied an arbitrary two-qubit state.

DOI
10.1103/physreva.103.052207

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We aim to explore the scenario that a single Alice steers multiple Bobs by using weak measurements. The multiple violations of two different Einstein-Podolsky-Rosen steering inequalities were discussed. Based on unbiased inputs, a single Alice can steer two Bobs when the double violation of the Clauser-Horne-Shimony-Holt-like inequality is realized. Meanwhile, three Bobs can be steered by a single Alice when the triple violation of the three-settings linear steering inequality is observed. More importantly, we theoretically inferred the analytical formula of the maximal violation for the corresponding inequality, which applied an arbitrary two-qubit state.
We propose a scheme for the sharing of quantum steering among three observers, Alice, Bob, and Charlie, using standard projective measurements. We show that in the unilateral sequential scenario, Alice can steer Bob's and Charlie's states and conversely, Bob and Charlie can steer Alice's state. Unlike the quantum steering sharing achieved through weak measurements, we use the standard projective measurements to enable quantum steering sharing. Quantum steering is demonstrated by the violations of the linear steering inequality among different observer combinations. We find that Alice can simultaneously steer both Bob's and Charlie's states, and Bob and Charlie can simultaneously steer Alice's state, regardless of whether they are in maximally entangled states or partially entangled states. The maximum double violation of the linear steering inequalities obtained from partially entangled states may not be greater in some cases than that obtained from maximally entangled states. Additionally, we verify hybrid quantum correlation sharing through the double violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality and the linear steering inequality. Our results provide an alternative perspective for studying quantum steering and hybrid quantum correlation and may lead to applications in quantum random access code, randomness certification, and self-testing processes.
Alice and Bob each have half of a pair of entangled qubits. Bob measures his half and then passes his qubit to a second Bob who measures again and so on. The goal is to maximize the number of Bobs that can have an expected violation of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality with the single Alice. This scenario was introduced in [R. Silva et al., Phys. Rev. Lett. 114, 250401 (2015)PRLTAO0031-900710.1103/PhysRevLett.114.250401] where the authors mentioned evidence that when the Bobs act independently and with unbiased inputs then at most two of them can expect to violate the CHSH inequality with Alice. Here we show that, contrary to this evidence, arbitrarily many independent Bobs can have an expected CHSH violation with the single Alice. Our proof is constructive and our measurement strategies can be generalized to work with a larger class of two-qubit states that includes all pure entangled two-qubit states. Since violation of a Bell inequality is necessary for device-independent tasks, our work represents a step towards an eventual understanding of the limitations on how much device-independent randomness can be robustly generated from a single pair of qubits.
We study a recently proposed Einstein–Podolsky–Rosen steering inequality (Cavalcanti et al 2015 J. Opt. Soc. Am. B 32 A74–A81 ). Analogous to Clauser–Horne–Shimony–Holt (CHSH) inequality for Bell nonlocality, in the simplest scenario, i.e., two parties, two measurements per party and two outcomes per measurement, this newly proposed inequality has been proved to be necessary and sufficient for steering. In this article we find the optimal violation amount of this inequality in quantum theory. Interestingly, the optimal violation amount matches with optimal quantum violation of CHSH inequality, i.e., Cirel'son quantity. We further study the optimal violation of this inequality for different classes of 2-qubit quantum states.
A well-known property of quantum nonlocality is monogamy. However, recent research by Silva et al. shows that multiple observers can share the nonlocality by using weak measurements [Phys. Rev. Lett. 114, 250401 (2015)]. In this paper, we deeply discuss nonlocality sharing for an arbitrary two-qubit state via weak measurements. Two types of nonlocality sharing, passive nonlocality sharing and active nonlocality sharing, which is another type of double violation of Clauser-Horne-Shimony-Holt (CHSH) inequality with many counterintuitive results existing, are fully analyzed. The two types are quite distinct. Active nonlocality sharing can be observed when Bob1 performs strong measurements only except for the ideal strong measurements. Also, a double violation still can be observed in the active nonlocality sharing even using a square pointer type for weak measurements of Bob1, which is impossible in the passive one. As a special example, for a Bell state, a double violation of Clauser-Horne-Shimony-Holt inequality can be always observed in a broad range whether the weak measurement of Bob1 is with an optimal pointer or a square pointer.
We address the recently posed question as to whether the nonlocality of a single member of an entangled pair of spin 1 / 2 particles can be shared among multiple observers on the other wing who act sequentially and independently of each other. We first show that the optimality condition for the trade-off between information gain and disturbance in the context of weak or non-ideal measurements emerges naturally when one employs a one-parameter class of positive operator valued measures (POVMs). Using this formalism we then prove analytically that it is impossible to obtain violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality by more than two Bobs in one of the two wings using unbiased input settings with an Alice in the other wing.
We derive an inequality that is necessary and sufficient to show Einstein-Podolsky-Rosen (EPR) steering in a scenario employing only correlations between two arbitrary dichotomic measurements on each party. Thus the inequality is a complete steering analogy of the Clauser-Horne-Shimony-Holt (CHSH) inequality, a generalization of the result of Cavalcanti et al. [E. G. Cavalcanti, C. J. Foster, M. Fuwa, and H. M. Wiseman, JOSA B 32, A74 (2015)]. We show that violation of the inequality only requires measuring over equivalence classes of mutually unbiased measurements on the trusted party and that in fact assuming a general two-qubit system arbitrary pairs of distinct projective measurements at the trusted party are equally useful. Via this it is found that for a given state the maximum violation of our EPR-steering inequality is equal to that for the CHSH inequality, so all states that are EPR steerable with CHSH-type correlations are also Bell nonlocal.