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Journal of Physics A Mathematical and Theoretical

Relating EPR steering with the fidelity of quantum teleportation for two- and three-qubit states

Yi Fan, Liang Qiu, Chuan-Lei Jia, Yiyue Gu

Published Apr 5, 2023• 3 citations
Publisher

Abstract

Abstract Einstein–Podolsky–Rosen (EPR) steering is an important quantum resource in quantum information and computation. The relationships between EPR steering and the fidelity of quantum teleportation for two- and three-qubit states are investigated in this paper. For two-qubit states, the upper bound of the average teleportation fidelity based on the EPR steering is derived. When the receiver can only perform the identity or the Pauli rotation operations, the two-qubit X-type states which violate the three-setting measurements steering inequality could be used for teleportation. In the end, the steering observables and the average teleportation fidelities of two-qubit reduced states for three-qubit pure states maintain the same ordering. The complementary relations between the steering observables and the average teleportation fidelities for three-qubit pure states are also established.

DOI
10.1088/1751-8121/accabe

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Abstract Einstein–Podolsky–Rosen (EPR) steering is an important quantum resource in quantum information and computation. The relationships between EPR steering and the fidelity of quantum teleportation for two- and three-qubit states are investigated in this paper. For two-qubit states, the upper bound of the average teleportation fidelity based on the EPR steering is derived. When the receiver can only perform the identity or the Pauli rotation operations, the two-qubit X-type states which violate the three-setting measurements steering inequality could be used for teleportation. In the end, the steering observables and the average teleportation fidelities of two-qubit reduced states for three-qubit pure states maintain the same ordering. The complementary relations between the steering observables and the average teleportation fidelities for three-qubit pure states are also established.
EPR steering is an important quantum resource in quantum information and computation. In this paper, its applications in quantum teleportation are investigated. First of all, the upper bound of the average teleportation fidelity based on the EPR steering is derived. When the receiver can only perform the identity or the Pauli rotation operations, the X-type states which violate the three-setting linear steering inequality could be used for teleportation. In the end, the steering observables and the average teleportation fidelities of the two-qubit reduced states for three-qubit pure states maintain the same ordering. The complementary relations between the steerable observables and the average teleportation fidelities for three-qubit pure states are also established.
The demonstration of quantum teleportation of a photonic qubit from Alice to Bob usually relies on data conditioned on detection at Bob's location. I show that Bohm's Einstein-Podolsky-Rosen (EPR) paradox can be used to verify that the quantum benchmark for qubit teleportation has been reached, without postselection. This is possible for scenarios insensitive to losses at the generation station, and with efficiencies of ${\ensuremath{\eta}}_{B}>1/3$ for the teleportation process. The benchmark is obtained if it is shown that Bob can ``steer'' Alice's record of the qubit as stored by Charlie. EPR steering inequalities involving mmm measurement settings can also be used to confirm quantum teleportation, for efficiencies ${\ensuremath{\eta}}_{B}>1/m$, if one assumes trusted detectors for Charlie and Alice. Using proofs of monogamy, I show that two-setting EPR steering inequalities can signify secure teleportation of the qubit state.
Einstein-Podolsky-Rosen (EPR) steering refers to the type of correlations described in the EPR paradox, where one observer seems to affect (”steer”) the state of other observer by using local measurements. There have been several works regarding characterization and quantification of EPR steering. One characteristic of this non-local correlation is that it can be asymmetric, while entanglement is symmetric. This asymmetric property is relevant for potential applications of EPR steering to quantum information, in particular to quantum cryptography and quantum teleportation. This latter refers to the process where one observer sends an unknown quantum state to Bob, who is in a different location. They communicate by classical means. Here we will show that EPR steering is a necessary resource to obtain secure continuous variable teleportation. We will also consider NOON states, which is an example of an entangled state. For this state, we will present a steering signature. This contribution reviews the work derived in Refs. [1] and [2], which was presented as an invited talk in ELAF 2017.
Quantum nonlocality is an essential resource in quantum information and is commonly classified into three distinct forms: quantum entanglement, Einstein-Podolsky-Rosen (EPR) steering, and Bell's nonlocality. Gisin's theorem shows that pure-state entanglement implies Bell nonlocality, and it motivates the question of how entanglement and EPR steering are related beyond pure states. Here we prove that every entangled rank-2 two-qubit state is EPR steerable, with rank-1 states included as a special case. The proof combines a local-unitary parametrization of rank-2 two-qubit states, a necessary and sufficient separability condition, and a state-dependent nonlinear steering inequality. This gives an analytical characterization of EPR steerability for rank-2 two-qubit entangled states. Thus, rank-2 two-qubit entangled states provide certifiable EPR-steering resources in quantum information.
Collective Einstein-Podolsky-Rosen (EPR) steering is a unique multipartite resource that enables the realization of more secure communication protocols. However, due to the lack of suitable experimental criteria, research on collective EPR steering has so far been limited to continuous variable systems. Here, we employ the coherence-based collective EPR steering criteria proposed by Mondal et al. [Phys. Rev. A 99, 012312 (2019)] and develop a semidefinite programming method to find the optimal measurement settings. Experimentally, we take the three-qubit generalized Greenberger-Horne-Zeilinger states and generalized WWW states as examples, demonstrating that collective EPR steering can be observed by detecting the average l1{l}_{1}l1​ norm of coherence of the steered subsystem. We also explored the effect of measurement settings on complementarity relations between different collective steering criteria, and established a connection between collective steering and coherence complementarity. Our paper fills a gap in the verification of collective steerability in discrete variable systems and reveals a profound connection between EPR steering and quantum coherence.
Abstract Einstein-Podolsky-Rosen (EPR) steering is a quantum nonlocality between entanglement and Bell’s nonlocality emphasizing the incompatibility of the local-hidden-state (LHS) model with quantum theory. It is well established that the EPR steering paradox is an essential method for determining steering, which reveals the nature of steering by logical contradiction. However, previous studies on the EPR steering paradox have only examined cases where the conditional states of the steered party are pure. In this work, we present the EPR steering paradox about the case when the conditional states of the steered party are mixed, which can be expressed as a paradoxical equality ‘ 2 Q = 2 − δ C ’ 0 < δ < 1 given by quantum (Q) and classical (C) theories. For any N-qubit state, in the two-setting steering protocol, the contradiction ‘ 2 Q = 2 − δ C ’ can be obtained if and only if a particular set of projective measurements of the steering party makes all the conditional states of the steered party pure, which also implies that the N-qubit state is steerable. Moreover, considering the conditional states of the steered party as mixed states, we are able to identify the full range of steerable states detectable through the EPR steering paradox method. This also means that, so far, one has found all the steerable states that can be recognized by the EPR steering paradox, which is significant for some typical quantum schemes such as quantum teleportation and quantum key distribution.