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Zhongguo kexue. Wulixue Lixue Tianwenxue

Detecting ${\boldsymbol~A}{\boldsymbol~B}{\boldsymbol~\rightarrow}{\boldsymbol~C}$ steering in tripartite quantum systems

Cao Huai-xin, Yang Ying, XIAO Shu, LIU Jie

Published Nov 7, 2019• 2 citations
Publisher

Abstract

In this paper, the detection problem of $AB\\rightarrow~C$ steering is discussed. Firstly, we give some relationships between the $AB\\rightarrow~C$ steering of $\\rho^{ABC}$ and $A\\rightarrow~C$ steering of the reduced state $\\rho^{AC}$ as well as $B\\rightarrow~C$ steering of the reduced state $\\rho^{BC}$ based ontheir definitions. Secondly, by deducing an inequality (called an steering inequality) for $AB\\rightarrow~C$ unsteerable states, a necessary condition for a quantum state to be $AB\\rightarrow~C$ unsteerable is obtained. Furthermore, the steering radius $S(\\rho)$ of a state $\\rho$ is defined and it is proved that when $\\rho$ is $AB\\rightarrow~C$ unsteerable, $S(\\rho)\\le1$ and that when $S(\\rho)>1$, $t\\rho+(1-t)\\eta$ is $AB\\rightarrow~C$ steerable for each $t_0<t\\le1$ and for each $AB\\rightarrow~C$ unsteerable state $\\eta$ where $t_0=\\frac{2}{S(\\rho)+1}$.Finally, $AB\\rightarrow~C$ steerability of the “partial convex combinations of the GHZ state, generalized GHZ state and the W-state with a $AB\\rightarrow~C$ unsteerable state are detected, respectively by using the obtained steering inequality.

DOI
10.1360/sspma-2019-0261

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Quantum steering is a kind of quantum feature between quantum entanglement and Bell non-locality. Due to its special asymmetry, quantum steering plays an important role in many aspects of quantum information processing and thus has been widely concerned. There are many contributions on quantum steering in bipartite systems up to now, while tripartite systems have more complex structures than bipartite systems and then have more diversified steering scenarios. In this paper, two types of quantum steering scenarios are introduced for a tripartite quantum system, named “one-sided device-independent steering" and “two-sided device-independent steering". Each of them contains two levels according to the separability of the local hidden states and the independence of the local hidden variables, respectively. Based on giving the mathematical definitions of these steering scenarios, some necessary and sufficient conditions for a state to be unsteerable are obtained and sufficient conditions for a state to be steerable are also established.

Zhongguo kexue. Wulixue Lixue Tianwenxue•Aug 31, 2018•7 citations
Detecting  A B → C{\boldsymbol~A}{\boldsymbol~B}{\boldsymbol~\rightarrow}{\boldsymbol~C} A B → C steering in tripartite quantum systems
In this paper, the detection problem of ABrightarrow CAB\\rightarrow~CABrightarrow C steering is discussed. Firstly, we give some relationships between the ABrightarrow CAB\\rightarrow~CABrightarrow C steering of rhoABC\\rho^{ABC}rhoABC and Arightarrow CA\\rightarrow~CArightarrow C steering of the reduced state rhoAC\\rho^{AC}rhoAC as well as Brightarrow CB\\rightarrow~CBrightarrow C steering of the reduced state rhoBC\\rho^{BC}rhoBC based ontheir definitions. Secondly, by deducing an inequality (called an steering inequality) for ABrightarrow CAB\\rightarrow~CABrightarrow C unsteerable states, a necessary condition for a quantum state to be ABrightarrow CAB\\rightarrow~CABrightarrow C unsteerable is obtained. Furthermore, the steering radius S(rho)S(\\rho)S(rho) of a state rho\\rhorho is defined and it is proved that when rho\\rhorho is ABrightarrow CAB\\rightarrow~CABrightarrow C unsteerable, S(rho)le1S(\\rho)\\le1S(rho)le1 and that when S(rho)>1S(\\rho)>1S(rho)>1, trho+(1−t)etat\\rho+(1-t)\\etatrho+(1−t)eta is ABrightarrow CAB\\rightarrow~CABrightarrow C steerable for each t0<tle1t_0<t\\le1t0​<tle1 and for each ABrightarrow CAB\\rightarrow~CABrightarrow C unsteerable state eta\\etaeta where t0=frac2S(rho)+1t_0=\\frac{2}{S(\\rho)+1}t0​=frac2S(rho)+1.Finally, ABrightarrow CAB\\rightarrow~CABrightarrow C steerability of the “partial convex combinations of the GHZ state, generalized GHZ state and the W-state with a ABrightarrow CAB\\rightarrow~CABrightarrow C unsteerable state are detected, respectively by using the obtained steering inequality.
Quantum steering, also called Einstein-Podolsky-Rosen steering, is the intriguing phenomenon associated with the ability of spatially separated observers to steer---by means of local measurements---the set of conditional quantum states accessible by a distant party. In the light of quantum information, all steerable quantum states are known to be resources for quantum information processing tasks. Here, via a quantity dubbed steering fraction, we derive a simple, but general criterion that allows one to identify quantum states that can exhibit quantum steering (without having to optimize over the measurements performed by each party), thus making an important step towards the characterization of steerable quantum states. The criterion, in turn, also provides upper bounds on the largest steering-inequality violation achievable by arbitrary finite-dimensional maximally entangled states. For the quantification of steerability, we prove that a strengthened version of the steering fraction is a convex steering monotone and demonstrate how it is related to two other steering monotones, namely, steerable weight and steering robustness. Using these tools, we further demonstrate the superactivation of steerability for a well-known family of entangled quantum states, i.e., we show how the steerability of certain entangled, but unsteerable quantum states can be recovered by allowing joint measurements on multiple copies of the same state. In particular, our approach allows one to explicitly construct a steering inequality to manifest this phenomenon. Finally, we prove that there exist examples of quantum states (including some which are unsteerable under projective measurements) whose steering-inequality violation can be arbitrarily amplified by allowing joint measurements on as little as three copies of the same state. For completeness, we also demonstrate how the largest steering-inequality violation can be used to bound the largest Bell-inequality violation and derive, analogously, a simple sufficient condition for Bell nonlocality from the latter.
Quantum steering can be detected via the violation of steering inequalities, which provide sufficient conditions for the steerability of quantum states. Here we discuss the converse problem, namely, ensuring that an entangled state is unsteerable and hence Bell local. We present a simple criterion, applicable to any two-qubit state, that guarantees that the state admits a local hidden state model for arbitrary projective measurements. Specifically, we construct local hidden state models for a large class of entangled states, which thus cannot violate any steering or Bell inequality. In turn, this leads to sufficient conditions for a state to be only one-way steerable and provides the simplest possible example of one-way steering. Finally, by exploiting the connection between steering and measurement incompatibility, we give a sufficient criterion for a continuous set of qubit measurements to be jointly measurable.
EPR steering is a form of bipartite quantum correlation that is intermediate between entanglement and Bell nonlocality. It is detected via the violation of steering inequalities, which provide sufficient conditions for the steerability of quantum states. Furthermore, there is a sufficient criterion for guaranteeing that a two-qubit state is unsteerable. But there are some two qubit states that can not determine the steerability of the steering inequality and the sufficient criteria. Here we introduce a way of quantifying this phenomenon and use it to study the steerability of two qubit states. This article recommend a platonic solid measurement schemes. And then discuss the problem, namely expanding the scope of steerable, which guarantees that the state is steerable.
Einstein-Podolsky-Rosen (EPR) steering is the ability that an observer persuades a distant observer to share entanglement by making local measurements. Determining a quantum state is steerable or unsteerable remains an open problem. Here, we derive a new steering inequality with infinite measurements corresponding to an arbitrary two-qubit T state, from consideration of EPR steering inequalities with N projective measurement settings for each side. In fact, the steering inequality is also a sufficient criterion for guaranteering that the T state is unsteerable. Hence, the steering inequality can be viewed as a necessary and sufficient criterion to distinguish whether the T state is steerable or unsteerable. In order to reveal the fact that the set composed of steerable states is the strict subset of the set made up of entangled states, we prove theoretically that all separable T states can not violate the steering inequality. Moreover, we put forward a method to estimate the maximum violation from concurrence for arbitrary two-qubit T states, which indicates that the T state is steerable if its concurrence exceeds 1/4.
Einstein-Podolsky-Rosen steering is a kind of powerful nonlocal quantum resource in quantum information processing such as quantum cryptography and quantum communication. Many criteria have been proposed in the past few years to detect steerability, both analytically and numerically, for bipartite quantum systems. We propose effective criteria for tripartite steerability and genuine tripartite steerability of three-qubit quantum states by establishing connections between the tripartite steerability (resp. genuine tripartite steerability) and the tripartite entanglement (resp. genuine tripartite entanglement) of certain corresponding quantum states. From these connections, tripartite steerability and genuine tripartite steerability can be detected without using any steering inequalities. The "complex cost" of determining tripartite steering and genuine tripartite steering can be reduced by detecting the entanglement of the newly constructed states in the experiment. Detailed examples are given to illustrate the power of our criteria in detecting the (genuine) tripartite steerability of tripartite states.
Quantum steering is a kind of quantum feature between quantum entanglement and Bell non-locality. Due to its special asymmetry, quantum steering plays an important role in many aspects of quantum information processing and thus has been widely concerned. There are many contributions on quantum steering in bipartite systems up to now, while tripartite systems have more complex structures than bipartite systems and then have more diversified steering scenarios. In this paper, two types of quantum steering scenarios are introduced for a tripartite quantum system, named “one-sided device-independent steering" and “two-sided device-independent steering". Each of them contains two levels according to the separability of the local hidden states and the independence of the local hidden variables, respectively. Based on giving the mathematical definitions of these steering scenarios, some necessary and sufficient conditions for a state to be unsteerable are obtained and sufficient conditions for a state to be steerable are also established.