Optimal quantum violation of ClauserHorneShimonyHolt like steering inequality
Abstract
We study a recently proposed EinsteinPodolskyRosen steering inequality [arXiv 1412.8178 (2014)]. Analogous to ClauserHorneShimonyHolt (CHSH) inequality for Bell nonlocality, in the simplest scenario, i.e., 2 parties, 2 measurements per party and 2 outcomes per measurement, this newly proposed inequality has been proved to be necessary and sufficient for steering. In this article, using an equivalence between measurement incompatibility (non joint measurability) and steering, 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 bipartite quantum states. To our surprise we find that optimal violation amount is different for different qubit pure entangled states, which is not the case for all other existing steering inequalities.
I Introduction
The phenomenal argument by Einstein, Podolsky and Rosen (EPR) in 1935 epr () to demonstrate the incompleteness of quantum mechanics, struck Schrödinger with the concept of ‘steering’ schr (). However, only recently, Wiseman et al. have formalized the concept of steering in the form of a task wisemanprl07 (); wisemanpra07 (). The task of steering can be seen as one’s inability to construct a local hidden variable local hidden state (LHVLHS) model that reproduces a given bipartite correlation. The work of Wiseman et al. has generated an immense interest in the study of this steering phenomenon jevtic2014 (); rudolph2014 (); jevtic12014 (); brunneroneway2014 (); wittman2014 (); wiseman2014 (). On the other hand, the concept of steering has been extended for multipartite case and the idea of npartite genuine mutipartite steering have been explored reidg (). Unlike the two well studied non classical correlations, namely nonlocality wehner2014 () and entanglement horodecki2009 (), there is an inherent asymmetry in the task of steering. This is because in case of steering, on one subsystem (which is being ‘steered’) the statistics must arise out of a valid measurement on a valid quantum state but no such constraint is required for the other subsystem. Here in addition to the simplicity of Bell’s assumptions of local causality one must also perform trusted measurements on one subsystem, whereas the other subsystem need not be trusted wiseman2013 ().
Apart form the foundational interest, the study of steering also finds applications in semi device independent scenario where only one party has trust on his/her quantum device but the other party’s device is untrusted. As a concrete example it has been shown that steering allows for secure quantum key distribution (QKD) when one of the parties cannot trust their devices branciard2012 (). One big advantage in this direction is that such scenarios are experimentally less demanding than fully deviceindependent protocols (where both of the parties distrust their devices) branciard2012 () and, at the same time, require less assumptions than standard quantum cryptographic scenarios Bennett1992 ().
In 1964, Bell sought a way to demonstrate that certain correlations appearing in quantum mechanics are incompatible with the notions of locality and reality aka localrealism (LR), through an inequality involving measurement statistics Bell64 (). A violation of such inequality implies the usefulness of correlations for EPR argument. In 1969, ClauserHorneShimonyHolt (CHSH) proposed a set of simple Bell inequalities which are easy to realize experimentally CHSH69 (). In the same spirit of Bell’s inequality in nonlocality, several steering inequalities (SI) have been proposed reid (); ou (); caval (); wal (); wal1 (), so that a violation of any such steering inequalities can render a correlation to be steerable. But an unavoidable hindrance to formalize such SIs follows from the fact that steering scenario is deviceindependent only on oneside.
Recently Cavalcanti et al. have proposed a CHSHlike inequality for quantum steering caval15 (). They have derived an EPRsteering inequality that is necessary and sufficient for a set of correlations in the simplest twoparty scenario involving two measurement settings per site and two outcomes per measurement, with mutually unbiased measurements at trusted party. In this article we derived the tight optimal quantum violation of the EPRsteering inequality proposed in caval15 (). We have also studied the violation amount of this inequality for different wellknown classes of qubit states. Interestingly, we find that for different qubit pure entangled states the optimal violation amount of this inequality differ. Note that, for all other existing SIs such difference is not observed.
The organization of this article goes as follows: we first briefly review few existing steering inequalities along with the newly proposed CHSH like steering inequality. Then we show that the optimal violation of this CHSH like steering inequality in quantum theory is restricted to . Then we study how the optimal violation amount of the CHSH like inequality varies for different entangled states.
Ii Steering Inequalities
To test EPR steering Reid first proposed a testable formulation for continuousvariable systems based on positionmomentum uncertainty relation reid () which was experimentally tested by Ou et al ou (). Cavalcanti et al. developed a general construction of experimental EPRsteering criteria based on the assumption of existence of LHS model caval (). Importantly this general construction is applicable to both the discrete as well as the continuousvariable observables and Reid’s criterion appears as a special case this general formulation. On the other hand, Walborn et al. formulated a steering inequality based on BialynickiBirula and Mycielski’s entropic positionmomentum uncertainty relation wal (); wal1 (). As the entropic uncertainty relation implies Heisenberg’s uncertainty relation, hence the set of states violating Walborn et al.’s steering inequality contains all the states violating Reid’s inequality. Thus Walborn et al.’s steering criterion is more powerful than Reid’s one. However this is true for continuous variable case only, not for the discrete case. In all instances the violation of the aforementioned inequalities by different pure entangled states are the same.
All of these steering inequalities have been proved to be as sufficient conditions for witnessing steering in bipartite quantum systems. But none of these condition is supposed to be a necessary and sufficient condition for steering. Search for such an necessary and sufficient condition has been culminated in a recent development by Cavalcanti et al. caval15 (). They have proposed an CHSH like EPRsteering inequality that is necessary and sufficient for the set of correlations in the simplest scenario involving two settings and two outcomes per setting, with mutually unbiased measurements at trusted end. At this point it is interesting to ask the following questions:

what is the optimal violation of this newly proposed CHSH like steering inequality in quantum theory?

how the violation amount of the concerned inequality depends on the state?
We provide a definite answer for the first question and study the second one for some classes of states.
Iii Maximum violation of CHSHlike steering inequality
Let us first briefly review the steering scenario as introduced by Wiseman et al. wisemanprl07 (); wisemanpra07 (). Given a pair of systems at Alice and Bob, denote and the sets of observables in the Hilbert space of Alice’s and Bob’s system, respectively. An element of is denoted by , with a set of outcomes labeled by , and similarly for Bob. The joint state of the system is steerable by Alice iff it is not the case that for all , , , , the joint probability distributions can be written in the form
(1) 
where denotes an arbitrary probability distribution and denotes the quantum probability of outcome given measurement on state . In other words the state will be called steerable if it does not satisfy a LHVLHS model. Note that, if for a given measurement strategy the correlations have a LHV–LHS model, this does not imply that the underlying state is not steerable, since there could be another strategy that does not. In the simplest scenario where Alice and Bob each have a choice between two dichotomic measurements to perform: , , and outcomes of are labeled and similarly for the other measurements, the authors of ref.caval15 () derived a necessary and sufficient criterion for steering which reads as:
(2) 
We know that, in the simplest Bell scenario which involves two observers with dichotomic measurements per site, the set of local correlations lies in a polytope (LHV polytope) with CHSH inequality providing the non trivial facets of the LHV polytope. In the steering scenario with similar settings the set of correlations having LHVLHS description also form a polytope.
One of the authors of this article, along with other collaborators, have shown that measurement incompatibility limits the BellCHSH inequality violation in quantum theory to Cirel’son bound manik (). Adopting similar approach and using a recently established connection between measurement incompatibility and steering, we derive the optimal quantum violation of the steering inequality (2). Before establishing this result we first briefly review the concept of measurement incompatibility and the concept of unsharp measurement in quantum theory.
Measurement incompatibility: In the case of projective measurements, compatibility is uniquely captured by the notion of commutativity. Non commutative projective measurements in quantum mechanics do not admit unambiguous joint measurement Varadarajan'85 (). With the introduction of the generalized measurement i.e. positive operatorvalued measures (POVMs) Kraus'83 (); Nielsen'10 (), it was shown that observables which do not admit perfect joint measurement, may allow joint measurement if the measurements are made sufficiently fuzzy Busch'85 (); Busch'86 (). Therefore, for general measurements there is no unique notion of compatibility. In this article measurement incompatibility is captured by non joint measurability Busch'96 ().
Mathematically, a POVM consists of a collection of operators which are positive, , and sum up to the identity, . Here denotes measurement outcome and denotes measurement choices. Physically, any POVM can be realized by first letting the physical system interact with an auxiliary system and then measuring an ordinary observable on the auxiliary system. A set of POVMs is called jointly measurable if there exists a measurement with outcome where gives the outcome of measurement, i.e.,
(3) 
where stands for the elements of except for . Hence, all POVM elements are recovered as marginals of the mother observable .
Unsharp measurement: Let us consider two dichotomic quantum measurements and , which are not jointly measurable. Denoting eigenvalues of these operator as , the expectation value over some state can be expressed as:
where , with being the POVM elements corresponding to . The unsharp of fuzzy observable is given by , with
Here is known as “unsharpness parameter" and the fraction is called “degree of reality" Busch'86 (). It may happen that the observables and do not allow any joint measurement, but with introduction of sufficient amount of unsharpness, there unsharp versions and may allow joint measurement. In ref.manik (), the authors have proved that given any ddimensional quantum system, joint measurement for unsharp versions of any two dichotomous observables and of the system is possible with the largest allowed value of the unsharpness parameter . Note that the expectation value of an unsharp observable over some quantum state is related to the expectation value of its sharp version in the following manner,
(4) 
Similarly, if Alice performs unsharp measurement on her part and Bob performs sharp measurement on his part of a bipartite shared state than we have,
(5) 
Except from quantum entanglement, another necessary ingredient which is necessary for study of quantum nonlocality is the existence of incompatible set of measurements. In the simplest bipartite scenario Wolf et al. showed that any set of two incompatible POVMs with binary outcomes can always lead to violation of the CHSHBell inequality Wolf2009 (). But, recently in refs.bru (); ghu () the authors have proved that this result does not hold in the general scenario where numbers of POVMs and outcomes are arbitrary. However in this general settings the authors of bru (); ghu () have established a connection between measurement incompatibility and a weaker form of quantum nonlocality i.e., EPRSchrödinger steering. They have shown that for any set of incompatible POVMs (i.e. not jointly measurable), one can find an entangled state, such that the resulting statistics violates a steering inequality. Please note that, one of the authors of this article has recently proved that connection between measurement incomparability and steering holds for a more general class of tensor product theories rather than just Hilbert space quantum theory Manik'2015 ().
Let Alice performs a measurement assemblage on her part of a bipartite shared quantum state . Upon performing measurement , and obtaining outcome , the (unnormalized) state held by Bob is given by . The normalized state on Bob side is given by . Also we have , which actually ensure no signaling from Alics to Bob. The state assemblage is unsteerable iff it admits a decomposition of the form
(6) 
where . Existence of such decomposition for state assemblage on Bob’s side ensures that the statistics obtained from the state admits a combined LHVLHS model of the form of Eq.(1). The authors in refs.bru (); ghu () have shown that the assemblage , with , is unsteerable for any state acting on if and only if the set of POVMs acting on is jointly measurable. As a corollary of this result we can say that
Corollary 1: The assemblage , with and , is unsteerable for any state acting in if and only if the set of POVMs acting on is jointly measurable.
At this stage, we are now in a position to prove our main result, which is described in the following theorem.
Theorem 2: Consider a composite quantum system composed of two subsystem with state spaces and , respectively. For any pair of dichotomic observables , for the first system and the mutually unbiased dichotomic observables , for the second system and the joint state acting on , we have the following inequality:
(7) 
where is the optimal unsharpness parameter that allows joint measurement for any two dichotomic quantum observables.
Proof: Let us consider two arbitrary dichotomic observables on Alice side, and . These two observables in general may not allow joint measurement. However, introduction of unsharpness makes it possible to measure the unsharp version of these two observables jointly. Let the optimal unsharpness is which allows joint measurement for any two dichotomic observables.
Now according to Corollary , as far as observables on Alice’s side are jointly measurable, they will not violate any steering inequality and hence the steering inequality (2). Thus we have
Now using the expressing from Eq.(5) we get,
The value of in quantum theory is proved to be manik (). Therefore the upper bound of the steering inequality (2) in quantum theory is , i.e., , which is numerically equal to the celebrated Cirel’son value. Naturally the question arises whether this value is tight or not. In the following section we answer this question affirmatively by showing that sharing a maximally two qubit entangled state with suitable choice of measurement, this value can be achieved in quantum theory.
Iv Violation of steering inequality by different states
In this section we study the optimal violation of the steering inequality (2) for different given entangled states.
Observation 1: is different for different qubit pure entangled states.
Consider that an arbitrary 2qubit pure entangled state is shared between Alice and Bob. Alice performs measurements and on her part of the entangled particle where . Similarly Bob performs measurements and on his part of the entangled particle where and , i.e. are mutually unbiased qubit measurements. Varying over the measurement directions of Alice and Bob we have numerically found the optimal violation amount of the steering inequality (2) for a given qubit pure entangled state and in Fig.1 we have plotted it with respect to state parameter .
From Fig.1 it is clear that all qubit pure entangled states violate the steering inequality (2). It is important to note that the optimal violation amount is different for different states with maximally entangled state providing the maximum violation where Alice performs measurement in the directions and and Bob’s MUB measurements look like and . Please note that except from the steering inequality considered here, for all other existing steering inequalities the violation amount does not change with respect to different pure entangled states.
Observation 2: for qubit Werner class of states.
Consider that Alice and Bob share the Werner state ; where is the singlet state. Likewise in the previous case, varying over the measurement setup of Alice and Bob we find the optimal violation amount of the steering inequality (2) for different Werner state and plot this values in Fig.2.
From Fig.2 it is clear that Werner states violate the steering inequality for and the violation amount increase with the parameter .
V Conclusion
There have been several attempts to quantify quantum steering. Two most recent instances are ‘steerable weight’ sky (), ‘steering robustness’ pia () and ‘relative entropy of steering’ gal (). But all of these proposed quantifiers assign same value for all pure entangled states gal (). At this point our observation becomes interesting. Since the amount of violation can also be shown to be non increasing under steering nonincreasing operations (SNIOs) gal (), one can take this amount of violation to be a valid quantifier of quantum steering. Our study stipulates further studies whether some semi device independent protocol(s) can be deigned whose payoff scales with violation amount of the steering inequality considered here. In such case different violation amount of the steering inequality by different pure entangles states will have an operational explanation.
We also find the violation of the CHSH like steering inequality by qubit Werner states. The inequality is violated by Werner states if and beyond this value it follows a trait the more entangled the state, more is the violation. We also find that the violation of this steering inequality in QM is tightly upper bounded by , the well known Cirel’son quantity.
Acknowledgment: It is a pleasure to thank Prof. Guruprasad Kar for various simulating discussions and useful suggestions. We also thank Dr. Ashutosh Rai for useful discussions. AM acknowledge support from the CSIR project 09/093(0148)/2012EMRI. n. MB thankfully acknowledges discussions with Prof. Sibasish Ghosh during recent visit to the Institute of Mathematical Sciences, Chennai, India.
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