Steering bound for qubits and POVMs

Cite this problem as Problem 39.

Problem

In steering [1] one asks for the possibility to model the correlations in a quantum state $\rho$ by a classical model, in which one party (say Bob) is constrained to be described by quantum mechanics. This is demanded for some class ${\cal F}$[/latex] of observables for the other party (say Alice). That is, we need (1) a probability space $X$ with probability measure $P$, (2) a family of “hidden” states $\sigma(x)$, $x\in X$, for Bob’s system, and (3) for any observable $F\in{\cal F}$, a classical observable $f^F$. That is to every POVM element $F_a$ corresponds a response probability function $f_a^F$ on $X$, so that $f_a^F(x)\geq0$ and \sum_af_a^F(x)=1 for all $x$. The model is valid if

    \begin{equation*} tr\rho F_a\otimes M=\int P(dx) f_a^F(x)\ tr\sigma(x) M.\end{equation*}

Consider a two-qubit state of the form $\rho=((2-f)1+(2f-1)F)/6$, where $F$ is the swap operator, and -1\leq f\leq1. These states are separable for $f=tr\rho F\geq0$. The problem is to find the critical value of f_s, so that a classical model exists for f>f_s, and does not exist (i.e., the state is “steering”) for f<f_s.

When the class ${\cal F}$of observables is taken to be the projection valued ones (in this case necessarily two-valued), the bound is known [2] to be $f_s^{PVM}=-1/3$. So the question is to determine the critical value for general (POVM) measurements, i.e., $f_s^{POVM}\geq f_s^{PVM}$. What is known so far is consistent with equality here, but that remains to be shown.

The problem has an obvious generalization to higher dimensional “Werner states”, which where introduced in [2], together with the optimal steering models for PVMs, long before steering was formalized.

References

[1] H. M. Wiseman, S. J. Jones, and A. C. Doherty: Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox.
Phys. Rev. Lett., 98:140402, 2007.

[2] R.F. Werner: Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model,
Phys. Rev. A 40(1989) 4277-4281.