# How to show that Werner states produce correlations explainable via local hidden variable models?

Werner states can be written as $$\rho_W= p\frac{\Pi_+}{\binom{n+1}{2}} +(1-p)\frac{\Pi_-}{\binom{n}{2}},$$ with $$\Pi_\pm\equiv\frac12(I\pm\mathrm{SWAP})$$ projectors onto the $$\pm1$$ eigenspaces of the swap operator, defined as the one acting on basis states as $$\mathrm{SWAP}(e_a\otimes e_b)=e_b\otimes e_a$$, and $$n$$ the dimension of the underlying spaces.

These are known to be separable for $$p\ge 1/2$$, as seen e.g. via PPT criterion. However, as shown in Werner's 1989 paper, these states admit local hidden variable (LHV) models also for $$p<1/2$$ corresponding to nonseparable states. In this context, by "admits LHV model" I mean that the correlation it produces can be explained by such a model, i.e. $$p(a,b|x,y)\equiv{\rm Tr}[(\Pi^x_a\otimes \Pi^y_b)W_{p}] = \sum_\lambda p_\lambda p_\lambda(a|x)p_\lambda(b|y),$$ for some probability distribution $$p_\lambda$$ (which can be continuous, in which case the sum above becomes an integral over some measure), and conditional probability distributions $$p_\lambda(a|x)$$ and $$p_\lambda(b|y)$$. The collections $$\{\Pi^x_a\}_a,\{\Pi^b_b\}$$ are spanning sets of orthogonal unit-trace projections, so that $$x,y$$ denote the "measurement choice" made by Alice and Bob when measuring, while $$a,b$$ are the corresponding measurement outcomes. More explicitly, we are thus calling $$\rho_W$$ "local" if $$\operatorname{Tr}[(\Pi_a\otimes\Pi'_b)\rho_W ] = \sum_\lambda p_\lambda p_\lambda(a|\{\Pi_a\}_a) p_\lambda(b|\{\Pi'_b\}_b),$$ for any pair of projective measurements $$\{\Pi_a\}_a,\{\Pi'_b\}_b$$, and (some) conditional probability distributions $$p_\lambda(a|\{\Pi_a\}_a),p_\lambda(b|\{\Pi'_b\}_b)$$.

Werner's paper shows this by explicitly building such a model, but the construction is not the easiest to follow. Is there an easier/alternative/"better" way to construct such LHV models?

• By local hidden variable model you mean it is a separable state? Or are you talking about LHV models in the context of nonlocality? Oct 13, 2021 at 18:18
• @Rammus the latter. As in, the correlations produced by Werner states can be reproduced via local hidden variables. I made this point more explicit
– glS
Oct 13, 2021 at 18:44
• It seems that the problem of whether the Werner state with parameter $p$ admits LHV is not completely solved. Look at this reference arxiv.org/abs/1609.06114. And also at this slides ncts.ncku.edu.tw/phys/qis/151210/speech/… Oct 14, 2021 at 19:22