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?

  • $\begingroup$ By local hidden variable model you mean it is a separable state? Or are you talking about LHV models in the context of nonlocality? $\endgroup$
    – Rammus
    Oct 13, 2021 at 18:18
  • $\begingroup$ @Rammus the latter. As in, the correlations produced by Werner states can be reproduced via local hidden variables. I made this point more explicit $\endgroup$
    – glS
    Oct 13, 2021 at 18:44
  • 1
    $\begingroup$ 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/… $\endgroup$
    – Danylo Y
    Oct 14, 2021 at 19:22


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