Consider a generic bipartite pure state $\newcommand{\ket}[1]{\lvert #1\rangle}\ket\Psi\equiv \sum_k \sqrt{p_k}\ket{u_k}\otimes\ket{v_k}\in\mathcal X\otimes\mathcal Y$, where $p_k\ge0$ are the Schmidt coefficients, and $\{\ket{u_k}\}_k\subset\mathcal X,\{\ket{v_k}\}_k\subset\mathcal Y$ are orthonormal sets of states.

We known that, for any pair of unitary operations $U,V$, the state $(U\otimes V)\ket{\Psi}$ has the same amount of entanglement as $\ket\Psi$, as reflected by the invariance of the Schmidt coefficients under such operation.

Consider now a local projection operation. More precisely, suppose $\mathcal X$ also has a bipartite structure, $\mathcal X=\mathcal X_1\otimes\mathcal X_2$, take some state $\ket\gamma\in\mathcal X_1$, and consider the postselected state $\ket{\Psi'}\equiv \langle \gamma\rvert\Psi\rangle/\|\langle \gamma\rvert\Psi\rangle\|\in\mathcal X_2\otimes\mathcal Y$. If I were to describe this as an operation, I guess this would amount to applying some non-unitary linear operator $A$ to $\ket\Psi$.

Can the amount of entanglement of $\ket{\Psi'}$ in the "residual bipartition" $\mathcal X_1\otimes\mathcal Y$ be larger than the initial entanglement in $\ket\Psi$? If so, is there some kind of known characterisation of when this is possible?

Intuitively, this would mean that an initially low amount of entanglement can be "enhanced", or somehow "activated", conditionally to some observation (i.e. finding $\ket\gamma$) of one party. Such a situation seems strange to me but I'm not sure how to rule out the possibility.

  • $\begingroup$ I believe that I had an example of this at some point. I think it went along the lines of the two parties share the two qubit state $\cos(\theta) |00 \rangle + \sin(\theta) |11\rangle$ and one of them performs an unsharp POVM and postselects on the outcome (with the Lüder's update rule). I think one could find situations where the post-selected state was closer to maximally entangled than the initial state -- the catch was that these events only occurred with very small probability. $\endgroup$ – Rammus Sep 20 '20 at 18:35

Of course this is possible. By LOCC you can probabilistically convert any (pure) state to a maximally entangled state with the same Schmidt rank, using a "filtering" POVM.

When this can be done, and at which optimal rate, is covered by the theory of majorization. This is covered very well in e.g. Nielsen and Chuang, or there is a review by Nielsen and Vidal.

(I realize you don't talk about POVMs but projections, but Stinespring/Naimark tells us that this is equivalent.)

  • $\begingroup$ thanks. Where are you saying this is discussed in N&C? Chapter 12 discusses some aspects of majorization and LOCCs, but I don't see where they talk about probabilistic schemes or filtering POVMs. $\endgroup$ – glS Sep 20 '20 at 22:13
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    $\begingroup$ @glS True, it is not there. Well, then Thm. 15 and 16 in the Nielsen+Vidal-review. It is also in my lecture notes ;) $\endgroup$ – Norbert Schuch Sep 23 '20 at 14:50

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