Undistillability of the canonical NPT family of DiVincenzo et al. and the existence of bound entanglement with negative partial transpose

Whether every bipartite entangled state with negative partial transpose (NPT) can be distilled, or whether NPT bound entangled states exist, is one of the oldest open problems in quantum information theory. DiVincenzo, Shor, Smolin, Terhal and Thapliyal (Phys. Rev. A 61, 062312 (2000)) reduced the question to a canonical two-parameter family $\rho_{bc}$ in $d\otimes d$ and conjectured that the states $\rho(c,\epsilon)$ immediately outside the separable boundary are undistillable for sufficiently small $\epsilon>0$. The problem is to decide this conjecture: equivalently, whether the pseudo-$n$-copy undistillability threshold $\epsilon_0(c,n)$ admits a strictly positive lower bound independent of the copy number $n$, or, in the language of positive linear maps, whether the tensor powers $\Lambda^{\otimes n}$ of the associated map $\Lambda=T\circ\mathcal{S}$ remain 2-positive for all $n$. The asymptotic core remains open.

Background

Two distant parties sharing many copies of a mixed bipartite quantum state $\rho$ can attempt, by local quantum operations and classical communication (LOCC), to convert them into shared maximally entangled pairs of qubits. The state $\rho$ is called distillable if some protocol acting on finitely many copies succeeds, and bound entangled if it is entangled yet undistillable. The partial transpose $\rho^{T_B}$ (transposition applied to one party's subsystem) must be positive semidefinite ('PPT') for $\rho$ to be separable. Entangled PPT states exist in $d\otimes d$ for $d\ge3$ and are all known to be bound entangled; in $2\otimes2$ and $2\otimes3$ every entangled state is NPT and distillable. Whether some NPT state is undistillable, equivalently whether bound entanglement is exhausted by the PPT states, has been open since 1999 and is posed, with an announced award, as one of the 'Five Open Problems in Quantum Information Theory' (Horodecki, Rudnicki and Życzkowski, PRX Quantum 3, 010101 (2022)).

The standard finite-copy criterion reduces distillability to a matrix inequality: $\rho$ is distillable from $n$ copies if and only if there exists a bipartite vector $|\psi\rangle$ of Schmidt rank at most two (its Schmidt decomposition across the bipartition has at most two terms, so local projections can carry it onto a $2\otimes2$ subspace) such that $\langle\psi|(\rho^{T_B})^{\otimes n}|\psi\rangle<0$. The state is called pseudo $n$-copy undistillable when no such vector exists for that $n$; $\rho$ is pseudo $n$-copy undistillable precisely when $\rho^{\otimes n}$ is pseudo one-copy undistillable.

DiVincenzo, Shor, Smolin, Terhal and Thapliyal introduced a symmetry-reduced testbed for the problem. Twirling any NPT state (averaging it over the diagonal-unitary and permutation groups by random local operations) yields, by stochastic LOCC (local operations that succeed with nonzero probability, coordinated by classical communication), an NPT state of a two-parameter canonical family on $d\otimes d$:

$$\rho_{bc}=a\sum_i|ii\rangle\langle ii|+b\sum_{i<j}|\psi^-_{ij}\rangle\langle\psi^-_{ij}|+c\sum_{i<j}|\psi^+_{ij}\rangle\langle\psi^+_{ij}|,$$

where $|\psi^{\pm}_{ij}\rangle=(|ij\rangle\pm|ji\rangle)/\sqrt2$, the coefficients $a,b,c\ge0$ obey $da+\frac{d(d-1)}{2}(b+c)=1$, and the Werner states (the permutation-invariant family $(I+\alpha F)/(d^2+\alpha d)$, with $F$ the swap operator) form the line $FH$ of the $(b,c)$ plane. Because any NPT state reduces into this family, it is a faithful testing ground: if some canonical state is undistillable, NPT bound entanglement exists. The paper established: (i) the quadrilateral region $BCGK$, with $G=(\frac{3}{d(2d-1)},\frac{1}{d(2d-1)})$ the one-copy-distillability threshold of the Werner line, is pseudo one-copy undistillable; (ii) a propagation lemma: if the state at $G$ is pseudo $n$-copy undistillable, then so is every state in the triangle $BGK$; and (iii) their strongest result: for every finite $n$ and every $0\le c<1/(d(d-1))$ there exists $\epsilon_0(c,n)>0$ such that the states $\rho(c,\epsilon)$ (Eq. (3.24) of the paper), the subfamily lying immediately to the right of the separable boundary $BK$ whose partial transpose has a single negative eigenvalue $-(d-1)\epsilon$, are pseudo $n$-copy undistillable for all $0\le\epsilon\le\epsilon_0(c,n)$. What is unknown is the behavior as $n\to\infty$: if $\epsilon_0(c,n)$ stays above some positive $\bar\epsilon(c)$, the states $\rho(c,\epsilon)$ with small positive $\epsilon$ are genuinely undistillable, NPT bound entangled states; if $\epsilon_0(c,n)\to0$ for every $c$, then every NPT state is distillable (any NPT state can first be mixed toward the boundary by LOCC). The authors conjectured the former, numerically supported in $d=3$ only up to two or three copies (the companion analysis of Dür, Cirac, Lewenstein and Bruß, Phys. Rev. A 61, 062313 (2000), covers the same questions for the Werner line).

The same paper recast the question in the language of positive linear maps. Through the Choi–Jamiołkowski correspondence (the standard isomorphism between bipartite states and linear maps on $d\times d$ matrices), each canonical state determines a map $\Lambda=T\circ\mathcal{S}$, the composition of matrix transposition $T$ with a completely positive map $\mathcal{S}$ (a map is completely positive if it and all its tensor extensions are positive, equivalently if its Choi matrix is positive; $k$-positivity of $\Lambda$ means that $\mathrm{id}_k\otimes\Lambda$ is positive). Their main theorem states that $\rho$ is undistillable if and only if $\Lambda^{\otimes n}$ is 2-positive for every $n$. Since 2-positivity is not in general preserved under tensor products, the conjecture is equivalent to asking whether the special structure $\Lambda=T\circ\mathcal{S}$ with $\mathcal{S}$ completely positive forces 2-positivity of all tensor powers; no general criterion for deciding this, or for deciding failure of 2-positivity at some finite $n$, is known.

Problem Statement

Decide the conjecture of DiVincenzo, Shor, Smolin, Terhal and Thapliyal (Phys. Rev. A 61, 062312 (2000)) for their canonical two-parameter family of NPT states: are the states $\rho(c,\epsilon)$ (Eq. (3.24) of the paper), lying immediately to the right of the separable boundary $BK$ in the $(b,c)$ plane, undistillable for sufficiently small positive $\epsilon$? Equivalently, and with the same generality as the source, either of the following two reformulations suffices on its own to settle the conjecture:

1. Determine whether the pseudo-$n$-copy undistillability thresholds $\epsilon_0(c,n)$ admit a strictly positive lower bound $\bar\epsilon(c)>0$ independent of $n$ for some $c$ (positive answer: the states $\rho(c,\epsilon)$ with $0\le\epsilon\le\bar\epsilon(c)$ are NPT bound entangled), or whether $\epsilon_0(c,n)\to0$ as $n\to\infty$ for all $c$ (negative answer: every NPT state is distillable, since any NPT state can be mixed toward the boundary by LOCC).

2. In the positive-map formulation, decide whether the map $\Lambda=T\circ\mathcal{S}$ associated with such states (transposition composed with a completely positive map) is 2-positive together with all of its tensor powers $\Lambda^{\otimes n}$, and more generally whether there exist structural criteria for maps of this composed form that decide failure of 2-positivity at some finite $n$, which would settle the additivity or nonadditivity of distillability for the undistillable side.

The next item, by contrast, is an auxiliary sub-goal of the source's program rather than an equivalent restatement of the headline question; settling the conjecture does not require it.

3. Determine the finite-copy distillability structure of the one-copy-undistillable region of the canonical plane: rigorously locate, in the $(b,c)$ plane, the boundary separating NPT states undistillable from all finite-copy protocols (if any exist) from states that are distillable but require increasingly many copies, including the multi-copy (three and higher) status of the triangle $BGK$ and of the Werner threshold point $G$.

This is the canonical, symmetry-reduced form of the general open question: does bound entanglement with negative partial transpose exist?

References

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