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Distillable entanglement of the antisymmetric state

Unsolved ID op_09dc815c8a194fb7 Last edited 8 September 2026
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What is the exact LOCC distillable entanglement \(E_D(\rho_{\mathrm{as}}^{(d)})\) of the \(d\times d\) antisymmetric state, for each integer \(d\ge3\)?

Let \(F\) be the swap operator on \(\mathbb C^d\otimes\mathbb C^d\), acting by \(F\,|ij\rangle=|ji\rangle\) on the computational basis. Define the antisymmetric projector and the normalized antisymmetric state

\begin{equation} P_{\mathrm{as}}:=\tfrac12(\mathbb 1-F), \qquad \rho_{\mathrm{as}}^{(d)}:=\frac{2}{d(d-1)}\,P_{\mathrm{as}}, \tag{1} \end{equation}

so Eq. (1) is the extreme point \(\alpha=-1\) of the Werner family \(\rho_\alpha\propto\mathbb 1+\alpha F\) and is not of positive partial transpose; for \(d=2\) it is the singlet, with \(E_D=1\), and the question concerns \(d\ge3\).

Let \(\Phi_2\) be a maximally entangled state of Schmidt rank two. The LOCC distillable entanglement of the state in Eq. (1) is

\begin{equation} E_D\bigl(\rho_{\mathrm{as}}^{(d)}\bigr):=\sup\Bigl\{r\ge0:\ \exists\ \mathrm{LOCC}\ \Lambda_n\ \mathrm{with}\ \lim_{n\to\infty}\bigl\|\Lambda_n\bigl((\rho_{\mathrm{as}}^{(d)})^{\otimes n}\bigr) -\Phi_2^{\otimes\lfloor rn\rfloor}\bigr\|_1=0\Bigr\}, \tag{2} \end{equation}

and replacing LOCC in Eq. (2) by completely PPT-preserving operations defines the relaxation \(E_{D,\mathrm{PPT}}(\rho_{\mathrm{as}}^{(d)})\), which is exactly known — see Progress — so the open quantity is the LOCC rate itself. Determine \(E_D(\rho_{\mathrm{as}}^{(d)})\) exactly for every \(d\ge3\). A complete answer consists of an explicit LOCC achievability protocol together with a matching converse bound.

Source

Implicit in Christandl, Schuch, and Winter’s analysis of the antisymmetric state: they bound its distillable key, and hence its distillable entanglement, by \(O(1/d)\) while its entanglement cost stays above a \(d\)-independent constant, leaving the exact rate open [CSW12]; announced in [CSW10]. Yura’s exact evaluation of the entanglement cost at \(d=3\) settles the cost side of the same question [Yur03].

Progress

Reports do not certify correctness or automatically change the problem's status. Progress policy.

  • DiVincenzo, Shor, Smolin, Terhal, and Thapliyal determined the one-copy-distillable region of the Werner family: a Schmidt-rank-two projection witnesses entanglement in a single copy precisely on one side of an explicit threshold, and the state in Eq. (1) lies in that region, so \(E_D(\rho_{\mathrm{as}}^{(d)})>0\) [DST+00].

  • A direct LOCC protocol quantifies the achievability side. Partition the local computational basis \(\{1,\dots,d\}\) into \(\lfloor d/2\rfloor\) disjoint pairs, together with one unpaired letter when \(d\) is odd; both parties perform the local coarse measurement that projects onto the two-dimensional support \(\operatorname{span}\{\lvert i\rangle,\lvert j\rangle\}\) of each pair; an outcome identifies the pair but not which of its two letters, and the parties keep the runs in which both obtain the same pair. Conditioned on a common pair, the post-measurement state is the singlet \((\lvert ij\rangle-\lvert ji\rangle)/\sqrt{2}\) on that pair, so every kept run is one ebit, and the \(\lfloor d/2\rfloor\) common-pair outcomes, each of unconditional probability \(2/(d(d-1))\), occur with total probability

    \begin{equation} p_d:=\frac{2\lfloor d/2\rfloor}{d(d-1)} =\begin{cases} \dfrac{1}{d-1}, & d\ \text{even},\\[4pt] \dfrac{1}{d}, & d\ \text{odd}. \end{cases} \tag{3} \end{equation}

    Independent repetition converts each copy into a singlet with probability \(p_d\), so

    \begin{equation} E_D\bigl(\rho_{\mathrm{as}}^{(d)}\bigr)\ \ge\ p_d=\Omega(1/d), \tag{4} \end{equation}

    an \(\Omega(1/d)\) achievable LOCC rate that coincides with the \(1/d\) lower bound on \(E_D\) and \(K_D\) recorded in [CSW12] for odd \(d\) and strengthens it to \(1/(d-1)\) for even \(d\). The protocol extends the one-copy Schmidt-rank-two projection witnessing distillability in [DST+00], which retains a single basis pair and succeeds with probability \(2/(d(d-1))\), to all \(\lfloor d/2\rfloor\) disjoint basis pairs at once; the quantitative bound Eq. (3) was recorded by Naixu Guo in the review discussion of this catalog entry (September 2026) and is not stated in the references above.

  • The logarithmic negativity upper-bounds distillable entanglement [VW02], and the Rains bound upper-bounds distillation by completely PPT-preserving operations [Rai01]. The partial transpose of the state in Eq. (1) has the single negative eigenvalue \(-1/d\) and the eigenvalue \(1/(d(d-1))\) with multiplicity \(d^2-1\),

    \begin{equation} \operatorname{spec}\bigl(\bigl(\rho_{\mathrm{as}}^{(d)}\bigr)^{\Gamma}\bigr) =\bigl\{-\tfrac1d\bigr\}\cup\bigl\{\tfrac{1}{d(d-1)}\ \text{with multiplicity}\ d^2-1\bigr\}, \tag{5} \end{equation}

    so its trace norm is \((d+2)/d\). The PPT relaxation itself is exactly solved: Rains’s semidefinite program evaluates distillation under the completely PPT-preserving operations defined in Sec. 2 of his paper, and his Corollary 5.6 (Eq. (5.40)) determines, with matching achievability and converse, the distillable entanglement \(D_\Gamma(W_d(1))=\log_2\bigl(\frac{d+2}{d}\bigr)\) of the antisymmetric Werner state \(W_d(1)\) — his Eq. (5.34) at \(p=1\), which is exactly the state of Eq. (1) — with one-shot achievability \(F_\Gamma(W_d(1);\frac{d+2}{d})=1\) (his Eq. (5.39)) and the trace-norm converse \(D_\Gamma\le \log_2\operatorname{Tr}\lvert\rho^\Gamma\rvert\) evaluated through Eq. (5). Combining with Eq. (4), the known rates are bracketed by

    \begin{equation} p_d\ \le\ E_D\bigl(\rho_{\mathrm{as}}^{(d)}\bigr)\ \le\ E_{D,\mathrm{PPT}}\bigl(\rho_{\mathrm{as}}^{(d)}\bigr) =\log_2\bigl(1+\tfrac2d\bigr), \tag{6} \end{equation}

    whose final inequality is Rains’s exact equality for every \(d\), including \(d=3\); the open gap is the middle one, LOCC versus completely PPT-preserving distillation. The same value was obtained independently as the regularized PPT relative entropy of entanglement: \(E_{R,\mathrm{PPT}}^{\infty}(\rho_{\mathrm{as}}^{(d)}) =\log_2\bigl(1+\tfrac2d\bigr)\) [AEA+01], and the chain \(E_{R,\mathrm{PPT}}^{\infty}=E_{\mathrm{Rains}}=E_N =\log_2\bigl(1+\tfrac2d\bigr)\), with \(E_{\mathrm{Rains}}\) the single-copy Rains bound, is recorded in Sec. III of [CSW12].

  • On the cost side, Yura computed \(E_C(\rho_{\mathrm{as}}^{(3)})=1\) ebit exactly [Yur03]. Christandl, Schuch, and Winter then showed that the distillable key obeys \(K_D(\rho_{\mathrm{as}}^{(d)})=O(1/d)\), hence \(E_D(\rho_{\mathrm{as}}^{(d)})\le K_D(\rho_{\mathrm{as}}^{(d)})\to0\) as \(d\to\infty\), while the entanglement cost stays above a \(d\)-independent constant, making asymptotic manipulation of the family irreversible in the limit \(d\to\infty\) — the cited bounds separate \(E_D\) from \(E_C\) only for \(d\ge7\) (and at \(d=3\), via Yura’s \(E_C=1\)), leaving \(d=4\), \(5\), and \(6\) open. Together with the lower bound Eq. (4) and the \(1/d\) bound on \(E_D\) and \(K_D\) recorded there, the \(O(1/d)\) upper bound is optimal up to a constant; the exact distillable entanglement is left open [CSW12].

  • For any rank-two mixed state supported on the \(3\otimes3\) antisymmetric subspace, Wang and Duan showed that the distillable entanglement under completely PPT-preserving operations is strictly smaller than one ebit, while the PPT entanglement cost is exactly one ebit, a strict separation between the two PPT measures; the inequality does not evaluate the rate in Eq. (2) [WD17].

  • Beyond LOCC and its PPT relaxation, other operation classes admit exact general characterizations, none of which evaluates the LOCC rate posed here. Lami and Regula proved that the distillable entanglement under dually non-entangling operations equals the separably measured regularized relative entropy of entanglement for every state — the second exact regularized distillation formula after the Devetak–Winter formula for one-way LOCC — and in particular is strictly positive for every entangled state [LR24]. Dually non-entangling operations do not contain the completely PPT-preserving class, nor vice versa: a constant channel preparing a PPT entangled state is completely PPT-preserving and CPTP yet not non-entangling, since it outputs entanglement on separable inputs, while completely PPT-preserving operations distill at zero rate from every PPT state [Rai01], excluding the strict positivity that holds under dually non-entangling operations. The Lami–Regula formula therefore determines neither rate in Eq. (6). Chen, Wang, Zhang, and Zhu established reversible exact state transfer under trace-preserving PPT-preserving maps that need not be completely positive, governed by logarithmic negativity, a zero-error non-CPTP setting distinct from the vanishing-error rates posed here [CWZ+25].

  • Kondra, Brinster, Kampermann, Bruß, and Wyderka constructed analytically solvable families built from antisymmetric states exhibiting exponential strong-converse irreversibility under completely PPT-preserving operations [KBK+26]. The results concern cost and error exponents rather than the vanishing-error rate in Eq. (2), and the LOCC-side analogues are stated there as open.

  • Sen and Lami computed the main entanglement measures of the flower-state family, whose distillable entanglement is exactly one ebit independent of local dimension, citing the antisymmetric state as the standard example of vanishing distillable entanglement at nonvanishing cost; the value for the antisymmetric state itself is not computed [SL26].

Comment

Even at \(d=3\) the exact LOCC value is unknown: Eq. (6) leaves \(\tfrac13\le E_D(\rho_{\mathrm{as}}^{(3)})\le\log_2(5/3)\), while the LOCC entanglement cost equals one ebit, so both the rate and the sharpness of the irreversibility are undetermined. Asymptotically, the sharp question is the leading constant: \(E_D(\rho_{\mathrm{as}}^{(d)})=O(1/d)\) by the key bound of [CSW12] and \(E_D(\rho_{\mathrm{as}}^{(d)})\ge p_d\) by Eq. (3), while the PPT relaxation is exactly \(\log_2\bigl(1+\tfrac2d\bigr)\sim 2/(d\ln2)\), so neither the existence nor the value of \(\lim_{d\to\infty}d\,E_D(\rho_{\mathrm{as}}^{(d)})\) is established; in particular whether the LOCC rate attains the PPT constant \(2/\ln2\) or falls below it by a definite factor is open. Determining the existence and value of this limit is a secondary asymptotic question, not a complete answer to the exact finite-dimensional problem. A positive LOCC–PPT gap alone does not determine the limit: a gap of order \(1/d^2\) leaves the leading constant unchanged. Even a gap bounded below by \(c/d\) for a fixed \(c>0\) would only exclude the PPT leading constant, without establishing existence or the exact value of the LOCC limit. A complete solution requires matching LOCC achievability and converse bounds for every finite \(d\ge3\); the PPT side itself is closed. This record asks for the value of a rate on one explicit family, unlike Problem , the LOCC distillable entanglement of Bell-diagonal qubit states, and Problem , whether the Rains bound of Bell-diagonal states is attained by a PPT-preserving channel; Problem  asks for a closed form of the regularized exact zero-error PPT distillable entanglement of an arbitrary support, a different quantity from the vanishing-error rate in Eq. (2).

References

[DST+00]
D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal, and A. V. Thapliyal, “Evidence for Bound Entangled States with Negative Partial Transpose,” Physical Review A 61, 062312 (2000).arXiv
[AEA+01]
K. Audenaert, J. Eisert, E. Jané, M. B. Plenio, S. Virmani, and B. De Moor, “The Asymptotic Relative Entropy of Entanglement,” Physical Review Letters 87, 217902 (2001).DOIarXiv
[Rai01]
E. M. Rains, “A Semidefinite Program for Distillable Entanglement,” IEEE Transactions on Information Theory 47, 2921–2933 (2001).arXiv
[VW02]
G. Vidal and R. F. Werner, “Computable Measure of Entanglement,” Physical Review A 65, 032314 (2002).arXiv
[Yur03]
F. Yura, “Entanglement Cost of Three-Level Antisymmetric States,” Journal of Physics A: Mathematical and General 36, L237 (2003).DOIarXiv
[CSW10]
M. Christandl, N. Schuch, and A. Winter, “Highly Entangled States With Almost No Secrecy,” Physical Review Letters 104, 240405 (2010).DOI
[CSW12]
M. Christandl, N. Schuch, and A. Winter, “Entanglement of the Antisymmetric State,” Communications in Mathematical Physics 311, 397–422 (2012).DOIarXiv
[WD17]
X. Wang and R. Duan, “Irreversibility of Asymptotic Entanglement Manipulation Under Quantum Operations Completely Preserving Positivity of Partial Transpose,” Physical Review Letters 119, 180506 (2017).DOI
[LR24]
L. Lami and B. Regula, “Distillable Entanglement under Dually Non-Entangling Operations,” Nature Communications 15, 10120 (2024).DOIarXiv
[CWZ+25]
Yu-Ao Chen, Xin Wang, Lei Zhang, and Chenghong Zhu, “Reversible Entanglement beyond Quantum Operations,” Physical Review Research 7, 013297 (2025).DOI
[KBK+26]
T. V. Kondra, R. Brinster, H. Kampermann, D. Bruß, and N. Wyderka, “Very Strong Irreversibility of Quantum Entanglement,” arXiv preprint (2026).arXiv
[SL26]
S. Sen and L. Lami, “Entanglement of Flower States,” arXiv preprint (2026).arXiv

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@incollection{qiqcop_op_09dc815c8a194fb7,
  title = {Distillable entanglement of the antisymmetric state},
  booktitle = {Quantum Information and Quantum Computation Open Problem Zoo (QIQCOP Zoo)},
  year = {2026},
  howpublished = {\url{https://qiqc-op.com/problem/op_09dc815c8a194fb7/}},
  note = {Stable ID op_09dc815c8a194fb7; status: Unsolved; accessed 2026-09-27}
}

Plain text

“Distillable entanglement of the antisymmetric state,” Quantum Information and Quantum Computation Open Problem Zoo (QIQCOP Zoo), https://qiqc-op.com/problem/op_09dc815c8a194fb7/, ID op_09dc815c8a194fb7, accessed 2026-09-27.

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op_09dc815c8a194fb7
01M208GXYXHWA3C1M5CSAN6VAF