Minimal dimensions for strict transpose degradability

Problem

What are the componentwise-minimal dimension triples \((d_A,d_B,d_E)\) that admit a transpose-degradable but nondegradable channel? Let \(d_X:=\dim X\) and let \(V:A\to B\otimes E\) be a support-minimal isometry, meaning that the channels

\begin{equation} \Phi_V(X):=\operatorname{Tr}_E(VXV^\dagger), \qquad \Phi_V^c(X):=\operatorname{Tr}_B(VXV^\dagger) \tag{1} \end{equation}

satisfy \(\operatorname{supp}(\Phi_V(I_A))=B\) and \(\operatorname{supp}(\Phi_V^c(I_A))=E\). Equation (1) is strictly transpose degradable when, for a fixed-basis transpose \(\mathsf T_E\), its factorization properties are

\begin{equation} \begin{aligned} &\exists\ \mathcal D:\mathcal L(B)\to\mathcal L(E)\ \text{CPTP}, &&\mathsf T_E\circ\Phi_V^c=\mathcal D\circ\Phi_V,\\ &\nexists\ \widetilde{\mathcal D}:\mathcal L(B)\to\mathcal L(E)\ \text{CPTP}, &&\Phi_V^c=\widetilde{\mathcal D}\circ\Phi_V. \end{aligned} \tag{2} \end{equation}

A feasible triple is componentwise minimal if no distinct feasible \((d'_A,d'_B,d'_E)\) satisfies \(d'_X\leq d_X\) for every \(X\in\{A,B,E\}\). Determine all minimal triples satisfying Eq. (2).

Source

This dimension-refined problem is implicit in Singh and Datta’s explicit question about whether transpose degradability differs from degradability and in their support-minimal dimension formalism [SD22].

Progress

Support minimality in Eq. (1) gives the Choi-rank identities

\begin{equation} d_E=\operatorname{rank}J(\Phi_V), \qquad d_B=\operatorname{rank}J(\Phi_V^c). \tag{3} \end{equation}

Equation (3) removes artificial output or environment dimensions introduced by nonminimal dilations [SD22].
The first line of Eq. (2) makes \(J(\Phi_V^c)\) PPT, whereas separability of this Choi operator would make \(\Phi_V\) degradable. The low-rank PPT separability theorem therefore gives the necessary inequality

\begin{equation} d_B=\operatorname{rank}J(\Phi_V^c) >\max\{d_A,d_E\}. \tag{4} \end{equation}

Equation (4) is only an obstruction: a PPT-entangled Choi operator need not satisfy the channel factorization in Eq. (2) [Bra15], [HLVC00].
PPT is equivalent to separability on \(2\otimes2\) and \(2\otimes3\). Combining this fact with Eq. (4), the componentwise-minimal triples not excluded by the known tests are

\begin{equation} (d_A,d_B,d_E)\in \{(2,5,4),\ (3,4,3),\ (4,5,2)\}. \tag{5} \end{equation}

No triple in Eq. (5) is known to be realizable [HHH96], [HLVC00].

Comment

The feasible set may be empty because existence of any strict transpose-degradable channel is unresolved in Problem 53. Under channel complementation, the triples for the equivalent strict transpose-antidegradable formulation are obtained by interchanging \(d_B\) and \(d_E\).

References

[SD22]: S. Singh and N. Datta, “ Detecting Positive Quantum Capacities of Quantum Channels,” npj Quantum Information 8, 50 (2022).DOI arXiv
[Bra15]: K. Brádler, “ The Pitfalls of Deciding Whether a Quantum Channel Is (Conjugate) Degradable and How to Avoid Them,” Open Systems & Information Dynamics 22, 1550026 (2015).DOI arXiv
[HLVC00]: P. Horodecki, M. Lewenstein, G. Vidal, and I. Cirac, “ Operational Criterion and Constructive Checks for the Separability of Low-Rank Density Matrices,” Physical Review A 62, 032310 (2000).DOI arXiv
[HHH96]: M. Horodecki, P. Horodecki, and R. Horodecki, “ Separability of Mixed States: Necessary and Sufficient Conditions,” Physics Letters A 223, 1–8 (1996).DOI arXiv

Minimal dimensions for strict transpose degradability

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