Structural criterion for classical–entanglement trade-off advantage

Unsolved ID op_ad12295d8fdfb02a Last edited 4 September 2026

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Fields: Quantum Communication
Quantum Resource Theory
Topics: Classical capacity
Entanglement-assisted communication
Additivity and regularization

Problem

Characterize the finite-dimensional quantum channels for which joint classical–entanglement coding strictly outperforms time sharing between unassisted and unlimited-entanglement classical communication. For a channel $\mathcal N:A'\to B$, let $C_{\rm CE}(\mathcal N,e)$ be the supremum of asymptotically achievable classical rates when at most $e$ ebits per channel use are consumed. Operationally,

\begin{equation} C_{\rm CE}(\mathcal N,e) :=\sup\left\{ R:\begin{array}{l} \text{there are length-$n$ classical-message codes with}\\ n^{-1}\log_2M_n\to R,\quad \limsup_{n\to\infty}n^{-1}\log_2K_n\leq e,\quad P_{\rm err}^{(n)}\to0 \end{array} \right\}, \tag{1} \end{equation}

where $K_n$ in Eq. (1) is the Schmidt rank of the preshared maximally entangled resource. Define the unassisted and unlimited-entanglement endpoints by

\begin{equation} C_0(\mathcal N):=C_{\rm CE}(\mathcal N,0), \qquad C_{\rm EA}(\mathcal N):=\sup_{e\geq0}C_{\rm CE}(\mathcal N,e), \qquad e_{\rm EA}(\mathcal N) :=\inf\{e:C_{\rm CE}(\mathcal N,e)=C_{\rm EA}(\mathcal N)\}. \tag{2} \end{equation}

For $e_{\rm EA}(\mathcal N)>0$, endpoint time sharing gives

If the infimum in Eq. (2) is not attained, interpret Eq. (3) through its operational closure; if $e_{\rm EA}(\mathcal N)=0$, set $C_{\rm TS}(\mathcal N,e)=C_{\rm EA}(\mathcal N)$. Give intrinsic necessary and sufficient conditions for strict suboptimality of this benchmark:

\begin{equation} \exists e>0:\qquad C_{\rm CE}(\mathcal N,e)>C_{\rm TS}(\mathcal N,e). \tag{4} \end{equation}

Source

Wilde explicitly asks how to determine which channels benefit from trade-off coding rather than time sharing in Section 22.5 of his text [Wil17]. Brádler, Hayden, Touchette, and Wilde likewise call for a general method that determines the gain over time sharing [BHTW10].

Progress

Hsieh and Wilde proved a matching achievable region and multiletter converse for simultaneous classical communication, quantum communication, and entanglement. The classical–entanglement slice determines $C_{\rm CE}(\mathcal N,e)$ through a regularized optimization, but it does not provide an intrinsic channel-level criterion for Eq. (4) [HW10].
Brádler, Hayden, Touchette, and Wilde derived exact trade-off regions for Hadamard channels. They exhibit strict improvement over endpoint time sharing in nontrivial parameter regimes, including finite-dimensional dephasing and $1\to N$ cloning channels, and introduce a quantitative measure of the gain. The trivial parameter endpoints need not show strict advantage, and the examples do not characterize all finite-dimensional channels [BHTW10].

Comment

Strict trade-off advantage itself is known to occur. The unresolved problem is a necessary-and-sufficient structural characterization, including any regularization across tensor powers, that identifies when endpoint time sharing is optimal. The statement is deliberately restricted to the precisely defined classical–entanglement trade-off; it does not use an informal full-dynamic time-sharing region.

References

[Wil17]: M. M. Wilde, Quantum Information Theory, 2nd ed., Cambridge University Press (2017), Section 22.5.DOI arXiv
[HW10]: M.-H. Hsieh and M. M. Wilde, “Trading Classical Communication, Quantum Communication, and Entanglement in Quantum Shannon Theory,” IEEE Trans. Inf. Theory 56, 4705–4730 (2010).DOI arXiv
[BHTW10]: K. Brádler, P. Hayden, D. Touchette, and M. M. Wilde, “Trade-Off Capacities of the Quantum Hadamard Channels,” Physical Review A 81, 062312 (2010).DOI arXiv

Structural criterion for classical–entanglement trade-off advantage

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