Superactivation of two-way secret-key capacity
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- Topics
Problem
Can two quantum channels with zero two-way-assisted secret-key capacity have positive secret-key capacity when used jointly? Let \(\mathcal N:A_0\to B_0\) and \(\mathcal M:A_1\to B_1\) be independent finite-dimensional memoryless quantum channels. For a channel \(\mathcal C\), let \(K_{\leftrightarrow}(\mathcal C)\) denote its secret-key capacity, in secret bits per channel use, when arbitrary adaptive local operations and unlimited authenticated two-way public communication are allowed between channel uses. Alice and Bob start without shared entanglement or secret key of positive rate. Eve receives every complementary-channel output and the full public transcript; the probability that Alice’s and Bob’s keys differ, and the trace distance between the key and a uniform key independent of Eve, must vanish asymptotically. One use of \(\mathcal N\otimes\mathcal M\) means one use of each channel, with independent environments and ordinary causal ordering. The question is whether there is a pair satisfying
Source
This formulation is derived. It combines the two-way-assisted capacity framework of [PLOB17] with the open existence question for entangled zero-key states stated in Sec. 1 of [HSDW26]. The unassisted private-capacity counterpart is described as a longstanding open problem, and resolved, in [ZW26]; the two-way-assisted question in Eq. (1) is not stated as a separate problem in the cited papers.
Progress
A channel is entanglement breaking when it outputs a separable state whenever it acts on one part of any input state; equivalently, its Choi state is separable [HSR03]. Such channels have \(K_{\leftrightarrow}=0\): the two-way-assisted private capacity of every channel is bounded by its squashed entanglement [TGW14], which vanishes when every output is separable. Tensor products of entanglement-breaking channels remain entanglement breaking, hence
\begin{equation} K_{\leftrightarrow}(\mathcal N\otimes\mathcal M)=0 \tag{2} \end{equation}when both factors are entanglement breaking. By Eq. (2), any pair satisfying Eq. (1) needs at least one zero-key channel outside this class.
For the \(d\)-dimensional erasure channel
\begin{equation} \mathcal E_{d,p}(\rho):=(1-p)\rho\oplus p\operatorname{Tr}(\rho)|e\rangle\langle e|, \qquad d\geq2,\quad0\leq p\leq1, \tag{3} \end{equation}with erasure flag \(|e\rangle\) orthogonal to the input space, Pirandola, Laurenza, Ottaviani, and Banchi determine the exact capacity [PLOB17]
\begin{equation} K_{\leftrightarrow}(\mathcal E_{d,p})=(1-p)\log_2d. \tag{4} \end{equation}In particular, the qubit erasure channel with \(p=1/2\), which is antidegradable [ZW26], has \(K_{\leftrightarrow}=1/2\), not zero. Antidegradability therefore does not certify the vanishing two-way key capacity required in Eq. (1).
Let \(J_{\mathcal N}:=(\operatorname{id}\otimes\mathcal N)(|\Phi_d\rangle\langle\Phi_d|)\) be the normalized Choi state, where \(d:=\dim A_0\) and \(|\Phi_d\rangle:=d^{-1/2}\sum_{j=0}^{d-1}|jj\rangle\). Sending halves of \(|\Phi_d\rangle\) through the channel and then distilling key from the resulting copies of \(J_{\mathcal N}\) gives
\begin{equation} K_{\leftrightarrow}(\mathcal N)\geq K_D(J_{\mathcal N}), \tag{5} \end{equation}where \(K_D\) is the distillable key of a state against an adversary holding a purification; the same Choi-state distribution underlies the lower bounds of [PLOB17]. The Choi state of a channel that is not entanglement breaking is entangled [HSR03], so by Eq. (5) such a channel with \(K_{\leftrightarrow}=0\) would give an entangled state with zero distillable key. The existence of such states is explicitly unresolved in [HSDW26].
Consequently, faithfulness of state distillable key, meaning that every entangled state has \(K_D>0\), would imply
\begin{equation} K_{\leftrightarrow}(\mathcal N)=0 \quad\Longrightarrow\quad J_{\mathcal N}\in\mathrm{SEP} \quad\Longrightarrow\quad \mathcal N\ \text{is entanglement breaking}, \tag{6} \end{equation}where \(\mathrm{SEP}\) denotes separable states. Together with Eq. (2), Eq. (6) would exclude Eq. (1). This is a conditional deduction, not an established characterization of zero key capacity [HSR03], [HSDW26].
Zhu and Wang’s September 2026 preprint instead constructs a four-level channel and a qubit erasure channel with \(1/2\leq p<1\), each with zero unassisted private capacity \(P\), whose joint use has positive \(P\) (Theorem 1 of [ZW26]). At half erasure, its helper obeys
\begin{equation} P(\mathcal E_{2,1/2})=0 \quad\text{but}\quad K_{\leftrightarrow}(\mathcal E_{2,1/2})=\frac12 \tag{7} \end{equation}by Eq. (4). That construction therefore does not provide a pair satisfying Eq. (1) under two-way public communication.
Comment
No pair satisfying Eq. (1), and no proof that the set of channels with \(K_{\leftrightarrow}=0\) is closed under tensor products, is known. Zero unassisted private capacity, or a zero rate for a chosen key-distribution protocol, is insufficient.
The record Secret key from every entangled state asks whether every entangled state has positive distillable key. An affirmative answer there would settle this question negatively through Eqs. (2) and (6). Conversely, by Eq. (2), a pair satisfying Eq. (1) would contain a zero-key channel that is not entanglement breaking, and its Choi state would answer that record negatively. The two questions are not claimed to be equivalent. The record Superactivation of distillable secret key asks the corresponding question for states; no implication between the state and channel versions is claimed. Superactivation of the unassisted quantum and private capacities with antidegradable helpers is the subject of the record on channels superactivated by antidegradable helpers. Literature checked through 15 September 2026.
References
- [PLOB17]
- S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, “Fundamental Limits of Repeaterless Quantum Communications,” Nature Communications 8, 15043 (2017).DOIarXiv
- [HSR03]
- M. Horodecki, P. W. Shor, and M. B. Ruskai, “Entanglement Breaking Channels,” Reviews in Mathematical Physics 15, 629–641 (2003).DOIarXiv
- [TGW14]
- M. Takeoka, S. Guha, and M. M. Wilde, “The Squashed Entanglement of a Quantum Channel,” IEEE Transactions on Information Theory 60, 4987–4998 (2014).DOIarXiv