Chernoff exponents under separable and LOCC tests across copies

Problem

Determine the optimal symmetric discrimination exponents for two faithful noncommuting states when measurements must be separable across copies or implemented by local operations and classical communication (LOCC) among the copies. In particular, do both classes have the same asymptotic exponent as the best fixed single-copy measurement?

Let \(\rho,\sigma>0\) be density operators on a finite-dimensional Hilbert space \(\mathcal H\), with \([\rho,\sigma]\neq0\). The hypotheses \(\rho^{\otimes n}\) and \(\sigma^{\otimes n}\) have equal prior probability. Each of \(n\) parties holds exactly one tensor factor. For a class \(\mathcal C_n\) of two-outcome POVMs, define the optimal error by Eq. (1):

\begin{equation} p_e^{\mathcal C}(n):=\inf_{\{T,I-T\}\in\mathcal C_n}\frac12\left[\operatorname{Tr}\rho^{\otimes n}(I-T)+\operatorname{Tr}\sigma^{\otimes n}T\right]. \tag{1} \end{equation}

Here \(\mathrm{SEP}_n\) consists of tests for which both effects are fully separable: each effect has the form \(\sum_j A_{1,j}\otimes\cdots\otimes A_{n,j}\) with \(A_{k,j}\geq0\). The class \(\mathrm{LOCC}_n\) consists of tests exactly implementable by a finite number of rounds of local operations and classical messages, with arbitrary two-way communication and no shared entangled resource. The number of rounds can depend on \(n\). No party may jointly operate on two input copies.

To avoid assuming convergence, define lower and upper exponents as in Eq. (2):

\begin{equation} \underline\xi_{\mathcal C}(\rho,\sigma):=\liminf_{n\to\infty}-\frac1n\log p_e^{\mathcal C}(n), \qquad \overline\xi_{\mathcal C}(\rho,\sigma):=\limsup_{n\to\infty}-\frac1n\log p_e^{\mathcal C}(n). \tag{2} \end{equation}

All logarithms are natural. For a finite-outcome single-copy POVM \(M=\{M_\omega\}_\omega\), put \(P^M_\rho(\omega):=\operatorname{Tr}(\rho M_\omega)\) and define the fixed-measurement Chernoff exponent in Eq. (3):

\begin{equation} C_M(\rho,\sigma):=\sup_M\max_{0\leq s\leq1}\left[-\log\sum_\omega P^M_\rho(\omega)^{1-s}P^M_\sigma(\omega)^s\right]. \tag{3} \end{equation}

The question is whether Eq. (4) holds for every such pair, and, if it fails, what the correct exponents are:

\begin{equation} \underline\xi_{\mathrm{LOCC}}=\overline\xi_{\mathrm{LOCC}} =\underline\xi_{\mathrm{SEP}}=\overline\xi_{\mathrm{SEP}} =C_M(\rho,\sigma). \tag{4} \end{equation}

Source

Hayashi explicitly identifies the two-way LOCC and separable-measurement exponents beyond the Stein setting as unresolved in Section IX, p. 16 of his arXiv paper [Hay09]. This statement isolates the symmetric Chernoff question and makes the separation across individual copies explicit.

Progress

For faithful states, Hayashi’s Section V, p. 9, reduces one-way adaptive single-copy measurements to classical channel discrimination. The optimal Chernoff exponent equals \(C_M(\rho,\sigma)\), so adaptation in this one-way model does not improve the best fixed-measurement exponent. Section IX leaves the corresponding two-way and separable cases open [Hay09].
With unrestricted collective measurements, the quantum Chernoff theorem gives Eq. (5):

\begin{equation} \xi_{\mathrm{ALL}}(\rho,\sigma)=C_Q(\rho,\sigma):=-\log\min_{0\leq s\leq1}\operatorname{Tr}(\rho^{1-s}\sigma^s). \tag{5} \end{equation}

Thus \(C_Q(\rho,\sigma)\) is an upper bound on both upper exponents in Eq. (2); this collective result does not establish the restricted equality in Eq. (4) [ACM+07].
For mixed qubits, Calsamiglia et al. obtain finite-copy lower bounds on LOCC error from positivity under partial transposition. Their Figure 1 and asymptotic fits using \(25\leq n\leq35\) are consistent with the fixed-measurement exponent, but do not exclude a small asymptotic improvement by two-way LOCC or separable tests [CdVMB10].

Comment

The locality constraint is across the \(n\) individual input copies. A model with two fixed spatial parties holding \(A^{\otimes n}\) and \(B^{\otimes n}\) and allowed collective operations within each laboratory is different. The one-way Chernoff result and numerical evidence do not settle arbitrary two-way communication or fully separable tests. The lower/upper-exponent formulation includes existence of the ordinary limit requested in the source note instead of assuming it. No exact duplicate was found in the authored catalog, ledger proposals, or private pool. Primary sources and later-work searches were checked through 9 September 2026; the audit was not exhaustive. Hayashi’s explicit originating research-paper passage was inspected directly.

References

[Hay09]: M. Hayashi, “Discrimination of Two Channels by Adaptive Methods and Its Application to Quantum System,” IEEE Transactions on Information Theory 55, 3807–3820 (2009).DOI arXiv
[ACM+07]: K. M. R. Audenaert, J. Calsamiglia, Ll. Masanes, R. Muñoz-Tapia, A. Acín, E. Bagan, and F. Verstraete, “Discriminating States: The Quantum Chernoff Bound,” Physical Review Letters 98, 160501 (2007).DOI arXiv
[CdVMB10]: J. Calsamiglia, J. I. de Vicente, R. Muñoz-Tapia, and E. Bagan, “Local Discrimination of Mixed States,” Physical Review Letters 105, 080504 (2010).DOI arXiv

Chernoff exponents under separable and LOCC tests across copies

Problem

Source

Progress

Comment

References

Cite this problem

BibTeX

Plain text

Problem

Source

Progress

Comment

References

Related problems