Single-copy identity testing of local Gibbs states
- Fields
- Topics
Problem
What is the minimax copy complexity of identity testing two unknown \(k\)-local Gibbs states using only measurements on individual copies?
For fixed \(k\geq2\), let
where \(P\) ranges over nonidentity \(n\)-qubit Pauli strings and \(|P|\) is Pauli weight. No bounded-degree or geometric promise is imposed. Independent copies of unknown \(\rho,\sigma\in\mathcal G_{n,k,\beta}\) are supplied under the promise
Let \(N^*_{\mathrm{sc}}(n,k,\beta,\varepsilon)\) be the least worst-case total copy count for success probability at least \(2/3\), allowing adaptive measurements on individual copies but no joint measurement across copies. Determine \(N^*_{\mathrm{sc}}\) up to logarithmic factors, including its dependence on \(n\), \(\beta\), and \(\varepsilon\), for the class in Eq. (1) and the test in Eq. (2).
Source
This precise formulation is editor wording based on the unresolved direction and limitations documented in the cited primary literature [Bluhm26]; it is not presented as a verbatim conjecture of those authors.
Progress
Reports do not certify correctness or automatically change the problem's status. Progress policy.
Published provenance and progress. Theorem 17 of [Bluhm26] supplies a polynomial-time, single-copy protocol; it includes the case where both states are unknown. After harmless constant rescaling of the promise gap, its upper bound gives
\begin{equation} N^*_{\mathrm{sc}} \leq\widetilde O\!\left( 3^k k\beta^2n^{2k}\varepsilon^{-4} \right). \tag{3} \end{equation}The displayed definitions, constraints, and target bounds are recorded in Eqs. (3).
The same paper explicitly identifies the absence of a matching lower bound as an open question in Section 1.3. [Bluhm26]
Comment
Retained as an explicit quantitative open problem. The precise minimax function is the research target; the displayed upper bound is not claimed to be optimal.
The Gibbs promise applies to both states. Removing that promise from the device state produces a different certification problem. Likewise, a theorem with a known reference Hamiltonian should not silently be substituted for two-unknown-state comparison; the two-unknown extension is expressly included in the source used here.
As a simple baseline, a one-qubit classical subfamily already exhibits ordinary statistical estimation/testing costs in the error parameter. Such a baseline does not constitute a matching many-body lower bound. Establishing a genuinely many-body hard family is a natural first step.
References
- [Bluhm26]
- Andreas Bluhm, Matthias C. Caro, Francisco Escudero Gutiérrez, Junseo Lee, Aadil Oufkir, Cambyse Rouzé, and Myeongjin Shin, Certifying and learning local quantum Hamiltonians. March 31, 2026. Locate: Theorems 16 and 17; Section 1.3, “Time-efficient Gibbs state learning” and “Optimal Gibbs state certification.”arXiv