bibkey: hiaimosonyihayashi2009groupsymmetry
authors: Fumio Hiai and Milan Mosonyi and Masahito Hayashi
year: 2009
title: Quantum hypothesis testing with group symmetry
doi: 10.1063/1.3234186
url: https://arxiv.org/abs/0904.0704v2
claim: Section 7 asks whether the asymptotic logarithmic overlap is differentiable throughout (0,1) for group-invariant state discrimination.
strata_touched:
  - D5/S3/Quantum/Information/GroupInvariantPsi/Defs
  - D5/S3/Quantum/Information/GroupInvariantPsi/Mixture
  - D5/S3/Quantum/Information/GroupInvariantPsi/Sandwich
  - D5/S3/Quantum/Information/GroupInvariantPsi/Result
license: "not verified; bibliographic attribution and a short attributed quotation, no source proof code copied"
triage: anchor

Quantum hypothesis testing with group symmetry

Verified locator

DOI: 10.1063/1.3234186

https://arxiv.org/abs/0904.0704v2

F. Hiai, M. Mosonyi and M. Hayashi, Quantum hypothesis testing with group symmetry, J. Math. Phys. 50 (2009) 103304. Section 7 gives the main open question about differentiability. Equation (2.1) defines the conditional expectation as the Haar average of tensor conjugates; equation (3.1) defines the normalized asymptotic logarithmic trace overlap.

Source objects and formal interpretation

For states on a finite-dimensional complex Hilbert space and a unitary representation of a compact group, the tensor states are averaged under the tensor representation. The finite overlap is the logarithm of the trace of their spectral powers with exponents s and 1-s. Its normalized limit is denoted by psi. The source proves bounds without assuming differentiability and asks whether psi is differentiable on (0,1).

Defs retains the source objects for finite groups, whose normalized Haar measure is uniform. It restricts to positive definite trace-one matrices and makes convergence an explicit premise of the differentiability claim. A finite-group example in this faithful domain is sufficient to refute the compact-group assertion. Tensor coordinates are functions from Fin n to Fin d, and matrix powers use the continuous functional calculus.

Mathematical consequence

Mixture identifies diagonal permutation twirls with classical orbit mixtures. Sandwich.psi_eq_max proves convergence and identifies the limit with the logarithm of the largest relative orbit overlap for strictly positive diagonal inputs. Result.result refutes the differentiability claim using p=(16,9,4,1)/30, q=(4,1,9,16)/30 and the involution (12)(34). The exact left and right slopes are respectively -(2/7)log(4/3) and (2/7)log(4/3).

Mosonyi–Szilágyi–Weiner, arXiv:2011.04645, Remark IV.9 addresses the separate conjecture psi=psi° with an invariant alternative hypothesis. That statement is distinct from the differentiability assertion treated here.