pith. sign in

arxiv: 2604.12808 · v1 · submitted 2026-04-14 · 🪐 quant-ph

Distinguishability of locally diagonal orthogonally invariant quantum states

Nathaniel Johnston , Vincent Russo This is my paper

Pith reviewed 2026-05-10 16:16 UTC · model grok-4.3

classification 🪐 quant-ph
keywords LDOI statesLOCC distinguishabilityPPT measurementsseparable measurementsquantum state discriminationWerner statessymmetric quantum states
0
0 comments X

The pith

For LDOI quantum states, optimal PPT and separable measurements can always be chosen LDOI, reducing optimization from n^4 to O(n^2) variables.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Quantum states invariant under local diagonal orthogonal twirling form an important class that includes Werner states, isotropic states, X-states, and Dicke states. The paper shows that when distinguishing these states, the best positive-partial-transpose and separable measurements can themselves be restricted to the same invariant form. Local operations with classical communication can approach the same performance using invariant measurements. This symmetry restriction shrinks the size of the optimization problems from n to the fourth power down to order n squared. For many families of such states, including every two-qubit case, the three measurement types achieve identical performance, while the gap between PPT and LOCC is bounded by (n-2)/(2n^2) in general.

Core claim

We show that optimal PPT and separable measurements for distinguishing LDOI states can always be taken to be LDOI, and the LOCC supremum can be approached by LDOI LOCC POVMs, enabling a dimensional reduction from n^4 to O(n^2) in the associated optimization problems. We establish efficiently computable bounds on the distinguishability of orthonormal LDOI bases and prove that for a broad class of such bases -- including all two-qubit cases -- the LOCC supremum equals the PPT and separable optima. More generally, we show the gap between PPT and LOCC distinguishability is at most (n-2)/(2n^2) for local dimension n.

What carries the argument

Local diagonal orthogonal invariance (LDOI), the property that states are unchanged under local diagonal orthogonal twirling; this invariance lets optimal measurements be restricted to the same class without losing performance.

If this is right

  • Optimal PPT and separable measurements for any LDOI states can be chosen to be LDOI.
  • LOCC performance for LDOI states can be approached arbitrarily closely by LDOI LOCC POVMs.
  • Optimization problems for LDOI distinguishability reduce from n^4 to O(n^2) variables.
  • Efficiently computable upper and lower bounds exist for the distinguishability of orthonormal LDOI bases.
  • The gap between PPT and LOCC distinguishability is at most (n-2)/(2n^2) for any local dimension n.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same symmetry reduction might apply to distinguishability problems for other classes of symmetric states in quantum information.
  • Numerical searches for optimal measurements become feasible for larger local dimensions that were previously intractable.
  • The small explicit gap bound indicates PPT measurements are often nearly optimal for LOCC when states share this symmetry.

Load-bearing premise

The states must remain invariant under local diagonal orthogonal twirling in a manner that keeps the distinguishability value unchanged when measurements are restricted to the invariant subclass.

What would settle it

An explicit orthonormal LDOI basis in local dimension n=3 where the LOCC distinguishability is strictly less than the PPT value by more than 1/18.

read the original abstract

We study the distinguishability of quantum states under local operations with classical communication (LOCC), separable, and positive-partial-transpose (PPT) measurements, focusing on locally diagonal orthogonally invariant (LDOI) states -- those invariant under local diagonal orthogonal twirling. This class includes many important families such as Werner states, isotropic states, X-states, and Dicke states. We show that optimal PPT and separable measurements for distinguishing LDOI states can always be taken to be LDOI, and the LOCC supremum can be approached by LDOI LOCC POVMs, enabling a dimensional reduction from $n^4$ to $O(n^2)$ in the associated optimization problems. We establish efficiently computable bounds on the distinguishability of orthonormal LDOI bases and prove that for a broad class of such bases -- including all two-qubit cases -- the LOCC supremum equals the PPT and separable optima. More generally, we show the gap between PPT and LOCC distinguishability is at most $(n-2)/(2n^2)$ for local dimension $n$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper studies distinguishability of locally diagonal orthogonally invariant (LDOI) quantum states under LOCC, separable, and PPT measurements. It shows that optimal PPT and separable measurements for LDOI states can be taken LDOI, that the LOCC supremum can be approached by sequences of LDOI LOCC POVMs (enabling reduction from n^4 to O(n^2) variables), provides efficiently computable bounds on distinguishability of orthonormal LDOI bases, proves LOCC=PPT=separable for a broad class including all two-qubit cases, and bounds the PPT-LOCC gap by at most (n-2)/(2n^2).

Significance. If the claims hold, the work offers a useful symmetry-based simplification for optimization problems involving a large class of states (Werner, isotropic, X-states, Dicke). The dimensional reduction, explicit gap bound, and equality result for two-qubit cases are concrete advances that facilitate both analytical and numerical work in quantum information. The paper correctly credits the standard group-averaging technique for the convex cases (PPT, separable) and attempts an extension to LOCC.

major comments (1)
  1. [§4] §4 (LOCC approximation theorem): the assertion that the LOCC distinguishability supremum can be approached by LDOI LOCC POVMs is load-bearing for both the O(n^2) reduction and the (n-2)/(2n^2) gap bound, yet the argument relies on an approximation step whose preservation of the LOCC property is not fully detailed. Group averaging yields a separable POVM, but because LOCC is non-convex, an explicit construction, density argument, or reference showing that LDOI LOCC approximants exist and achieve the same value is required; without it the reduction does not rigorously apply to the LOCC case.
minor comments (2)
  1. [§2] Notation for the local diagonal orthogonal twirling operator is introduced without a displayed equation in the preliminaries; adding an explicit formula would improve readability.
  2. [Abstract and §1] The statement of the gap bound in the abstract and introduction should clarify whether it applies to the worst-case orthonormal LDOI basis or to a specific family.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for their thorough review and for recognizing the potential utility of our symmetry-based simplifications for LDOI states. We address the single major comment below and have made revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: [§4] §4 (LOCC approximation theorem): the assertion that the LOCC distinguishability supremum can be approached by LDOI LOCC POVMs is load-bearing for both the O(n^2) reduction and the (n-2)/(2n^2) gap bound, yet the argument relies on an approximation step whose preservation of the LOCC property is not fully detailed. Group averaging yields a separable POVM, but because LOCC is non-convex, an explicit construction, density argument, or reference showing that LDOI LOCC approximants exist and achieve the same value is required; without it the reduction does not rigorously apply to the LOCC case.

    Authors: We thank the referee for pointing out this important detail. Upon re-examination, we agree that the preservation of the LOCC property under the approximation requires a more explicit justification. In the revised manuscript, we have expanded §4 with a detailed construction: we first note that the set of LDOI LOCC POVMs is dense in the sense that any LOCC protocol can be approximated by one that is invariant under the local diagonal orthogonal group by averaging the local operations at each communication round. Since the states are LDOI, the success probability remains unchanged under this averaging. This provides the required density argument, allowing the supremum to be approached by LDOI LOCC POVMs. Consequently, the O(n^2) reduction and the gap bound hold for the LOCC case as well. We have also added a remark clarifying why group averaging works here despite non-convexity, by performing it locally. revision: yes

Circularity Check

0 steps flagged

No circularity: results follow from invariance and convexity without self-referential reduction

full rationale

The derivation proceeds by applying the local diagonal orthogonal twirling operator to the states (which fixes them by definition of LDOI) and to candidate measurements. For PPT and separable classes the twirled POVM remains inside the class because both are convex and closed under local operations; the figure of merit is linear, so the averaged value is at least as good, yielding the claimed optimality and the O(n^2) reduction directly from the invariance. For LOCC the paper asserts only that the supremum is approachable by a sequence of LDOI LOCC POVMs, which is a standard density statement inside the (non-convex) LOCC set and does not equate any quantity to itself or rename a fitted parameter. No self-citation is invoked as a uniqueness theorem or load-bearing premise, and no ansatz is smuggled in. All steps are self-contained against the definitions of LOCC, PPT, and separability.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard axioms of quantum mechanics and information theory with no free parameters or new postulated entities.

axioms (2)
  • standard math Quantum states are density operators on finite-dimensional Hilbert spaces.
    Invoked implicitly when defining LDOI states and their distinguishability.
  • domain assumption Measurements are POVMs; LOCC, separable, and PPT classes are defined by their standard operational constraints.
    Used throughout to compare the three measurement types.

pith-pipeline@v0.9.0 · 5479 in / 1584 out tokens · 48816 ms · 2026-05-10T16:16:35.438097+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [1]

    CVXOPT : Convex Optimization

    Martin Andersen, Joachim Dahl, and Lieven Vandenberghe. CVXOPT : Convex Optimization . Astrophysics Source Code Library , page ascl:2008.017, 2020. ADS Bibcode: 2020ascl.soft08017A

    work page 2008

  2. [2]

    Dense quantum coding and a lower bound for 1-way quantum automata

    Andris Ambainis, Ashwin Nayak, Amnon Ta-Shma, and Umesh Vazirani. Dense quantum coding and a lower bound for 1-way quantum automata. In Proceedings of the thirty-first annual ACM symposium on Theory of computing , pages 376--383. ACM, 1999

    work page 1999

  3. [3]

    More nonlocality with less purity

    Somshubhro Bandyopadhyay. More nonlocality with less purity. Physical Review Letters , 106(21):210402, 2011

    work page 2011

  4. [4]

    Limitations on separable measurements by convex optimization

    Somshubhro Bandyopadhyay, Alessandro Cosentino, Nathaniel Johnston, Vincent Russo, John Watrous, and Nengkun Yu. Limitations on separable measurements by convex optimization. IEEE Transactions on Information Theory , 61(6):3593--3604, 2015

    work page 2015

  5. [5]

    Quantum nonlocality without entanglement

    Charles H Bennett, David P DiVincenzo, Christopher A Fuchs, Tal Mor, Eric Rains, Peter W Shor, John A Smolin, and William K Wootters. Quantum nonlocality without entanglement. Physical Review A , 59(2):1070, 1999

    work page 1999

  6. [6]

    Tight bounds on the distinguishability of quantum states under separable measurements

    Somshubhro Bandyopadhyay and Michael Nathanson. Tight bounds on the distinguishability of quantum states under separable measurements. Physical Review A , 88(5):052313, 2013

    work page 2013

  7. [7]

    Bennett and Stephen J

    Charles H. Bennett and Stephen J. Wiesner. Communication via one- and two-particle operators on E instein- P odolsky- R osen states. Physical Review Letters , 69(20):2881--2884, 1992

    work page 1992

  8. [8]

    Class of positive partial transposition states

    Dariusz Chru \'s ci \'n ski and Andrzej Kossakowski. Class of positive partial transposition states. Physical Review A—Atomic, Molecular, and Optical Physics , 74(2):022308, 2006

    work page 2006

  9. [9]

    Everything you always wanted to know about LOCC (but were afraid to ask)

    Eric Chitambar, Debbie Leung, Laura Man c inska, Maris Ozols, and Andreas Winter. Everything you always wanted to know about LOCC (but were afraid to ask). Communications in Mathematical Physics , 328(1):303--326, 2014

    work page 2014

  10. [10]

    Positive-partial-transpose-indistinguishable states via semidefinite programming

    Alessandro Cosentino. Positive-partial-transpose-indistinguishable states via semidefinite programming. Physical Review A , 87(1):012321, 2013

    work page 2013

  11. [11]

    Small sets of locally indistinguishable orthogonal maximally entangled states

    Alessandro Cosentino and Vincent Russo. Small sets of locally indistinguishable orthogonal maximally entangled states. Quantum Information and Computation , 14(13--14):1098--1106, 2014

    work page 2014

  12. [12]

    Distinguishability of quantum states by separable operations

    Runyao Duan, Yuan Feng, Yu Xin, and Mingsheng Ying. Distinguishability of quantum states by separable operations. IEEE Transactions on Information Theory , 55(3):1320--1330, 2009

    work page 2009

  13. [13]

    Local indistinguishability of orthogonal pure states by using a bound on distillable entanglement

    Sibasish Ghosh, Guruprasad Kar, Anirban Roy, Debasis Sarkar, Aditi Sen, Ujjwal Sen, et al. Local indistinguishability of orthogonal pure states by using a bound on distillable entanglement. Physical Review A , 65(6):062307, 2002

    work page 2002

  14. [14]

    Helstrom

    Carl W. Helstrom. Quantum detection and estimation theory. Journal of Statistical Physics , 1(2):231--252, 1969

    work page 1969

  15. [15]

    Reduction criterion of separability and limits for a class of distillation protocols

    Micha Horodecki and Pawel Horodecki. Reduction criterion of separability and limits for a class of distillation protocols. Physical Review A , 59:4206--4216, 1999

    work page 1999

  16. [16]

    One-parameter family of indecomposable optimal entanglement witnesses arising from generalized C hoi maps

    Kil-Chan Ha and Seung-Hyeok Kye. One-parameter family of indecomposable optimal entanglement witnesses arising from generalized C hoi maps. Physical Review A , 84:024302, 2011

    work page 2011

  17. [17]

    Bounds for the quantity of information transmitted by a quantum communication channel

    Alexander Semenovich Holevo. Bounds for the quantity of information transmitted by a quantum communication channel. Problemy Peredachi Informatsii , 9(3):3--11, 1973

    work page 1973

  18. [18]

    Local indistinguishability: M ore nonlocality with less entanglement

    Micha Horodecki, Aditi Sen, Ujjwal Sen, and Karol Horodecki. Local indistinguishability: M ore nonlocality with less entanglement. Physical review letters , 90(4):047902, 2003

    work page 2003

  19. [19]

    Pairwise completely positive matrices and conjugate local diagonal unitary invariant quantum states

    Nathaniel Johnston and Olivia MacLean. Pairwise completely positive matrices and conjugate local diagonal unitary invariant quantum states. Electronic Journal of Linear Algebra , 35:156--180, 2019

    work page 2019

  20. [20]

    Harold W. Kuhn. The H ungarian method for the assignment problem. Naval Research Logistics Quarterly , 2:83--97, 1955

    work page 1955

  21. [21]

    Nielsen and Isaac L

    Michael A. Nielsen and Isaac L. Chuang. Quantum C omputation and Q uantum I nformation, 2002

    work page 2002

  22. [22]

    A graphical calculus for integration over random diagonal unitary matrices

    Ion Nechita and Satvik Singh. A graphical calculus for integration over random diagonal unitary matrices. Linear Algebra and its Applications , 613:46--86, 2021

    work page 2021

  23. [23]

    Nicolás Quesada, Asma Al-Qasimi, and Daniel F. V. James. Quantum properties and dynamics of X states. Journal of Modern Optics , 59(15):1322--1329, 2012

    work page 2012

  24. [24]

    toqito -- Theory of quantum information toolkit: A Python package for studying quantum information

    Vincent Russo. toqito -- Theory of quantum information toolkit: A Python package for studying quantum information. Journal of Open Source Software , 6(61):3082, 2021

    work page 2021

  25. [25]

    Diagonal unitary and orthogonal symmetries in quantum theory

    Satvik Singh and Ion Nechita. Diagonal unitary and orthogonal symmetries in quantum theory. Quantum , 5:519, 2021

    work page 2021

  26. [26]

    PICOS : A Python interface to conic optimization solvers

    Guillaume Sagnol and Maximilian Stahlberg. PICOS : A Python interface to conic optimization solvers. Journal of Open Source Software , 7(70):3915, 2022

    work page 2022

  27. [27]

    Separability of diagonal symmetric states: A quadratic conic optimization problem

    Jordi Tura, Albert Aloy, Ruben Quesada, Maciej Lewenstein, and Anna Sanpera. Separability of diagonal symmetric states: A quadratic conic optimization problem. Quantum , 2:45, 2018

    work page 2018

  28. [28]

    Optimal local discrimination of two multipartite pure states

    Shashank Virmani, Massimiliano F Sacchi, Martin B Plenio, and Damian Markham. Optimal local discrimination of two multipartite pure states. Physics Letters A , 288(2):62--68, 2001

    work page 2001

  29. [29]

    The T heory of Q uantum I nformation

    John Watrous. The T heory of Q uantum I nformation . Cambridge University Press, 2018

    work page 2018

  30. [30]

    Reinhard F. Werner. Quantum states with E instein- P odolsky- R osen correlations admitting a hidden-variable model. Physical Review A , 40:4277--4281, 1989

    work page 1989

  31. [31]

    Nonlocality, asymmetry, and distinguishing bipartite states

    Jonathan Walgate and Lucien Hardy. Nonlocality, asymmetry, and distinguishing bipartite states. Physical Review Letters , 89(14):147901, 2002

    work page 2002

  32. [32]

    Mark M. Wilde. Quantum I nformation T heory . Cambridge University Press, 2013

    work page 2013

  33. [33]

    Local distinguishability of multipartite orthogonal quantum states

    Jonathan Walgate, Anthony J Short, Lucien Hardy, and Vlatko Vedral. Local distinguishability of multipartite orthogonal quantum states. Physical Review Letters , 85(23):4972, 2000

    work page 2000

  34. [34]

    Four locally indistinguishable ququad-ququad orthogonal maximally entangled states

    Nengkun Yu, Runyao Duan, and Mingsheng Ying. Four locally indistinguishable ququad-ququad orthogonal maximally entangled states. Physical review letters , 109(2):020506, 2012

    work page 2012

  35. [35]

    Distinguishability of quantum states by positive operator-valued measures with positive partial transpose

    Nengkun Yu, Runyao Duan, and Mingsheng Ying. Distinguishability of quantum states by positive operator-valued measures with positive partial transpose. IEEE Transactions on Information Theory , 60(4):2069--2079, 2014

    work page 2069

  36. [36]

    Separability of a mixture of D icke states

    Nengkun Yu. Separability of a mixture of D icke states. Physical Review A , 94:060101(R), 2016

    work page 2016