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arxiv: 2312.17709 · v2 · pith:RT2BSL6Nnew · submitted 2023-12-29 · 🪐 quant-ph · math.CO

Boson-fermion complementarity in a linear interferometer: An identity relating the determinant and permanent of a matrix

Michael G. Jabbour , Nicolas J. Cerf This is my paper

Pith reviewed 2026-05-24 04:50 UTC · model grok-4.3

classification 🪐 quant-ph math.CO
keywords boson-fermion complementaritylinear interferometerpermanent-determinant identitymultiparticle interferencetransition probabilitiesquantum statisticsboson bunchingfermion antibunching
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The pith

Bosonic and fermionic transition probabilities in any linear interferometer are linked by a single equation that also connects the squared moduli of the permanent and determinant of a matrix.

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

The paper establishes that bosonic and fermionic multiparticle interferences inside an arbitrary linear interferometer combine into one fundamental relation. This equation constrains the transition probabilities of the two particle types independently of the interferometer's specific form. For two particles the average of the bosonic and fermionic probabilities equals the probability that classical distinguishable particles would obey. The same relation supplies a previously unknown identity that equates the squared modulus of the permanent of any complex matrix to a combination involving the squared modulus of its determinant, thereby extending an identity obtained by Muir in the nineteenth century.

Core claim

The bosonic and fermionic transition probabilities appear together in a single equation which constrains their values, hence expressing a boson-fermion complementarity that is independent of the details of the interferometer; this also provides a heretofore unknown mathematical identity connecting the squared moduli of the permanent and determinant of an arbitrary complex matrix.

What carries the argument

The unifying equation that places bosonic and fermionic transition probabilities together for particles evolving under a unitary single-particle transformation.

If this is right

  • For two particles the average of bosonic and fermionic probabilities must coincide with the classical probability for any unitary interferometer.
  • The complementarity relation holds for every possible linear interferometer and is therefore independent of the concrete unitary matrix.
  • The same equation yields an identity relating the squared modulus of the permanent of an arbitrary complex matrix to the squared modulus of its determinant.
  • The identity extends Muir's nineteenth-century result to a new form connecting permanent and determinant.

Where Pith is reading between the lines

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

  • The relation could be used to verify the correct implementation of bosonic or fermionic statistics without precise knowledge of the interferometer unitary.
  • The mathematical identity may supply new bounds or computational relations between permanents and determinants that apply outside physics.
  • Similar complementarity equations might exist when more than two particles are involved or when the interferometer includes controlled nonlinear elements.

Load-bearing premise

The interferometer is described by a unitary matrix on the single-particle space and the many-particle states are formed by standard symmetrization for bosons or antisymmetrization for fermions with no extra interactions or loss.

What would settle it

Measure the two-particle bosonic and fermionic transition probabilities through any linear interferometer and check whether their average equals the corresponding classical distinguishable-particle probability; any systematic deviation falsifies the claimed relation.

Figures

Figures reproduced from arXiv: 2312.17709 by Michael G. Jabbour, Nicolas J. Cerf.

Figure 1
Figure 1. Figure 1: FIG. 1. Examples of the components of the fundamental [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

Bosonic and fermionic statistics are well known to give rise to antinomic behaviors, most notably boson bunching vs fermion antibunching. Here, we establish a fundamental relation that combines bosonic and fermionic multiparticle interferences in an arbitrary linear interferometer. The bosonic and fermionic transition probabilities appear together in a single equation which constrains their values, hence expressing a boson-fermion complementarity that is independent of the details of the interferometer. For two particles in any interferometer, for example, it implies that the average between the bosonic and fermionic probabilities must coincide with the probability obeyed by classical particles. Crucially, this fundamental relation also provides a heretofore unknown mathematical identity connecting the squared moduli of the permanent and determinant of an arbitrary complex matrix, hence extending an identity by Muir dating from the nineteenth century.

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

0 major / 3 minor

Summary. The manuscript claims to establish a relation combining bosonic and fermionic multiparticle transition probabilities in an arbitrary linear interferometer, yielding a complementarity constraint independent of interferometer details. For two particles this implies that the average of the bosonic and fermionic probabilities equals the classical probability. The same construction is asserted to produce a new algebraic identity relating |per(M)|^2 and |det(M)|^2 for an arbitrary complex matrix M, extending Muir's nineteenth-century identity.

Significance. If the central identity holds, the work supplies both a physically motivated constraint on boson-fermion interference and a previously unknown relation between the squared moduli of the permanent and determinant. The derivation from standard symmetrized/antisymmetrized amplitudes in a unitary interferometer, together with the explicit two-particle verification, constitutes a clear strength; the result is falsifiable by direct matrix computation for small dimensions.

minor comments (3)
  1. [Abstract and §1] The precise algebraic form of the claimed identity (e.g., the coefficient relating |per(M)|^2 and |det(M)|^2) is stated only qualitatively in the abstract and introduction; an explicit equation should appear in the main text before the generalization is asserted.
  2. [§2] Notation for the submatrix A extracted from the unitary interferometer should be defined once at the outset (including its dimensions relative to the number of modes and particles) to make the passage from the physical setup to the arbitrary-matrix statement transparent.
  3. [§4] A short appendix or remark verifying the identity by direct expansion for a generic 3×3 complex matrix would strengthen the generalization claim without altering the manuscript's scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the provided report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives the claimed identity directly from the standard definitions of bosonic (permanent) and fermionic (determinant) amplitudes under a unitary linear interferometer and symmetrization/antisymmetrization, without any fitted parameters, self-referential definitions, or load-bearing self-citations. The two-particle case reduces to an algebraically verifiable relation for arbitrary complex matrices, and the generalization is presented as following from the same construction. This makes the central result a mathematical consequence of the inputs rather than a reduction to them by construction, with the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard construction of bosonic and fermionic many-body states from a unitary single-particle transformation; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (1)
  • domain assumption Many-particle states are obtained by symmetrization (bosons) or antisymmetrization (fermions) of products of single-particle amplitudes under a unitary matrix describing the interferometer.
    Invoked implicitly when defining transition probabilities for bosons and fermions in a linear interferometer.

pith-pipeline@v0.9.0 · 5673 in / 1387 out tokens · 35577 ms · 2026-05-24T04:50:08.262778+00:00 · methodology

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Reference graph

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