Complementarity between bosonic and fermionic many-body interferences with partially distinguishable particles

Marco Robbio; Michael G. Jabbour; Nicolas J. Cerf

arxiv: 2604.23316 · v1 · submitted 2026-04-25 · 🪐 quant-ph

Complementarity between bosonic and fermionic many-body interferences with partially distinguishable particles

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

Pith reviewed 2026-05-08 08:18 UTC · model grok-4.3

classification 🪐 quant-ph

keywords bosonic interferencefermionic interferencepartial distinguishabilitylinear interferometercorrelation matricesquantum metrologysum rulemultiparticle interference

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The pith

Bosonic and fermionic particle-number correlation matrices sum to twice the classical matrix even with partial distinguishability.

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

The paper extends the known opposite interference behaviors of bosons and fermions to the realistic case where particles are only partially distinguishable. It derives a relation that holds for any linear interferometer and shows that the output correlation matrices for bosons and fermions add exactly to twice the classical correlation matrix. This sum rule immediately limits the quantum Fisher information available for phase estimation, producing a concrete trade-off in which greater indistinguishability improves bosonic metrology while reduced indistinguishability can improve fermionic schemes.

Core claim

We establish a relation that combines bosonic and fermionic multiparticle interferences in an arbitrary linear interferometer in the presence of partial distinguishability. The correlation matrices for bosonic and fermionic particle number distributions at the output obey the sum rule that their sum equals twice the correlation matrix for classical particles. This also supplies a new mathematical identity relating the permanent and determinant of tensors of order 3 and directly constrains achievable quantum Fisher information in phase-estimation protocols.

What carries the argument

The sum rule that equates the bosonic plus fermionic particle-number correlation matrices to twice the classical matrix, derived from an extended complementarity identity that incorporates a tensor of order 3 encoding partial distinguishability.

If this is right

Quantum Fisher information for phase estimation is bounded by the classical correlation matrix via the sum rule.
Bosonic metrology gains increase with greater indistinguishability while fermionic schemes gain from controlled distinguishability.
Any protocol relying on many-body interference must account for this bosonic-fermionic trade-off when particles are not perfectly identical.
The identity provides a direct test of whether an observed output distribution is consistent with the assumed linear interferometer and distinguishability model.

Where Pith is reading between the lines

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

The same sum-rule structure may apply to other quantum statistics or to mixtures of bosonic and fermionic particles.
Measuring deviations from the sum rule could serve as a practical diagnostic for unexpected sources of distinguishability or interferometer imperfections.
The relation suggests that optimal metrology strategies could alternate between bosonic and fermionic modes by tuning distinguishability rather than only the interferometer.

Load-bearing premise

Partial distinguishability is completely described by a fixed tensor of pairwise overlaps and the interferometer is strictly linear and lossless.

What would settle it

Measure the full output correlation matrices for three partially distinguishable photons (or atoms) in a known linear interferometer under both bosonic and fermionic statistics, add the two matrices, and test whether the result equals twice the classical matrix to within experimental error.

Figures

Figures reproduced from arXiv: 2604.23316 by Marco Robbio, Michael G. Jabbour, Nicolas J. Cerf.

**Figure 1.** Figure 1: FIG. 1. Schematics of the case of a 50:50 beam-splitter with single particle inputs (red for bosons and blue for fermions) given an overlap view at source ↗

**Figure 2.** Figure 2: FIG. 2. We consider a simple protocol where a set of particles (bosons, fermions, classical) interfere through a linear interferometer. The goal view at source ↗

read the original abstract

It is well known that bosons and fermions exhibit opposite behaviors when experiencing interference, in the sense that bosons have a tendency to bunch whereas fermions have a tendency to antibunch. Recently, this complementarity was mathematically characterized in [arXiv:2312.17709] by means of an identity relating the transition probabilities of both types of particles in a linear interferometer. Here, we show that such a complementarity still holds even when particles become partially distinguishable, for example, when they have slightly different polarizations or time delays. Namely, we establish a relation that combines bosonic and fermionic multiparticle interferences in an arbitrary linear interferometer, in the presence of partial distinguishability. Incidentally, this also provides a new mathematical identity relating the permanent and determinant of tensors of order 3. Importantly, this complementarity has direct operational consequences in quantum metrology. Indeed, we show that the correlation matrices for bosonic and fermionic particle number distributions at the output of the interferometer obey a simple sum rule: their sum equals twice the correlation matrix for classical particles. This, in turn, constraints the achievable quantum Fisher information in phase-estimation protocols, highlighting a trade-off whereby greater indistinguishability enhances bosonic sensitivity whereas reduced indistinguishability can benefit fermionic schemes.

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.

Desk Editor's Note private letter to a colleague

The paper extends the prior bosonic-fermionic complementarity identity to partial distinguishability via a 3-tensor overlap model and derives the B+F=2C sum rule for output correlations, with metrology implications.

read the letter

The main point is that this work shows how the bosonic and fermionic particle-number correlation matrices at the output of any linear interferometer add up to twice the classical distinguishable case, even when particles have partial distinguishability from overlaps like polarization or timing differences. They model that distinguishability with a fixed order-3 tensor whose slices capture the internal-state overlaps, then use it to combine the symmetrized and antisymmetrized amplitudes into a single relation that also gives a new permanent-determinant identity for such tensors. The metrology section follows directly: the sum rule limits the quantum Fisher information in phase estimation, with a trade-off where more indistinguishability helps bosons but can help fermions when distinguishability increases. That part is straightforward once the identity is granted. What they do well is keep the derivation compact and state the operational consequence clearly without extra assumptions beyond the linear lossless network. The math appears to follow from standard many-body amplitude construction once the tensor is plugged in. The soft spot is the modeling choice itself. The sum rule holds because cross terms cancel under the assumption that all distinguishability factors through that single 3-tensor in the usual way. If real systems have non-factorizable internal dynamics, continuous spectra, or losses that do not reduce to the same tensor, those cancellations fail and the relation does not apply. The paper presents the tensor as the given description rather than deriving it from a microscopic Hamiltonian, so the claimed generality stops at that modeling step. This is a minor limitation for the stated scope but worth noting for readers who want broader applicability. The paper is aimed at quantum optics groups working on multi-particle interference, boson sampling variants, or metrology with imperfect sources. A reader already using the 2023 identity will see the incremental step immediately and can check the new tensor identity against their own setups. It deserves a serious referee because the central relation is falsifiable, the assumptions are explicit, and the metrology bound is concrete enough to test. I would send it to review with a request to add one or two worked numerical examples that verify the sum rule for small particle numbers.

Referee Report

1 major / 0 minor

Summary. The paper establishes a complementarity between bosonic and fermionic many-body interferences for partially distinguishable particles in linear interferometers. It derives a sum rule stating that the bosonic and fermionic correlation matrices for the output particle number distributions sum to twice the corresponding matrix for classical distinguishable particles. This follows from a new mathematical identity relating the permanent and determinant of a third-order tensor that encodes the particles' internal degrees of freedom overlaps. The result is applied to quantum metrology, showing constraints on the achievable quantum Fisher information.

Significance. Assuming the derivation is rigorous, this work provides a clean and general relation that unifies bosonic and fermionic behaviors under partial distinguishability. The operational consequence for metrology is particularly noteworthy, as it quantifies a trade-off in sensitivity. The new tensor identity is a mathematical advance that may have broader applications. The manuscript includes reproducible aspects through the explicit identity, which is a strength.

major comments (1)

[the section introducing the overlap tensor (modeling of partial distinguishability)] The sum rule B + F = 2C is shown to hold when distinguishability is introduced via the 3-tensor of overlaps. However, the paper should address whether this is the most general model or if there exist physically relevant cases (e.g., continuous spectra or mode-dependent effects) where the cross terms do not cancel, potentially invalidating the sum rule. A brief discussion or counterexample would strengthen the claim of generality.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation for minor revision. We address the major comment below and will incorporate a clarifying discussion in the revised manuscript.

read point-by-point responses

Referee: The sum rule B + F = 2C is shown to hold when distinguishability is introduced via the 3-tensor of overlaps. However, the paper should address whether this is the most general model or if there exist physically relevant cases (e.g., continuous spectra or mode-dependent effects) where the cross terms do not cancel, potentially invalidating the sum rule. A brief discussion or counterexample would strengthen the claim of generality.

Authors: We thank the referee for this constructive comment. The third-order overlap tensor is the standard and most general model for partial distinguishability in linear interferometers, as it encodes all relevant overlaps between the internal states of the particles. Any physical source of distinguishability, including continuous spectra (via overlap integrals over frequency or time distributions) and mode-dependent effects (via mode-specific internal states), is captured by appropriately populating the tensor entries. The algebraic identity relating the permanent and determinant ensures that cross terms cancel independently of the specific form of these overlaps, so the sum rule remains valid. We will add a brief discussion in the section introducing the overlap tensor to explicitly address the generality of the model and confirm its applicability to these cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under stated model

full rationale

The paper cites arXiv:2312.17709 only to recall the fully distinguishable complementarity identity, then derives the partial-distinguishability extension and the B+F=2C sum rule from the 3-tensor overlap model and standard bosonic/fermionic symmetrization. No equation reduces to a fitted parameter renamed as prediction, no uniqueness theorem is imported from the authors' prior work to force the result, and the central identity is obtained by direct expansion of the many-body amplitudes under the given tensor factorization. The modeling assumption (overlap tensor as sole distinguishability source) is explicit and external to the derivation itself, so the claimed sum rule does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of permanents and determinants together with the definition of partial distinguishability via a fixed overlap tensor; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption Linear interferometers are described by unitary transformations on the single-particle modes.
Invoked when stating the setup for arbitrary linear interferometers.
domain assumption Partial distinguishability is fully captured by a symmetric tensor of pairwise overlaps of order equal to the number of particles.
Required for the order-3 tensor identity and the correlation-matrix sum rule.

pith-pipeline@v0.9.0 · 5529 in / 1371 out tokens · 30707 ms · 2026-05-08T08:18:45.411120+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

[1]

The symmetric characteristic function is defined as follows: χρ (ααα) =Tr(D(ααα)ρ).(59) We will exploit two facts:

Bosonic Characteristic function We start by recalling the definition of quantum characteristic function of a stateρ, by first defining (for every vector of complex numbersααα= (α 1, ...,αm)T ) the symmetric displacement operator: D(ααα) =exp( ˆaaa†ααα−ααα† ˆaaa) = m ∏ i=1 exp(αi ˆa† i −α ∗ i ˆai),(58) where we used the notation ˆaaa= (ˆa1, ...,ˆam)T with ...

work page
[2]

The effect of a linear interferometer is described by Eqs. (3) and (4), so that the evolution of an input stateρunder the effect of the linear interferometer can be described by the following characteristic function: χ ˆUρ ˆU†(α) =Tr D(ααα) ˆUρ ˆU † =Tr D(U†ααα)ρ .(60)

work page
[3]

We start from the case of indistinguishable particles, then later generalize the formalism in the presence of a Gram matrixS

The expectation value of the product of two operatorsAandB † can be directly computed from the characteristic function using the Parseval identity: Tr AB† = Z d2nα πn χA(ααα) χB(ααα),(61) where the bar over a complex number represents a complex-conjugate. We start from the case of indistinguishable particles, then later generalize the formalism in the pre...

work page
[4]

Quantentheorie des einatomigen idealen gases,

Fermionic Characteristic function In the fermionic case, phase space variables, which we denote byβi (whereiis the mode index), are so-called anti-commuting Grassmann variables. They satisfy the following: β k i =0,∀k≥2,{β i,β ∗ j }=0,∀i,j=1,· · ·,m.(89) Following the same procedure as in the bosonic case, we start by introducing the single mode stateρ(x)...

work page doi:10.1073/pnas.2010827117 1925

[1] [1]

The symmetric characteristic function is defined as follows: χρ (ααα) =Tr(D(ααα)ρ).(59) We will exploit two facts:

Bosonic Characteristic function We start by recalling the definition of quantum characteristic function of a stateρ, by first defining (for every vector of complex numbersααα= (α 1, ...,αm)T ) the symmetric displacement operator: D(ααα) =exp( ˆaaa†ααα−ααα† ˆaaa) = m ∏ i=1 exp(αi ˆa† i −α ∗ i ˆai),(58) where we used the notation ˆaaa= (ˆa1, ...,ˆam)T with ...

work page

[2] [2]

The effect of a linear interferometer is described by Eqs. (3) and (4), so that the evolution of an input stateρunder the effect of the linear interferometer can be described by the following characteristic function: χ ˆUρ ˆU†(α) =Tr D(ααα) ˆUρ ˆU † =Tr D(U†ααα)ρ .(60)

work page

[3] [3]

We start from the case of indistinguishable particles, then later generalize the formalism in the presence of a Gram matrixS

The expectation value of the product of two operatorsAandB † can be directly computed from the characteristic function using the Parseval identity: Tr AB† = Z d2nα πn χA(ααα) χB(ααα),(61) where the bar over a complex number represents a complex-conjugate. We start from the case of indistinguishable particles, then later generalize the formalism in the pre...

work page

[4] [4]

Quantentheorie des einatomigen idealen gases,

Fermionic Characteristic function In the fermionic case, phase space variables, which we denote byβi (whereiis the mode index), are so-called anti-commuting Grassmann variables. They satisfy the following: β k i =0,∀k≥2,{β i,β ∗ j }=0,∀i,j=1,· · ·,m.(89) Following the same procedure as in the bosonic case, we start by introducing the single mode stateρ(x)...

work page doi:10.1073/pnas.2010827117 1925