Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2307.06626 · Physical Review Letters

Indistinguishability of Identical Bosons from a Quantum Information Theory Perspective

Matthias Englbrecht, Tristan Kraft, Christoph Dittel, Andreas Buchleitner, G. Giedke, Barbara Kraus

Published Jan 30, 2024• 5 citations
arXivHTMLPDF

Abstract

Using tools from quantum information theory, we present a general theory of indistinguishability of identical bosons in experiments consisting of passive linear optics followed by particle number detection. Our results do neither rely on additional assumptions on the input state of the interferometer, such as, for instance, a fixed mode occupation, nor on any assumption on the degrees of freedom that potentially make the particles distinguishable. We identify the expectation value of the projector onto the N-particle symmetric subspace as an operationally meaningful measure of indistinguishability, and derive tight lower bounds on it that can be efficiently measured in experiments. Moreover, we present a consistent definition of perfect distinguishability and characterize the corresponding set of states. In particular, we show that these states are diagonal in the computational basis up to a permutationally invariant unitary. Moreover, we find that convex combinations of states that describe partially distinguishable and perfectly indistinguishable particles can lead to perfect distinguishability, which itself is not preserved under convex combinations.

DOI
10.1103/physrevlett.132.050201

Related papers

Indistinguishability Theory for Identical Particles with Invariant Degrees of Freedom

Valery Shchesnovich

Identical particles have dynamically invariant labels in all setups where their degrees of freedom can be partitioned into two parts, which we call internal and visible, with the visible part subject to unitary evolution, while the dynamically invariant internal part accounts for the partial distinguishability of particles. We give the explicit form of the visible state in terms of the indistinguishability function, generalizing the standard symmetrization/antisymmetrization postulate for bosons/fermions with internal modes. The indistinguishability function on the symmetric group accounts for the permutation symmetry of the dynamically invariant label state and gives a self-contained description of the symmetry spectrum of the visible state. It is shown that the indistinguishability of single particles emitted independently from a stable source is completely characterized by the projective measures on the symmetry spectrum. We reveal a hierarchy of successive upper bounds on the projective measure of indistinguishability in terms of marginal projective measures. We point out an experimentally feasible way to get the upper bounds on the indistinguishability of an arbitrarily large number of single photons by direct experimental readout of their marginal projective measures. We also analyze the indistinguishability of coherent superpositions and convex mixtures and resolve a recently posed problem.

quant-pharXiv:2609.14428•Sep 15, 2026•0 citations

Intrinsic Indistinguishability of Identical Particles and How Particle Labels Affect It

Valery Shchesnovich

We investigate indistinguishability of identical bosons and fermions undergoing arbitrary particle-number-preserving evolutions of their visible degrees of freedom. For the projective indistinguishability measure, defined by the projection of the visible state onto the symmetric/anti-symmetric subspace, we derive an equivalent expression in terms of the dynamically invariant internal state. We further generalize the textbook symmetrization/anti-symmetrization framework for bosons and fermions to arbitrary partial distinguishability by deriving an explicit reconstruction formula for the multiparticle visible state in terms of the indistinguishability function encoding the dynamical invariants. We give complete characterization of the class-functions of indistinguishability by projective measures on generalized symmetries. Finally, we reveal a strikingly counterintuitive effect: introducing additional particle label states can increase the multiparticle indistinguishability of identical particles. The effect originates from the cancellation of collective multiparticle phases.

quant-pharXiv:2606.29312•Jun 30, 2026•0 citations

Measuring multiparticle indistinguishability with the generalized bunching probability

Shawn Geller, Emanuel H. Knill

The indistinguishability of many bosons undergoing passive linear transformations followed by number basis measurements is fully characterized by the visible state of the bosons. However, measuring all the parameters in the visible state is experimentally demanding. In this work, we seek to perform partial characterization of the visible state by measuring properties of it that are available after randomization. First we study the case where the occupied visible modes are randomly permuted, and second we study the case where Haar random linear optical unitaries are applied. In each case, we find that the generalized bunching probability -- which is the probability that all the input bosons arrive in a given subset of the output modes -- obeys monotonicity with respect to some partial order of distinguishability of the input bosons. As an intermediate result, we show that Lieb's permanental-dominance conjecture for immanants is equivalent to the following statement: for states that are invariant under permutations of the occupied visible modes, the generalized bunching probability is maximized when the bosons are perfectly indistinguishable. We also prove that a consequence of the monotonicity of the generalized bunching probability after Haar averaging is that this average is maximized when the bosons are perfectly indistinguishable. Finally, we discuss applications of our results to thermometry of cold-atom systems.

arXiv:2509.04550•Physical Review A•Feb 3, 2026•0 citations

Distinguishability in quantum interference with multimode squeezed states

V. S. Shchesnovich

Distinguishability theory is developed for quantum interference of the squeezed vacuum states on unitary linear interferometers. It is found that the entanglement of photon pairs over the Schmidt modes is one of the sources of distinguishability. The distinguishability is quantified by the symmetric part of the internal state of $n$ pairs of photons over the spectral Schmidt modes, whose normalization ${q}_{2n}$ is the probability that $2n$ photons interfere as indistinguishable. For two pairs of photons ${q}_{4}=(1+2\mathbb{P})/3$, where $\mathbb{P}$ is the purity of the squeezed states ($K=1/\mathbb{P}$ is the Schmidt number). For a fixed purity $\mathbb{P}$, the probability ${q}_{2n}$ decreases exponentially fast in $n$. For example, in the experimental Gaussian boson sampling of H.-S. Zhong et al., [Science 370, 1460 (2020)], the achieved purity $\mathbb{P}\ensuremath{\approx}0.938$ for the average number of photons $2n\ensuremath{\ge}43$ gives ${q}_{2n}\ensuremath{\lesssim}0.5$, i.e., close to the middle line between $n$ indistinguishable and $n$ distinguishable pairs of photons. In derivation of all the results, the first-order quantization representation based on the particle decomposition of the Hilbert space of identical bosons serves as an indispensable tool. The approach can be applied also to the generalized (non-Gaussian) squeezed states, such as those recently generated in the three-photon parametric down-conversion.

arXiv:2109.01857•Physical Review A•Jun 2, 2022•8 citations

Optimal interferometric certification of multi-photon indistinguishability

Marco Robbio, Michał Oszmaniec, Nicolas J. Cerf +2 more

Multiphoton indistinguishability is a key resource for photonic quantum technologies, yet its characterization typically relies on resource-intensive methods. In this work, we develop two efficient and experimentally friendly protocols to estimate or bound the fidelity $F_{\mathrm{ind}}$ of an $N$-photon state to the closest perfectly indistinguishable state. The first protocol applies to sources preparing separable states, uses a single Fourier interferometer together with photon-number-resolving detection, and yields tight two-sided bounds on $F_{\mathrm{ind}}$. The second protocol combines randomized implementations of linear-optical interferometers with photon counting, enabling direct estimation of $F_{\mathrm{ind}}$ for arbitrary $N$-photon states. Both protocols can certify $F_{\mathrm{ind}} = 1 - \mathcal{O}(ε)$ using provably optimal $\mathcal{O}(1/ε)$ samples, in contrast to previous approaches which required prior assumptions on the model of partial distinguishability. Our methods, based on a multiphoton generalization of the Hong-Ou-Mandel test, bring the rigorous and operationally meaningful certification of multiphoton indistinguishability within reach of current photonic technologies.

quant-pharXiv:2609.37620•Sep 30, 2026•0 citations

Testing for multipartite indistinguishability

P. S. Turner, Toshi Sugiyama, Terry Rudolph +2 more

Indistinguishability of subsystems is a requirement for quantum interference, which is in turn the workhorse of extra‐classical demonstrations such as those proposed in quantum information protocols. However, indistinguishability does not arise in the standard axioms of quantum mechanics. Once the standard textbook approach is assumed, the pure states of the system in question are conventionally restricted to a Hilbert subspace with permutation symmetry. From the tomographic perspective, this is unacceptable, as such an assumption of symmetry can be too restrictive. Here we report two approaches to the problem of multiple bosons that admit both distinguishable and indistinguishable states of the same system. In doing so we define two notions of distinguishability. Both approaches are shown to give the same results in the sense that they predict the same restriction on density matrices as would be reconstructed tomographically. The number of unknown elements is polynomial in the number of particles, in stark contrast to the distinguishable case which scales exponentially. We use this picture to investigate possible measurement schemes which could detect and/or characterise indistinguishability.

AIP conference proceedings•Jan 1, 2011•0 citations
Using tools from quantum information theory, we present a general theory of indistinguishability of identical bosons in experiments consisting of passive linear optics followed by particle number detection. Our results do neither rely on additional assumptions on the input state of the interferometer, such as, for instance, a fixed mode occupation, nor on any assumption on the degrees of freedom that potentially make the particles distinguishable. We identify the expectation value of the projector onto the N-particle symmetric subspace as an operationally meaningful measure of indistinguishability, and derive tight lower bounds on it that can be efficiently measured in experiments. Moreover, we present a consistent definition of perfect distinguishability and characterize the corresponding set of states. In particular, we show that these states are diagonal in the computational basis up to a permutationally invariant unitary. Moreover, we find that convex combinations of states that describe partially distinguishable and perfectly indistinguishable particles can lead to perfect distinguishability, which itself is not preserved under convex combinations.
Identical particles have dynamically invariant labels in all setups where their degrees of freedom can be partitioned into two parts, which we call internal and visible, with the visible part subject to unitary evolution, while the dynamically invariant internal part accounts for the partial distinguishability of particles. We give the explicit form of the visible state in terms of the indistinguishability function, generalizing the standard symmetrization/antisymmetrization postulate for bosons/fermions with internal modes. The indistinguishability function on the symmetric group accounts for the permutation symmetry of the dynamically invariant label state and gives a self-contained description of the symmetry spectrum of the visible state. It is shown that the indistinguishability of single particles emitted independently from a stable source is completely characterized by the projective measures on the symmetry spectrum. We reveal a hierarchy of successive upper bounds on the projective measure of indistinguishability in terms of marginal projective measures. We point out an experimentally feasible way to get the upper bounds on the indistinguishability of an arbitrarily large number of single photons by direct experimental readout of their marginal projective measures. We also analyze the indistinguishability of coherent superpositions and convex mixtures and resolve a recently posed problem.
We investigate indistinguishability of identical bosons and fermions undergoing arbitrary particle-number-preserving evolutions of their visible degrees of freedom. For the projective indistinguishability measure, defined by the projection of the visible state onto the symmetric/anti-symmetric subspace, we derive an equivalent expression in terms of the dynamically invariant internal state. We further generalize the textbook symmetrization/anti-symmetrization framework for bosons and fermions to arbitrary partial distinguishability by deriving an explicit reconstruction formula for the multiparticle visible state in terms of the indistinguishability function encoding the dynamical invariants. We give complete characterization of the class-functions of indistinguishability by projective measures on generalized symmetries. Finally, we reveal a strikingly counterintuitive effect: introducing additional particle label states can increase the multiparticle indistinguishability of identical particles. The effect originates from the cancellation of collective multiparticle phases.
The indistinguishability of many bosons undergoing passive linear transformations followed by number basis measurements is fully characterized by the visible state of the bosons. However, measuring all the parameters in the visible state is experimentally demanding. In this work, we seek to perform partial characterization of the visible state by measuring properties of it that are available after randomization. First we study the case where the occupied visible modes are randomly permuted, and second we study the case where Haar random linear optical unitaries are applied. In each case, we find that the generalized bunching probability -- which is the probability that all the input bosons arrive in a given subset of the output modes -- obeys monotonicity with respect to some partial order of distinguishability of the input bosons. As an intermediate result, we show that Lieb's permanental-dominance conjecture for immanants is equivalent to the following statement: for states that are invariant under permutations of the occupied visible modes, the generalized bunching probability is maximized when the bosons are perfectly indistinguishable. We also prove that a consequence of the monotonicity of the generalized bunching probability after Haar averaging is that this average is maximized when the bosons are perfectly indistinguishable. Finally, we discuss applications of our results to thermometry of cold-atom systems.
Distinguishability theory is developed for quantum interference of the squeezed vacuum states on unitary linear interferometers. It is found that the entanglement of photon pairs over the Schmidt modes is one of the sources of distinguishability. The distinguishability is quantified by the symmetric part of the internal state of nnn pairs of photons over the spectral Schmidt modes, whose normalization q2n{q}_{2n}q2n​ is the probability that 2n2n2n photons interfere as indistinguishable. For two pairs of photons q4=(1+2P)/3{q}_{4}=(1+2\mathbb{P})/3q4​=(1+2P)/3, where P\mathbb{P}P is the purity of the squeezed states (K=1/PK=1/\mathbb{P}K=1/P is the Schmidt number). For a fixed purity P\mathbb{P}P, the probability q2n{q}_{2n}q2n​ decreases exponentially fast in nnn. For example, in the experimental Gaussian boson sampling of H.-S. Zhong et al., [Science 370, 1460 (2020)], the achieved purity $\mathbb{P}\ensuremath{\approx}0.938$ for the average number of photons $2n\ensuremath{\ge}43$ gives ${q}_{2n}\ensuremath{\lesssim}0.5$, i.e., close to the middle line between nnn indistinguishable and nnn distinguishable pairs of photons. In derivation of all the results, the first-order quantization representation based on the particle decomposition of the Hilbert space of identical bosons serves as an indispensable tool. The approach can be applied also to the generalized (non-Gaussian) squeezed states, such as those recently generated in the three-photon parametric down-conversion.
Multiphoton indistinguishability is a key resource for photonic quantum technologies, yet its characterization typically relies on resource-intensive methods. In this work, we develop two efficient and experimentally friendly protocols to estimate or bound the fidelity FindF_{\mathrm{ind}}Find​ of an NNN-photon state to the closest perfectly indistinguishable state. The first protocol applies to sources preparing separable states, uses a single Fourier interferometer together with photon-number-resolving detection, and yields tight two-sided bounds on FindF_{\mathrm{ind}}Find​. The second protocol combines randomized implementations of linear-optical interferometers with photon counting, enabling direct estimation of FindF_{\mathrm{ind}}Find​ for arbitrary NNN-photon states. Both protocols can certify Find=1−O(ε)F_{\mathrm{ind}} = 1 - \mathcal{O}(ε)Find​=1−O(ε) using provably optimal O(1/ε)\mathcal{O}(1/ε)O(1/ε) samples, in contrast to previous approaches which required prior assumptions on the model of partial distinguishability. Our methods, based on a multiphoton generalization of the Hong-Ou-Mandel test, bring the rigorous and operationally meaningful certification of multiphoton indistinguishability within reach of current photonic technologies.
Indistinguishability of subsystems is a requirement for quantum interference, which is in turn the workhorse of extra‐classical demonstrations such as those proposed in quantum information protocols. However, indistinguishability does not arise in the standard axioms of quantum mechanics. Once the standard textbook approach is assumed, the pure states of the system in question are conventionally restricted to a Hilbert subspace with permutation symmetry. From the tomographic perspective, this is unacceptable, as such an assumption of symmetry can be too restrictive. Here we report two approaches to the problem of multiple bosons that admit both distinguishable and indistinguishable states of the same system. In doing so we define two notions of distinguishability. Both approaches are shown to give the same results in the sense that they predict the same restriction on density matrices as would be reconstructed tomographically. The number of unknown elements is polynomial in the number of particles, in stark contrast to the distinguishable case which scales exponentially. We use this picture to investigate possible measurement schemes which could detect and/or characterise indistinguishability.