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.
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.
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.
Identical particles exhibit correlations even in the absence of inter-particle interaction, due to the exchange (anti)symmetry of the many-particle wavefunction. Two fermions obey the Pauli principle and anti-bunch, whereas two bosons favor bunched, doubly occupied states. Here, we show that the collective interference of three or more particles leads to much more diverse behavior than expected from the boson–fermion dichotomy known from quantum statistical mechanics. The emerging complexity of many-particle interference is tamed by a simple law for the strict suppression of events in the Bell multiport beam splitter. The law shows that counting events are governed by widely species-independent interference, such that bosons and fermions can even exhibit identical interference signatures, while their statistical character remains subordinate. Recent progress in the preparation of tailored many-particle states of bosonic and fermionic atoms promises experimental verification and applications in novel many-particle interferometers.
The commonly assumed straight link between boson bunching and particle indistinguishability in quantum interferometry has recently been challenged [Nat. Photon. 17, 702 (2023)]. Exploiting the connection between quantum optical interferences and matrix permanents, it appeared that bunching effects may arise that exceed the expected limit of fully indistinguishable particles by injecting peculiar polarization states of partially distinguishable photons in some interferometers. Surprisingly, all states giving rise to such an anomalous bunching were found to be far from the state of fully indistinguishable particles, raising the question of whether this intriguing phenomenon might even possibly exist with nearly indistinguishable particles. Here, we answer this question positively by relating it to a mathematical conjecture on matrix permanents dating from 1986, whose physical interpretation had not yet been unveiled. Using a recently found counterexample to this conjecture, we demonstrate that there is an optical interferometer involving 8 photons in 10 modes such that the probability that all photons bunch into two output modes can be enhanced by suitably perturbing the state of all photons having the same polarization. Such a finding reflects still another -- even less expected -- facet of anomalous boson bunching.
It is found that identical bosons (fermions) show generalized bunching (antibunching) property in linear networks: The absolute maximum (minimum) of probability that all N input particles are detected in a subset of K output modes of any nontrivial linear M-mode network is attained \textit{only} by completely indistinguishable bosons (fermions). For fermions K is arbitrary, for bosons it is either (i) arbitrary for only classically correlated bosons or (ii) satisfies K≥N (or K=1) for arbitrary input states of N particles. The generalized bunching allows to certify in a \textit{polynomial} in N number of runs that a physical device realizing Boson Sampling with \textit{an arbitrary} network operates in the regime of full quantum coherence compatible \textit{only} with completely indistinguishable bosons. The protocol needs \textit{only polynomial} classical computations for the standard Boson Sampling, whereas an \textit{analytic formula} is available for the scattershot version.
Progress in the reliable preparation, coherent propagation and efficient detection of many-body states has recently brought collective quantum phenomena of many identical particles into the spotlight. This tutorial introduces the physics of many-boson and many-fermion interference required for the description of current experiments and for the understanding of novel approaches to quantum computing. The field is motivated via the two-particle case, for which the uncorrelated, classical dynamics of distinguishable particles is compared to the quantum behaviour of identical bosons and fermions. Bunching of bosons is opposed to anti-bunching of fermions, while both species constitute equivalent sources of bipartite two-level entanglement. The realms of indistinguishable and distinguishable particles are connected by a monotonic transition, on a scale defined by the coherence length of the interfering particles. As we move to larger systems, any attempt to understand many particles via the two-particle paradigm fails: In contrast to two-particle bunching and anti-bunching, the very same signatures can be exhibited by bosons and fermions, and coherent effects dominate over statistical behaviour. The simulation of many-boson interference, termed Boson-Sampling, entails a qualitatively superior computational complexity when compared to fermions. The hierarchy between bosons and fermions also characterises multipartite entanglement generation, for which bosons again clearly outmatch fermions. Finally, the quantum-to-classical transition between many indistinguishable and many distinguishable particles features non-monotonic structures. While the same physical principles govern small and large systems, the deployment of the intrinsic complexity of collective many-body interference makes more particles behave differently.
The collective interference of partially distinguishable bosons in multi-mode networks is studied via double-sided Feynman diagrams. The probability for many-body scattering events becomes a multi-dimensional tensor-permanent, which interpolates between distinguishable particles and identical bosons, and easily extends to mixed initial states. The permanent of the distinguishability matrix, composed of all mutual scalar products of the single-particle mode-functions, emerges as a natural measure for the degree of interference: It yields a bound on the difference between event probabilities for partially distinguishable bosons and the idealized species, and exactly quantifies the degree of bosonic bunching.