The quantum central limit theorem derived by Cushen and Hudson provides the foundations for understanding how subsystems of large bosonic systems evolving unitarily do reach equilibrium. It finds important applications in the context of quantum interferometry, for example, with photons. A practical feature of current photonic experiments, however, is that photons carry their own internal degrees of freedom pertaining to, e.g., the polarization or spatiotemporal mode they occupy, which makes them partially distinguishable. The ensuing deviation from ideal indistinguishability is well known to have observable consequences, for example in relation with boson bunching, but an understanding of its role in bosonic equilibration phenomena is still missing. Here, we generalize the Cushen-Hudson quantum central limit theorem to encompass scenarios with partial distinguishability, implying an asymptotic convergence of the subsystem's reduced state towards a multimode Gaussian state defined over the internal degrees of freedom. While these asymptotic internal states may not be directly accessible, we show that particle number distributions carry important signatures of distinguishability, which may be used to diagnose experimental imperfections in large boson sampling experiments.
Abstract Various quantum analogues of the central limit theorem, which is one of the cornerstones of probability theory, are known in the literature. One such analogue, due to Cushen and Hudson, is of particular relevance for quantum optics. It implies that the state in any single output arm of an n-splitter, which is fed with n copies of a centred state
ρ ρ with finite second moments, converges to the Gaussian state with the same first and second moments as
ρ ρ . Here we exploit the phase space formalism to carry out a refined analysis of the rate of convergence in this quantum central limit theorem. For instance, we prove that the convergence takes place at a rate
O(n−1/2) O n - 1 / 2 in the Hilbert–Schmidt norm whenever the third moments of
ρ ρ are finite. Trace norm or relative entropy bounds can be obtained by leveraging the energy boundedness of the state. Via analytical and numerical examples we show that our results are tight in many respects. An extension of our proof techniques to the non-i.i.d. setting is used to analyse a new model of a lossy optical fibre, where a given m-mode state enters a cascade of n beam splitters of equal transmissivities
λ1/n λ 1 / n fed with an arbitrary (but fixed) environment state. Assuming that the latter has finite third moments, and ignoring unitaries, we show that the effective channel converges in diamond norm to a simple thermal attenuator, with a rate
O(n−2(m+1)1) O ( n - 1 2 ( m + 1 ) ) . This allows us to establish bounds on the classical and quantum capacities of the cascade channel. Along the way, we derive several results that may be of independent interest. For example, we prove that any quantum characteristic function
χρ χ ρ is uniformly bounded by some
ηρ<1 η ρ < 1 outside of any neighbourhood of the origin; also,
ηρ η ρ can be made to depend only on the energy of the state
ρ ρ .
Gaussian boson sampling (GBS) allows for a way to demonstrate quantum supremacy with the relatively modest experimental resources of squeezed light sources, linear optics, and photon detection. In a realistic experimental setting, numerous effects can modify the complexity of the sampling, in particular loss, partial distinguishability of the photons, and the use of threshold detectors rather than photon counting detectors. In this paper, we investigate GBS with partial distinguishability using an approach based on virtual modes and indistinguishability efficiency. We develop a model using these concepts and derive the probabilities of measuring a specific output pattern from partially distinguishable and lossy GBS for both types of detectors. In the case of threshold detectors, the probability as calculated by the Torontonian is a special case under our framework. By analyzing the expressions of these probabilities we propose an efficient auxiliary classical simulation algorithm which can be used to calculate the probabilities. Our model and the auxiliary algorithm provide foundations for an approximation method for the probability calculation and simulation algorithms not only compatible with existing state-of-the-art simulation algorithms for ideal GBS but also reduce their complexities exponentially depending on the indistinguishability. Using the approximation method and the simulation algorithms we show how the boundary of quantum supremacy in GBS can be pushed further by partial distinguishability.
Anomalous bunching is a paradoxical quantum interferometric phenomenon in which partially distinguishable photons exhibit a higher probability of bunching into two or more modes than fully indistinguishable photons [Nat. Photonics 17, 702 (2023)]. While this effect is directly linked to violations of certain conjectures on matrix permanents, the mechanism underlying the anomaly is not yet fully understood. Here, we show that boson bunching is governed not by internal indistinguishability alone, but by the total indistinguishability of the postselected output state. Total indistinguishability combines the internal degrees of freedom, such as polarization or arrival time, with the spatial degrees of freedom of their wave functions restricted to the measured output modes of the interferometer. This reformulation preserves the expected connection between boson bunching and total indistinguishability, while clarifying how anomalous bunching can arise. It also reveals several additional facets of the same phenomenon, which are captured within a unified framework. In particular, we exhibit situations in which adding an independent source of distinguishability, either internal or spatial, can enhance multimode or even single-mode bunching probabilities, offering new insights into the subtle role of distinguishability in multiphoton interference.
Abstract Gaussian boson sampling (GBS) allows for a way to demonstrate quantum supremacy with the relatively modest experimental resources of squeezed light sources, linear optics, and photon detection. In a realistic experimental setting, numerous effects can modify the complexity of the sampling, in particular loss, partial distinguishability of the photons, and the use of threshold detectors rather than photon counting detectors. In this paper, we investigate GBS with partial distinguishability using an approach based on virtual modes and indistinguishability efficiency. We develop a model using these concepts and derive the probabilities of measuring a specific output pattern from partially distinguishable and lossy GBS for both types of detectors. In the case of threshold detectors, the probability as calculated by the Torontonian is a special case under our framework. By analyzing the expressions of these probabilities we propose an efficient classical simulation algorithm which can be used to calculate the probabilities. Our model and algorithm provide foundations for an approximate method for calculating probabilities. It also allows for a way to design sampling algorithms that are not only compatible with existing algorithms for ideal GBS, but can also reduce their complexity exponentially, depending on the indistinguishability efficiency. Using this we show how the boundary of quantum supremacy in GBS can be affected by partial distinguishability.
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.
In this paper, we develop an extension of the Central Limit Theorem (CLT) to the setting of bosonic quantum channels. This extension provides a deeper understanding of Gaussian bosonic channels as extremal objects. Using our CLT for bosonic quantum channels, we recover both the classical CLT and the CLT for bosonic quantum states, thereby offering a unified perspective that connects classical probability theory with continuous-variable quantum systems. Moreover, using our result, we can provide necessary uncertainty relations that every physical (possibly non-Gaussian) bosonic quantum channel must satisfy. As another application of our limit theorems, we derive tight lower bounds on the energy-constrained quantum capacity of linear bosonic channels by relating it to the capacity of their associated Gaussian bosonic channels, further reinforcing the role of Gaussian channels as extremal.