Probabilities of photon counts at the output of a multiport optical device are generalised for optical sources of arbitrary quantum states in partially distinguishable optical modes. For the single-mode photon sources, the generating function for the probabilities is a linear combination of the matrix permanents of positive semi-definite Hermitian matrices, where each Hermitian matrix is a Hadamard product of a submatrix of the multiport matrix and a Hermitian matrix describing partial distinguishability. For the multi-mode sources the generating function is given by an integral of the Husimi functions of the sources. When each photon source outputs exactly a Fock state, the obtained expression reduces to the probability formula derived for partially distinguishable photons, \textit{Physical Review A \textbf{91}, 013844 (2015)}. The derived probability formula can be useful in analysing experiments with partially distinguishable sources and error bounds of experimental Boson Sampling devices.
Probabilities of photon counts at the output of a multiport optical device are generalised for optical sources of arbitrary quantum states in partially distinguishable optical modes. For the single-mode photon sources, the generating function for the probabilities is a linear combination of the matrix permanents of positive semi-definite Hermitian matrices, where each Hermitian matrix is a Hadamard product of a submatrix of the multiport matrix and a Hermitian matrix describing partial distinguishability. For the multi-mode sources the generating function is given by an integral of the Husimi functions of the sources. When each photon source outputs exactly a Fock state, the obtained expression reduces to the probability formula derived for partially distinguishable photons, \textit{Physical Review A \textbf{91}, 013844 (2015)}. The derived probability formula can be useful in analysing experiments with partially distinguishable sources and error bounds of experimental Boson Sampling devices.
We generalize an approach for description of multiphoton experiments with multiport unitary linear optical devices, started in Phys. Rev. A 89, 022333 (2014) with single photons in mixed spectral states, to arbitrary (multiphoton) input and arbitrary photon detectors. We show that output probabilities are always given in terms of the matrix permanents of the Hadamard product of a matrix built from the network matrix and matrices built from the spectral state of photons and spectral sensitivities of detectors. Moreover, in the case of input with up to one photon per mode, the output probabilities are given by a sum (or integral) with each term being the absolute value squared of such a matrix permanent. We conjecture that, for an arbitrary multiphoton input, zero output probability of an output configuration is always the result of an exact cancellation of quantum transition amplitudes of completely indistinguishable photons (a subset of all input photons) and, moreover, is independent of coherence between only partially indistinguishable photons. The conjecture is supported by examples. Furthermore, we propose a measure of partial indistinguishability of photons which generalizes Mandel's observation, and find the law of degradation of quantum coherence in a realistic boson-sampling device with increase of the total number of photons and/or their ``classicality parameter.''
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.
Boson sampling is the problem of generating a multiphoton state whose counting probability is the permanent of an $n\ifmmode\times\else\texttimes\fi{}n$ matrix. This is created as the output n-photon coincidence rate of a prototype quantum computing device with n input photons. It is a fundamental challenge to verify boson sampling, and therefore the question of how output count rates scale with matrix size n is crucial. Here we apply results from random matrix theory as well as the characteristic function approach from quantum optics to establish analytical scaling laws for average count rates. We treat boson sampling experiments with arbitrary inputs, outputs, and losses. Using the scaling laws we analyze grouping of channel outputs and the count rates for this case.
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.
This thesis presents results under the topics of quantum correlations and quantum information processing using linear-optical networks. The first part of this thesis concerns with quantum correlations, specially in Gaussian states, which play a central role as resource in quantum protocols. The second part is about quantum information processing using linear-optical networks that are of great practical interest due to their simple physical realization. In the first part, we present a measurement-based method for verifying nonzero quantum discord, a measure of quantum correlations, in bipartite systems that can be used for both continuous and discrete-variable systems. Based on this method, we show that the only bipartite Gaussian states with zero quantum discord are product states. We also present a simple and efficient experimental method for verifying quantum correlations in Gaussian states using homodyne measurements. Moreover, we introduce an operational measure for quantifying quantum correlations in Gaussian states that is purely based on Gaussian measurements. We then illustrate the operational significance of the measure in terms of a Gaussian quantum protocol. In the second part, we present results on linear quantum-optics experiments, specifically boson sampling, an intermediate model of quantum computation that can be implemented using single-photon states, linear-optical networks, and photodetectors. We first present an efficient experimental method for characterization of linear-optical networks, which is essential for practical implementation of boson sampling. We show that using coherent states one can directly measure all the nontrivial phases and moduli of the matrix describing the network. Next, we consider the problem of boson sampling with Gaussian states. We derive a general formula for calculating the output photon-counting probabilities. We show that for input thermal states, sampling from the output probability distribution can be efficiently classically simulated. Using this, we find that permanents of positive-semidefinite Hermitian matrices can be approximated to within a multiplicative error in BPPNP. We also discuss boson sampling with squeezed vacuum states. Finally, we provide a sufficient condition for the efficient classical simulation of the general sampling problem using linear-quantum optics, which involves a multimode input quantum state to a linear-optical network and set of measurements at the output. This condition is based on the phase-space quasiprobability distributions, and suggests that negativity is a quantum resource. We apply this condition to implementations of boson sampling using single-photon or spontaneous parametric down-conversion sources. We show that above some threshold for loss and noise in the experiment, boson sampling is classically simulatable.
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.