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Physical Review A

Means and covariances of photon numbers in multimode Gaussian states

Giuseppe Vallone, Gianfranco Cariolaro, Gianfranco L. Pierobon

Published Feb 8, 2019• 9 citations
Publisher

Abstract

In the field of continuous variables and particularly of Gaussian states an important role is played by the statistics of photon numbers. This issue has been considered in pioneering works for the one- and two-mode cases, but a systematic approach seems not to be available. In this paper, the most general multimode mixed Gaussian state, which is generated starting from a multimode thermal state and processed by the most general Gaussian unitary, is considered. In the $N$-mode the photon numbers are represented by $N$ random variables, one for each mode, and the target is the evaluation of the means and of the covariances. The means describe statistically the amount of photon numbers in each mode, while the covariances give the correlations between the photon numbers in the different modes. For both means and covariances a closed-form result is obtained, expressed in simple and compact formulas. The main tools are provided by the representation of Gaussian unitaries (given by a cascade of an $N$-mode squeeze operator, an $N$-mode rotation operator, and a parallel set of $N$ single-mode displacement operators) and by the corresponding Bogoliubov transformations.

DOI
10.1103/physreva.99.023817

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In the field of continuous variables and particularly of Gaussian states an important role is played by the statistics of photon numbers. This issue has been considered in pioneering works for the one- and two-mode cases, but a systematic approach seems not to be available. In this paper, the most general multimode mixed Gaussian state, which is generated starting from a multimode thermal state and processed by the most general Gaussian unitary, is considered. In the NNN-mode the photon numbers are represented by NNN random variables, one for each mode, and the target is the evaluation of the means and of the covariances. The means describe statistically the amount of photon numbers in each mode, while the covariances give the correlations between the photon numbers in the different modes. For both means and covariances a closed-form result is obtained, expressed in simple and compact formulas. The main tools are provided by the representation of Gaussian unitaries (given by a cascade of an NNN-mode squeeze operator, an NNN-mode rotation operator, and a parallel set of NNN single-mode displacement operators) and by the corresponding Bogoliubov transformations.
Gaussian states are ubiquitous in quantum optics and information processing, and it is essential to have effective tools for their characterization. One such tool is a photon-number-resolving detector, and the simplest configuration involves counting the total number of photons in the state to be characterized. This motivates the following question: What properties of a multimode Gaussian state are determined by the signal from one detector that measures the total number of photons up to some bound? We find that if the Gaussian state occupies SSS modes and the probabilities of nnn photons for all $n\ensuremath{\le}8S$ are known, then we can determine the spectrum of the Gaussian covariance matrix and the magnitude of the displacements in each eigenspace of the covariance matrix. Nothing more can be learned, even if all photon-number probabilities are known. When the state is pure, the covariance matrix spectrum determines the squeezing parameters of the state.
Quantum Gaussian states play a fundamental role in quantum communications and in quantum information. This paper deals with the implementation of multimode, and particularly of two-mode Gaussian unitaries and Gaussian states with primitive components (phase shifters, single-mode real squeezers, displacements, and beam splitters). The architecture thus obtained allows one to obtain an insight into the physical meaning of each variable involved. Moreover, following the implementation architecture, it is possible to formulate an easy algebra (radical free) for the main operations and transformations of Gaussian states.
The Fock expansion of multimode pure Gaussian states is derived starting from their representation as displaced and squeezed multimode vacuum states. The approach is new and appears to be simpler and more general than previous ones starting from the phase-space representation given by the characteristic or Wigner function. Fock expansion is performed in terms of easily evaluable two-variable Hermite–Kampé de Fériet polynomials. A relatively simple and compact expression for the joint statistical distribution of the photon numbers in the different modes is obtained. In particular, this result enables one to give a simple characterization of separable and entangled states, as shown for two-mode and three-mode Gaussian states.
We derive a computable analytical formula for the quantum fidelity between two arbitrary multimode Gaussian states which is simply expressed in terms of their first- and second-order statistical moments. We also show how such a formula can be written in terms of symplectic invariants and used to derive closed forms for a variety of basic quantities and tools, such as the Bures metric, the quantum Fisher information, and various fidelity-based bounds. Our result can be used to extend the study of continuous-variable protocols, such as quantum teleportation and cloning, beyond the current one-mode or two-mode analyses, and paves the way to solve general problems in quantum metrology and quantum hypothesis testing with arbitrary multimode Gaussian resources.
We investigate two special classes of two-mode Gaussian states of light that are important from both the experimental and theoretical points of view: the mode-mixed thermal states and the squeezed thermal ones. Aiming to a parallel study, we write the Uhlmann fidelity between pairs of states belonging to each class in terms of their defining parameters. The quantum Fisher information matrices on the corresponding four-dimensional manifolds are diagonal and allow insightful parameter estimation. The scalar curvatures of the Bures metric on both Riemannian manifolds of special two-mode Gaussian states are evaluated and discussed. They are functions of two variables, namely, the mean numbers of photons in the incident thermal modes. Our comparative analysis opens the door to further investigation of the interplay between geometry and statistics for Gaussian states produced in simple optical devices.
We consider a complete description of a multi-mode bosonic quantum state in the coherent-state basis (which in this paper is denoted as the ``KKK'' function), which---up to a phase---is the square root of the well-known Husimi QQQ representation. We express the KKK function of any NNN-mode Gaussian state as a function of its covariance matrix and displacement vector, and also that of a general continuous-variable cluster state in terms of the modal squeezing and graph topology of the cluster. This formalism lets us characterize the non-Gaussian state left over when one measures a subset of modes of a Gaussian state using photon number resolving detection, the fidelity of the obtained non-Gaussian state with any target state, and the associated heralding probability, all analytically. We show that this probability can be expressed as a Hafnian, reinterpreting the output state of a circuit claimed to demonstrate quantum supremacy termed Gaussian boson sampling. As an example application of our formalism, we propose a method to prepare a two-mode coherent-cat-basis Bell state with fidelity close to unity and success probability that is fundamentally higher than that of a well-known scheme that splits an approximate single-mode cat state---obtained by photon number subtraction on a squeezed vacuum mode---on a balanced beam splitter. This formalism could enable exploration of efficient generation of cat-basis entangled states, which are known to be useful for quantum error correction against photon loss.