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Physical Review A

Quantum-noise matrix for multimode systems: U(n) invariance, squeezing, and normal forms

R. Simon, Narasimhaiengar Mukunda, Biswadeb Dutta

Published Mar 1, 1994• 463 citations
Publisher

Abstract

We present a complete analysis of variance matrices and quadrature squeezing for arbitrary states of quantum systems with any finite number of degrees of freedom. Basic to our analysis is the recognition of the crucial role played by the real symplectic group Sp(2n,openR) of linear canonical transformations on n pairs of canonical variables. We exploit the transformation properties of variance (noise) matrices under symplectic transformations to express the uncertainty-principle restrictions on a general variance matrix in several equivalent forms, each of which is manifestly symplectic invariant. These restrictions go beyond the classically adequate reality, symmetry, and positivity conditions. Towards developing a squeezing criterion for n-mode systems, we distinguish between photon-number-conserving passive linear optical systems and active ones. The former correspond to elements in the maximal compact U(n) subgroup of Sp(2n,openR), the latter to noncompact elements outside U(n). Based on this distinction, we motivate and state a U(n)-invariant squeezing criterion applicable to any state of an n-mode system, and explore alternative ways of expressing it. The set of all possible quantum-mechanical variance matrices is shown to contain several interesting subsets or subfamilies, whose definitions are related to the fact that a general variance matrix is not diagonalizable within U(n). Definitions, characterizations, and canonical forms for variance matrices in these subfamilies, as well as general ones, and their squeezing nature, are established. It is shown that all conceivable variance matrices can be generated through squeezed thermal states of the n-mode system and their symplectic transforms. Our formulas are developed in both the real and the complex forms for variance matrices, and ways to pass between them are given.

DOI
10.1103/physreva.49.1567

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Invariant theoretic approach to uncertainty relations for quantum systems

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We present a general framework and procedure to derive uncertainty relations for observables of quantum systems in a covariant manner. All such relations are consequences of the positive semidefiniteness of the density matrix of a general quantum state. Particular emphasis is given to the action of unitary symmetry operations of the system on the chosen observables, and the covariance of the uncertainty relations under these operations. The general method is applied to the case of an $n$-mode system to recover the $Sp(2n,\,R)$-covariant multi mode generalization of the single mode Schrödinger-Robertson Uncertainty Principle; and to the set of all polynomials in canonical variables for a single mode system. In the latter situation, the case of the fourth order moments is analyzed in detail, exploiting covariance under the homogeneous Lorentz group $SO(2,\,1)$ of which the symplectic group $Sp(2,\,R)$ is the double cover.

arXiv:1205.5132•May 23, 2012•4 citations
We present a complete analysis of variance matrices and quadrature squeezing for arbitrary states of quantum systems with any finite number of degrees of freedom. Basic to our analysis is the recognition of the crucial role played by the real symplectic group Sp(2n,openR) of linear canonical transformations on n pairs of canonical variables. We exploit the transformation properties of variance (noise) matrices under symplectic transformations to express the uncertainty-principle restrictions on a general variance matrix in several equivalent forms, each of which is manifestly symplectic invariant. These restrictions go beyond the classically adequate reality, symmetry, and positivity conditions. Towards developing a squeezing criterion for n-mode systems, we distinguish between photon-number-conserving passive linear optical systems and active ones. The former correspond to elements in the maximal compact U(n) subgroup of Sp(2n,openR), the latter to noncompact elements outside U(n). Based on this distinction, we motivate and state a U(n)-invariant squeezing criterion applicable to any state of an n-mode system, and explore alternative ways of expressing it. The set of all possible quantum-mechanical variance matrices is shown to contain several interesting subsets or subfamilies, whose definitions are related to the fact that a general variance matrix is not diagonalizable within U(n). Definitions, characterizations, and canonical forms for variance matrices in these subfamilies, as well as general ones, and their squeezing nature, are established. It is shown that all conceivable variance matrices can be generated through squeezed thermal states of the n-mode system and their symplectic transforms. Our formulas are developed in both the real and the complex forms for variance matrices, and ways to pass between them are given.
A general anaylsis of squeezing transformations for two-mode systems is given based on the four-dimensional real symplectic group Sp(4,openR). Within the framework of the unitary (metaplectic) representation of this group, a distinction between compact photon-number-conserving and noncompact photon-number-nonconserving squeezing transformations is made. We exploit the U(2) invariant squeezing criterion to divide the set of all squeezing transformations into a two-parameter family of distinct equivalence classes with representative elements chosen for each class. Familiar two-mode squeezing transformations in the literature are recognized in our framework and seen to form a set of measure zero. Examples of squeezed coherent and thermal states are worked out. The need to extend the heterodyne detection scheme to encompass all of U(2) is emphasized, and known experimental situations where all U(2) elements can be reproduced are briefly described.
It is shown that any two Hamiltonians H(t) and H'(t) of N dimensional quantum systems can be related by means of time-dependent canonical transformations (CT). The dynamical symmetry group of system with Hamiltonian H(t) coincides with the invariance group of H(t). Quadratic Hamiltonians can be diagonalized by means of linear time-dependent CT. The diagonalization can be explicitly carried out in the case of stationary and some nonstationary quadratic H. Linear CT can diagonalize the uncertainty matrix σ(ρ) for canonical variables p_k, q_j in any state ρ, i.e., σ(ρ) is symplectically congruent to a diagonal uncertainty matrix. For multimode squeezed canonical coherent states (CCS) and squeezed Fock states with equal photon numbers in each mode σis symplectic itself. It is proved that the multimode Robertson uncertainty relation is minimized only in squeezed CCS.
The set of covariance matrices of a continuous-variable quantum system with a finite number of degrees of freedom is a strict subset of the set of real positive-definite matrices (PDMs) due to Heisenberg’s uncertainty principle. This has the implication that, in general, not every orthogonal transform of a quantum covariance matrix (CM) produces a PDM that obeys the uncertainty principle. A natural question thus arises, to find the set of quantum covariance matrices consistent with a given eigenspectrum. For the special class of pure Gaussian states the set of quantum covariance matrices with a given eigenspectrum consists of a single orbit of the action of the orthogonal symplectic group. The eigenspectrum of a CM of a state in this class is composed of pairs that each multiply to one. Our main contribution is finding a non-trivial class of eigenspectra with the property that the set of quantum covariance matrices corresponding to any eigenspectrum in this class are related by orthogonal symplectic transformations. We show that all non-degenerate eigenspectra with this property must belong to this class, and that the set of such eigenspectra coincides with the class of non-degenerate eigenspectra that identify the physically relevant thermal and squeezing parameters of a Gaussian state.
Gaussian pure states of systems with n degrees of freedom and their evolution under quadratic Hamiltonians are studied. The Wigner-Moyal technique together with the symplectic group Sp(2n,openR) is shown to give a convenient framework for handling these problems. By mapping these states to the set of n\ifmmode\times\else\texttimes\fi{}n complex symmetric matrices with a positive-definite real part, it is shown that their evolution under quadratic Hamiltonians is compactly described by a matrix generalization of the M\"obius transformation; the connection between this result and the ``abcd law'' of Kogelnik in the context of laser beams is brought out. An equivalent Poisson-bracket description over a special orbit in the Lie algebra of Sp(2n,openR) is derived. Transformation properties of a special class of partially coherent anisotropic Gaussian Schell-model optical fields under the action of Sp(4, openR) first-order systems are worked out as an example, and a generalization of the ``abcd law'' to the partially coherent case is derived. The relevance of these results to the problem of squeezing in multimode systems is noted.
Quantum continuous variable systems are quantum systems obeying the canonical commutation relations. As we have seen in the previous chapter, such systems always require the adoption of infinite-dimensional Hilbert spaces, even when a finite number of degrees of freedom is considered. Besides making their dynamics potentially intractable, this fact also implies that the study of properties related to the information and correlation content of continuous variable systems turns out to be particularly difficult. A widely known way to make continuous variable systems more tractable is to consider the restriction to Gaussian states. Already well trodden in older research on quantum optics, the study of Gaussian systems has more recently become a central theme of continuous variable quantum information. If open quantum systems, coupled to an external environment, are considered, the Gaussian regime applies when the system couples linearly to its environment and when system and environment are governed by Hamiltonians which are at most quadratic in the canonical operators (or, equivalently, when the overall Hamiltonian of system and environment is at most quadratic). These conditions may seem rather restrictive but, as we shall point out in the course of the book, are actually ordinarily met by a vast number of existing experimental set-ups in the areas of quantum optics, trapped ions, optomechanics, atomic ensembles and certain superconducting degrees of freedom (all of these platforms are reviewed in Chapter 9 ). What we will refer to as Gaussian measurements – POVMs that preserve the Gaussian character of initial states – are also customarily carried out with comparatively high efficiency, while the description of quantum noise can be incorporated seamlessly into the Gaussian picture. What is perhaps even more important to note is that the restriction to the Gaussian realm, while not exhaustive of the full wealth of dynamics and situations allowed in the whole Hilbert space, still lets one include most of the processes relevant to quantum technologies. Squeezing, whereby certain canonical quadratures have uncertainties below the vacuum state noise, and can hence be used in precision measurements, quantum entanglement, and cooling, where the entropy is drained out of a system in order to reach a pure quantum state and thus initialise information protocols or low noise experiments, may all be described within the Gaussian regime. Within the quantum information discourse, the Gaussian restriction has been often criticised on the grounds that no real genuinely quantum effects can ever be observed without leaving it at some point, since the statistics obtained from Gaussian states through Gaussian measurements can be reproduced by classical probability distributions. While technically correct, the reader should be reminded that such critiques often disregard the fact that the object described by a Gaussian state does entail a genuinely quantum description in a Hilbert space. Hence, for instance, the preparation of a pure Gaussian quantum state is the preparation of a pure vector of the Hilbert space. Likewise, a highly squeezed Gaussian state prepared for a metrological protocol is a state with very low noise in a certain physical observable, regardless of whether it can be mimicked by a classical distribution or not. The testing and exploitation of quantum non-locality (stronger than classical correlations that imply the presence of quantum entanglement) with Gaussian states does instead require a departure from Gaussian measurements. However, as we will see, certain non-Gaussian measurements are customarily implemented with current technology, and it is hence still relevant to study in detail the creation of such a resource for Gaussian states. This chapter will introduce Gaussian states from a somewhat unusual angle: as ground and thermal states of at most second-order Hamiltonians. Our treatment will highlight the privileged role played by quadratic canonical ( i.e. , ‘symplectic’) transformations in this context, which we will take care to describe in full generality, and will hinge on the reduction of Gaussian states into their normal modes, which allows for their systematic diagonalisation. We will then proceed to explore an equivalent characterisation in terms of statistical moments of the canonical operators, and to introduce the implications of non-commutativity on such statistics in the form of the Robertson–Schrödinger uncertainty relation.
We present a general framework and procedure to derive uncertainty relations for observables of quantum systems in a covariant manner. All such relations are consequences of the positive semidefiniteness of the density matrix of a general quantum state. Particular emphasis is given to the action of unitary symmetry operations of the system on the chosen observables, and the covariance of the uncertainty relations under these operations. The general method is applied to the case of an nnn-mode system to recover the Sp(2n, R)Sp(2n,\,R)Sp(2n,R)-covariant multi mode generalization of the single mode Schrödinger-Robertson Uncertainty Principle; and to the set of all polynomials in canonical variables for a single mode system. In the latter situation, the case of the fourth order moments is analyzed in detail, exploiting covariance under the homogeneous Lorentz group SO(2, 1)SO(2,\,1)SO(2,1) of which the symplectic group Sp(2, R)Sp(2,\,R)Sp(2,R) is the double cover.