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

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,R) 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,R), 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.

The Symplectic Camel and the Uncertainty Principle: The Tip of an Iceberg?

“…This property, which is implicit in the papers [39,40] by Simon et al, was proved by Narcowich in [24,25] (also see Narcowich and O'Connell [26]). It is easily checked using a characterization of the nonnegativity of + i 2 J .…”

“…Littlejohn [17]). To prove that the uncertainty relations (39) hold is now very easy: in view of the discussion of last subsection we have…”

“…at time t = 0, then we will have ( X j,t ) 2 ( P j,t ) 2 ≥ Cov(X j,t , P j,t ) 2 + 1 4 2 (39) for all times t, both future and past, where X j,t , etc. are defined by the new covariance ellipsoid.…”

Quantification of continuous variable entanglement with only two types of simple measurements

“…The first one is related to rotations of the quadratures of the quantized electromagnetic field and the last one to local squeezing operations. The new covariance matrixṼ is connected to V by the following relation [40,41],Ṽ = SV S † , (A. 4) which implies that…”

“…Unitary local operations preserving the Gaussian character of a state described by V are mapped to the following local symplectic transformation [40]:…”