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arXiv:2504.20979 · IEEE International Symposium on Information Theory (ISIT)

A Complete Characterization of Passive Unitary Normalizable (PUN) Gaussian States

Tiju Cherian John, Hemant K. Mishra, Saikat Guha

Published Jun 22, 2025• 0 citations
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Abstract

We provide a complete characterization of the class of multimode quantum Gaussian states that can be reduced to a tensor product of thermal states using only a passive unitary operator. We call these states passive unitary normalizable (PUN) Gaussian states. The characterization of PUN Gaussian states is given in three different ways: ($i$) in terms of their covariance matrices, (ii) using gauge-invariance (a special class of Glauber-Sudarshan p-functions), and (iii) with respect to the recently obtained$(A,\ \Lambda)$parametrization of Gaussian states in [J. Math. Phys. 62, 022102 (2021)]. In terms of the covariance matrix, our characterization states that an n-mode quantum Gaussian state is PUN if and only if its$2n\times 2n$quantum covariance matrix$S_{\mathrm{T}\mathfrak{i}}$is skew-Hamiltonian, i.e., it commutes with the standard symplectic matrix,$\left[\begin{array}{cc} 0 & I_n \\ -I_n & 0 \end{array}\right]$, with$I_{n}$being the$nxn$identity matrix. It is well-known that the so-called gauge-invariant Gaussian states are PUN, but whether the converse is true is not known in the literature to the best of our knowledge. We establish the converse in affirmation. Lastly, in terms of the$(A,\ \Lambda)$-parameterization, we show that a Gaussian state with parameters$(A,\ \Lambda)$is PUN if and only if$A=0$

DOI
10.1109/isit63088.2025.11195518

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Abstract. We investigate the action of local unitary operations on multimode (pure or mixed) Gaussian states and single out the minimal number of locally invariant parametres which completely characterise the covariance matrix of such states. For pure Gaussian states, central resources for continuous-variable quantum information, we investigate separately the parametre reduction due to the additional constraint of global purity, and the one following by the local-unitary freedom. Counting arguments and insights from the phase-space Schmidt decomposition and in general from the framework of symplectic analysis, accompany our description of the standard form of pure n-mode Gaussian states. In particular we clarify why only in pure states with n ≤ 3 modes all the direct correlations between position and momentum operators can be set to zero by local unitary operations. For any n, the emerging minimal set of parametres contains complete information about all forms of entanglement in the corresponding states. An efficient state engineering scheme (able to encode direct correlations between position and momentum operators as well) is proposed to produce entangled multimode Gaussian resources, its number of optical elements matching the minimal number of locally invariant degrees of freedom of general pure n-mode Gaussian states.
We address the decomposition of a multimode pure Gaussian state with respect to a bipartite division of the modes. For any such division the state can always be expressed as a product state involving entangled two-mode squeezed states and single-mode local states at each side. The character of entanglement of the state can therefore be understood modewise; that is, a given mode on one side is entangled with only one corresponding mode of the other, and therefore the total bipartite entanglement is the sum of the modewise entanglement. This decomposition is generally not applicable to all mixed Gaussian states. However, the result can be extended to a special family of ``isotropic'' states, characterized by a phase space covariance matrix with a completely degenerate symplectic spectrum.
Let Γ(H) be the boson Fock space over a finite dimensional Hilbert space H. It is shown that every Gaussian symmetry admits a Klauder–Bargmann integral representation in terms of coherent states. Furthermore, Gaussian states, Gaussian symmetries, and second quantization contractions belong to a weakly closed self-adjoint semigroup E2(H) of bounded operators in Γ(H). This yields a common parametrization for these operators. It is shown that the new parametrization for Gaussian states is a fruitful alternative to the customary parametrization by position–momentum mean vectors and covariance matrices. This leads to a rich harvest of corollaries: (i) every Gaussian state ρ admits a factorization ρ=Z1†Z1, where Z1 is an element of E2(H) and has the form Z1=cΓ(P)exp∑r=1nλrar+∑r,s=1nαrsaras on the dense linear manifold generated by all exponential vectors, where c is a positive scalar, Γ(P) is the second quantization of a positive contractive operator P in H, ar, 1 ≤ r ≤ n, are the annihilation operators corresponding to the n different modes in Γ(H), λr∈C, and [αrs] is a symmetric matrix in Mn(C); (ii) an explicit particle basis expansion of an arbitrary mean zero pure Gaussian state vector along with a density matrix formula for a general Gaussian state in terms of its E2(H)-parameters; (iii) a class of examples of pure n-mode Gaussian states that are completely entangled; (iv) tomography of an unknown Gaussian state in Γ(Cn) by the estimation of its E2(Cn) parameters using O(n2) measurements with a finite number of outcomes.
We present a novel approach to the separability problem for Gaussian quantum states of bosonic continuous variable systems. We derive a simplified necessary and sufficient separability criterion for arbitrary Gaussian states of m versus n modes, which relies on convex optimisation over marginal covariance matrices on one subsystem only. We further revisit the currently known results stating the equivalence between separability and positive partial transposition (PPT) for specific classes of Gaussian states. Using techniques based on matrix analysis, such as Schur complements and matrix means, we then provide a unified treatment and compact proofs of all these results. In particular, we recover the PPT-separability equivalence for: (i) Gaussian states of 1 versus n modes; and (ii) isotropic Gaussian states. In passing, we also retrieve (iii) the recently established equivalence between separability of a Gaussian state and and its complete Gaussian extendability. Our techniques are then applied to progress beyond the state of the art. We prove that: (iv) Gaussian states that are invariant under partial transposition are necessarily separable; (v) the PPT criterion is necessary and sufficient for separability for Gaussian states of m versus n modes that are symmetric under the exchange of any two modes belonging to one of the parties; and (vi) Gaussian states which remain PPT under passive optical operations can not be entangled by them either. This is not a foregone conclusion per se (since Gaussian bound entangled states do exist) and settles a question that had been left unanswered in the existing literature on the subject. This paper, enjoyable by both the quantum optics and the matrix analysis communities, overall delivers technical and conceptual advances which are likely to be useful for further applications in continuous variable quantum information theory, beyond the separability problem.
We present a novel approach to the separability problem for Gaussian quantum states of bosonic continuous variable systems. We derive a simplified necessary and sufficient separability criterion for arbitrary Gaussian states of m versus n modes, which relies on convex optimisation over marginal covariance matrices on one subsystem only. We further revisit the currently known results stating the equivalence between separability and positive partial transposition (PPT) for specific classes of Gaussian states. Using techniques based on matrix analysis, such as Schur complements and matrix means, we then provide a unified treatment and compact proofs of all these results. In particular, we recover the PPT-separability equivalence for: (i) Gaussian states of 1 versus n modes; and (ii) isotropic Gaussian states. In passing, we also retrieve (iii) the recently established equivalence between separability of a Gaussian state and and its complete Gaussian extendability. Our techniques are then applied to progress beyond the state of the art. We prove that: (iv) Gaussian states that are invariant under partial transposition are necessarily separable; (v) the PPT criterion is necessary and sufficient for separability for Gaussian states of m versus n modes that are symmetric under the exchange of any two modes belonging to one of the parties; and (vi) Gaussian states which remain PPT under passive optical operations can not be entangled by them either. This is not a foregone conclusion per se (since Gaussian bound entangled states do exist) and settles a question that had been left unanswered in the existing literature on the subject. This paper, enjoyable by both the quantum optics and the matrix analysis communities, overall delivers technical and conceptual advances which are likely to be useful for further applications in continuous variable quantum information theory, beyond the separability problem.
Let Γ(H)\Gamma(\mathcal{H})Γ(H) be the boson Fock space over a finite dimensional Hilbert space H\mathcal{H}H. It is shown that every gaussian symmetry admits a Klauder-Bargmann integral representation in terms of coherent states. Furthermore, gaussian symmetries, gaussian states and second quantization contractions, all of these operators belong to a weakly closed, selfadjoint semigroup E2(H)\mathcal{E}_2(\mathcal{H})E2​(H) of bounded operators in Γ(H)\Gamma(\mathcal{H})Γ(H). This yields, a new parametrization of gaussian states, which is a very fruitful alternative to the customary parametrization by position-momentum mean vectors and covariance matrices. This leads to a rich harvest of corollaries: (i) every gaussian state ρ\rhoρ admits a factorization ρ=Z1†Z1\rho = Z_{1}^{\dagger}Z_{1}ρ=Z1†​Z1​, where Z1Z_{1}Z1​ is an element of E2(H)\mathcal{E}_2(\mathcal{H})E2​(H) and has the form Z1=cΓ(Λ)exp⁡∑r=1nλrar+∑r,s=1nαrsarasZ_{1} = \sqrt{c}\Gamma(\sqrt{\Lambda})\exp{\sum_{r=1}^{n} \lambda_ra_r+\sum_{r,s=1}^{n} \alpha_{rs}a_{r}a_{s}}Z1​=c​Γ(Λ​)exp∑r=1n​λr​ar​+∑r,s=1n​αrs​ar​as​ on the dense linear manifold generated by all exponential vectors, Λ\LambdaΛ being a positive operator in H\mathcal{H}H, ar,1≤r≤na_{r}, 1\leq r \leq nar​,1≤r≤n are the annihilation operators corresponding to the nnn different modes in Γ(H)\Gamma(\mathcal{H})Γ(H), λr∈C\lambda_r\in \mathbb{C}λr​∈C and [αrs][\alpha_{rs}][αrs​] is a symmetric matrix in Mn(C)M_n(\mathbb{C})Mn​(C); (ii) an explicit particle basis expansion of an arbitrary mean zero pure gaussian state vector along with a density matrix formula for a general gaussian state in terms of its E2(H)\mathcal{E}_2(\mathcal{H})E2​(H)-parameters; (iii) an easy test for the entanglement of pure gaussian states and a class of examples of pure nnn-mode gaussian states which are completely entangled; (iv) tomography of an unknown gaussian state in Γ(Cn)\Gamma(\mathbb{C}^n)Γ(Cn) by the estimation of its E2(Cn)\mathcal{E}_2(\mathbb{C}^n)E2​(Cn)-parameters using O(n2)O(n^2)O(n2) measurements with a finite number of outcomes.