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Comptes Rendus Mathématique

On the disentanglement of Gaussian quantum states by symplectic rotations

Maurice A. de Gosson

Published Jul 28, 2020• 4 citations
Publisher

Abstract

We show that every Gaussian mixed quantum state can be disentangled by conjugation with a unitary operator corresponding to a symplectic rotation via the metaplectic representation of the symplectic group. The main tools we use are the Werner–Wolf condition for separability on covariance matrices and the symplectic covariance of Weyl pseudo-differential operators.

DOI
10.5802/crmath.57

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We show that every Gaussian mixed quantum state can be disentangled by conjugation with a unitary operator corresponding to a symplectic rotation via the metaplectic representation of the symplectic group. The main tools we use are the Werner–Wolf condition for separability on covariance matrices and the symplectic covariance of Weyl pseudo-differential operators.
Werner and Wolf have proven in Phys. Rev. Lett. 86(16) (2001) a very elegant necessary and sufficient condition for a bosonic continuous variable bipartite Gaussian mixed quantum state to be separable. This condition is, however, difficult to implement in practice. In the present Letter, we propose a simpler condition which only involves the calculation of the symplectic matrix in the Williamson diagonalization of the covariance matrix of the state under consideration. The main tool in our construction is the observation, proved in previous work, that the Wigner transform is covariant only under symplectic or antisymplectic linear transformations. We also give a geometric interpretation of our condition in terms of the orthogonal projections of "quantum blobs"..
Gaussian unitary transformations are generated by quadratic Hamiltonians, i.e., Hamiltonians containing quadratic terms in creations and annihilation operators, and are heavily used in many areas of quantum physics, ranging from quantum optics and condensed matter theory to quantum information and quantum field theory in curved spacetime. They are known to form a representation of the metaplectic and spin group for bosons and fermions, respectively. These groups are the double covers of the symplectic and special orthogonal group, respectively, and our goal is to analyze the behavior of the sign ambiguity that one needs to deal with when moving between these groups and their double cover. We relate this sign ambiguity to expectation values of the form ⟨0∣exp⁡(−iH^)∣0⟩\langle 0|\exp{(-i\hat{H})}|0\rangle⟨0∣exp(−iH^)∣0⟩, where ∣0⟩|0\rangle∣0⟩ is a Gaussian state and H^\hat{H}H^ an arbitrary quadratic Hamiltonian. We provide closed formulas for ⟨0∣exp⁡(−iH^)∣0⟩\langle 0|\exp{(-i\hat{H})}|0\rangle⟨0∣exp(−iH^)∣0⟩ and show how we can efficiently describe group multiplications in the double cover without the need of going to a faithful representation on an exponentially large or even infinite-dimensional space. Our construction relies on an explicit parametrization of these two groups (metaplectic, spin) in terms of symplectic and orthogonal group elements together with a twisted U(1) group.