Normal form decomposition for Gaussian-to-Gaussian superoperators. (English) Zbl 1316.81013

De Palma, Giacomo; Mari, Andrea; Giovannetti, Vittorio; Holevo, Alexander S.

J. Math. Phys. 56, No. 5, 052202, 19 p. (2015).

Similarly to the fact that a Gaussian probability distribution is optimal (has largest entropy) among all distributions with given first order moments, it has been proven that in many problems of quantum computation and quantum communication, optimal quantum states are Gaussian, i.e., density operators \(\hat\rho\) whose characteristic function \(\chi(k_x,k_p)=\text{Tr}(\hat\rho \cdot \exp(\text{i}\cdot (k_x^T\cdot \hat x+k_p^T\cdot \hat p))\) has the Gaussian form; here, as usual, \(\hat x\) and \(\hat p\) denote coordinates and momentum operators. By combining different Gaussian states with different probabilities, we can get convex combinations of Gaussian states. It is therefore important to know which states \(\hat\rho\) can be represented as such convex combinations and which cannot. Proving that a given state cannot be so represented is often difficult. However, once we find such non-representable states \(\hat\rho_i\), we can then prove that a state \(\hat\rho\) is non-representable if we show that \(\Phi\hat\rho=\hat\rho_i\) for some linear operator (“superoperator”) \(\Phi\) that maps Gaussian states into Gaussian states (and thus, maps their convex combinations into similar convex combinations).
In the past, researchers studied such linear operators, but only physically meaningful ones, with an additional limitation that \(\Phi\) maps every physical state into a physical state. The paper shows that without such a limitation, we get a wider class of operators \(\Phi\) and thus, more ways to prove non-representability. The authors provide a general characterization of all such Gaussian-to-Gaussian superoperators. For several physically important class of such superoperators – e.g., for the ones corresponding to single-mode results – they get a general representation of such superoperators as a composition of dilatation, addition of a Gaussian noise, and (possibly) transposition.

Reviewer: Vladik Ya. Kreinovich (El Paso)

MSC:

81P45	Quantum information, communication, networks (quantum-theoretic aspects)
94A40	Channel models (including quantum) in information and communication theory
81R30	Coherent states
46L07	Operator spaces and completely bounded maps

Keywords:

quantum computation; quantum communication; quantum cryptography; Gaussian quantum states; Gaussian-to-Gaussian superoperators; normal form decomposition

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