We use a generalized Bogoliubov transformation to construct a set of transformed occupation number states of the harmonic oscillator in the Liouville space. General expressions for the expansion coefficients of these states in the bare number basis are obtained and the properties of these states investigated. The transformation is non-dynamical in nature and divides the transformed basis into disconnected correlation subspaces under the transformation. The basis is complete and orthonormal. Elements of the basis are in general mixed states, and the state with the lowest number indices is the thermal vacuum. Since the Kossakowski–Lindblad (KL) equation remains form invariant under the same transformation, the transformation parameter can be identified to the thermal parameter of the master equation. The transformed states then acquire thermal property through this connection. We also work out the explicit expressions of the transformation matrices between the transformed states and the eigenstates of the KL equation for different temperatures.
Finding ways to transform a quantum state to another is fundamental to quantum information processing. In this paper, we apply the sparse matrix approach to the quantum state transformation problem. In particular, we present an approach for searching for unitary matrices for quantum state transformation by directly optimizing the objective problem using the alternating direction method of multipliers. Moreover, we consider the use of group sparsity as an alternative sparsity choice in quantum state transformation problems. Our approach incorporates sparsity constraints into quantum state transformation by formulating it as a nonconvex problem. It establishes a useful framework for efficiently handling complex quantum systems and achieving precise state transformations.
We examine the powers of entanglement-assisted transformation and multiple-copy entanglement transformation. First, we find a sufficient condition of when a given catalyst is useful in producing another specific target state. As an application of this condition, for any non-maximally entangled bipartite pure state and any integer
n not less than 4, we are able to explicitly construct a set of
n×n quantum states which can be produced by using the given state as a catalyst. Second, we prove that for any positive integer
k, entanglement-assisted transformation with
k×k-dimensional catalysts is useful in producing a target state if and only if multiple-copy entanglement transformation with
k copies of state is useful in producing the same target. Moreover, a necessary and sufficient condition for both of them is obtained in terms of the Schmidt coefficients of the target. This equivalence of entanglement-assisted transformation and multiple-copy entanglement transformation implies many interesting properties of entanglement transformation. Furthermore, these results are generalized to the case of probabilistic entanglement transformations.