We present an interpretation for reduced density matrices of secondary variables in relativistic systems via an analysis of Wigner's method to construct the irreducible unitary representations of the Poincar\'e group. We argue that the usual partial trace method used to obtain spin-reduced matrices is not fully rigorous, however, employing our interpretation, similar effective reduced density matrices can be constructed. In addition, we show that our proposal is more useful than the usual one since we are not restricted only to the reduced density matrices that could be obtained by the ordinary partial trace method.We present an interpretation for reduced density matrices of secondary variables in relativistic systems via an analysis of Wigner's method to construct the irreducible unitary representations of the Poincar\'e group. We argue that the usual partial trace method used to obtain spin-reduced matrices is not fully rigorous, however, employing our interpretation, similar effective reduced density matrices can be constructed. In addition, we show that our proposal is more useful than the usual one since we are not restricted only to the reduced density matrices that could be obtained by the ordinary partial trace method.

arXiv:1311.0159Physical Review ADec 31, 201314 citations

Properties of Reduced Density Matrices

Theoretical chemistryJan 1, 19760 citations

Invariance of reduced density matrices under Local Unitary operations

Ana María Guerra Martins

We derive necessary and sufficient conditions for local unitary (LU) operators to leave invariant the set of 1-qubit reduced density matrices of a multi-qubit state. LU operators with this property are tensor products of {\it cyclic local} operators, and form a subgroup, the centralizer subgroup of the set of reduced states, of the Lie group $SU(2)^{\otimes n}$ . The dimension of this subgroup depends on the type of reduced density matrices. It is maximum when all reduced states are maximally mixed and it is minimum when none of them is maximally mixed. For any given multi-qubit state, pure or mixed, we compute the LU operators that fix the corresponding reduced density matrices and determine the equivalence class of the given state.We derive necessary and sufficient conditions for local unitary (LU) operators to leave invariant the set of 1-qubit reduced density matrices of a multi-qubit state. LU operators with this property are tensor products of {\it cyclic local} operators, and form a subgroup, the centralizer subgroup of the set of reduced states, of the Lie group $SU(2)^{\otimes n}$. The dimension of this subgroup depends on the type of reduced density matrices. It is maximum when all reduced states are maximally mixed and it is minimum when none of them is maximally mixed. For any given multi-qubit state, pure or mixed, we compute the LU operators that fix the corresponding reduced density matrices and determine the equivalence class of the given state.

arXiv:1307.0948Jul 3, 20130 citations

Determining the N-representability of transition reduced density matrices

Gustavo E. Massaccesi, P. Capuzzi, Ofelia B. Oña +4 more

The N-representability problem of reduced density matrices represents a fundamental challenge in electronic structure theory. In this work, we focus on the N-representability of transition reduced density matrices and propose a practical approach to embed a p-body transition reduced density matrix (RDM) of an N-particle system into a (p + 1)-body RDM of an (N + 1)-particle system. This embedding allows us to apply a unitary evolution algorithm based on a recently developed adaptive derivative-assembled pseudo-Trotter variational quantum algorithm to determine the N-representability of reduced density matrices. The algorithm works by effectively applying a sequence of unitary transformations to a given (N + 1)-particle initial state in such a way that the distance of a projection of its corresponding (p + 1)-body RDM (embedded p-body transition RDM) to a target is minimized. Our methodology enables the purification of transition reduced density matrices and the reconstruction of approximate wave functions involved in the studied transitions, providing an effective strategy to correct and refine transition density matrices. We validate our approach with numerical simulations on N-particle systems, including a three-electron model system and the H3 molecule, demonstrating its robustness and accuracy.The N-representability problem of reduced density matrices represents a fundamental challenge in electronic structure theory. In this work, we focus on the N-representability of transition reduced density matrices and propose a practical approach to embed a p-body transition reduced density matrix (RDM) of an N-particle system into a (p + 1)-body RDM of an (N + 1)-particle system. This embedding allows us to apply a unitary evolution algorithm based on a recently developed adaptive derivative-assembled pseudo-Trotter variational quantum algorithm to determine the N-representability of reduced density matrices. The algorithm works by effectively applying a sequence of unitary transformations to a given (N + 1)-particle initial state in such a way that the distance of a projection of its corresponding (p + 1)-body RDM (embedded p-body transition RDM) to a target is minimized. Our methodology enables the purification of transition reduced density matrices and the reconstruction of approximate wave functions involved in the studied transitions, providing an effective strategy to correct and refine transition density matrices. We validate our approach with numerical simulations on N-particle systems, including a three-electron model system and the H3 molecule, demonstrating its robustness and accuracy.

The Journal of Chemical PhysicsJul 21, 20252 citations

SU(2)^{\otimes n}

. The dimension of this subgroup depends on the type of reduced density matrices. It is maximum when all reduced states are maximally mixed and it is minimum when none of them is maximally mixed. For any given multi-qubit state, pure or mixed, we compute the LU operators that fix the corresponding reduced density matrices and determine the equivalence class of the given state.