Constraints placed on a reduced density matrix or density by the requirement that it correspond to an eigenstate or ensemble of eigenstates of some operator are investigated. Linear conditions, including expectation value, commutator, hypervirial, and hierarchy equation conditions, may lead to linear constraints on lower-order reduced density matrices and densities or leave them unconstrained. A zero-dispersion condition for a $p$ -electron operator defines a parabolic trough on which reduced density matrices of order $2p$ must lie, but usually does not restrict lower-order reduced density matrices or the density. Some properties of symmetrized and antisymmetrized matrix products, related to their reduction behavior, are also investigated.Constraints placed on a reduced density matrix or density by the requirement that it correspond to an eigenstate or ensemble of eigenstates of some operator are investigated. Linear conditions, including expectation value, commutator, hypervirial, and hierarchy equation conditions, may lead to linear constraints on lower-order reduced density matrices and densities or leave them unconstrained. A zero-dispersion condition for a $p$-electron operator defines a parabolic trough on which reduced density matrices of order $2p$ must lie, but usually does not restrict lower-order reduced density matrices or the density. Some properties of symmetrized and antisymmetrized matrix products, related to their reduction behavior, are also investigated.

Physical Review AJul 1, 198418 citations

Multiple fidelities and joint numerical range

Pei Li, Bang-Hai Wang

We investigate the effectiveness of entanglement detection based on multiple fidelities via the geometry of the joint separable numerical range. When all reference states are product states, we derive a necessary and sufficient criterion for such detection: either some pair of reference states has nontrivial moduli of the local inner products on both subsystems, or the orthogonal complement of the span of the reference states is completely entangled. We further show that there exist sets of reference product states for which no proper subset is effective for entanglement detection, whereas the full set is. A typical example of this phenomenon is provided by unextendible product bases. Moreover, for a pair of reference product states on a bipartite system with arbitrary local dimensions, we characterize both the joint numerical range and the joint separable numerical range, showing that the joint separable numerical range is determined solely by their local fidelities, as illustrated by a representative two-qubit example. Our results offer a systematic approach to designing effective entanglement witnesses and lay the groundwork for extensions to higher-dimensional and multipartite scenarios.We investigate the effectiveness of entanglement detection based on multiple fidelities via the geometry of the joint separable numerical range. When all reference states are product states, we derive a necessary and sufficient criterion for such detection: either some pair of reference states has nontrivial moduli of the local inner products on both subsystems, or the orthogonal complement of the span of the reference states is completely entangled. We further show that there exist sets of reference product states for which no proper subset is effective for entanglement detection, whereas the full set is. A typical example of this phenomenon is provided by unextendible product bases. Moreover, for a pair of reference product states on a bipartite system with arbitrary local dimensions, we characterize both the joint numerical range and the joint separable numerical range, showing that the joint separable numerical range is determined solely by their local fidelities, as illustrated by a representative two-qubit example. Our results offer a systematic approach to designing effective entanglement witnesses and lay the groundwork for extensions to higher-dimensional and multipartite scenarios.

quant-pharXiv:2605.24360May 26, 20260 citations

The joint numerical range of three hermitian $4\times 4$ matricesThe joint numerical range of three hermitian $4\times 4$ matrices

Piotr Pikul, Ilya M. Spitkovsky, Konrad Szymański +2 more

We analyze the joint numerical range $W$ of three hermitian matrices of order four. In the generic case, this three-dimensional convex set has a smooth boundary. We analyze non-generic structures. Fifteen possible classes regarding the numbers of non-elliptic faces in the boundary of $W$ are identified and an explicit example is presented for each class. Secondly, it is shown that a nonempty intersection of three mutually distinct one-dimensional faces is a corner point. Thirdly, introducing a tensor product structure into $\mathbb C^4=\mathbb C^2\otimes\mathbb C^2$ , one defines the separable joint numerical range - a subset of $W$ useful in studies of quantum entanglement. The boundary of the separable numerical range is compared with that of $W$ .We analyze the joint numerical range $W$ of three hermitian matrices of order four. In the generic case, this three-dimensional convex set has a smooth boundary. We analyze non-generic structures. Fifteen possible classes regarding the numbers of non-elliptic faces in the boundary of $W$ are identified and an explicit example is presented for each class. Secondly, it is shown that a nonempty intersection of three mutually distinct one-dimensional faces is a corner point. Thirdly, introducing a tensor product structure into $\mathbb C^4=\mathbb C^2\otimes\mathbb C^2$, one defines the separable joint numerical range - a subset of $W$ useful in studies of quantum entanglement. The boundary of the separable numerical range is compared with that of $W$.

arXiv:2510.27670Oct 31, 20250 citations

Properties of Reduced Density Matrices

Theoretical chemistryJan 1, 19760 citations

p

-electron operator defines a parabolic trough on which reduced density matrices of order

2p

must lie, but usually does not restrict lower-order reduced density matrices or the density. Some properties of symmetrized and antisymmetrized matrix products, related to their reduction behavior, are also investigated.

The joint numerical range of three hermitian

4\times 4

matrices

We analyze the joint numerical range

W

of three hermitian matrices of order four. In the generic case, this three-dimensional convex set has a smooth boundary. We analyze non-generic structures. Fifteen possible classes regarding the numbers of non-elliptic faces in the boundary of

W

are identified and an explicit example is presented for each class. Secondly, it is shown that a nonempty intersection of three mutually distinct one-dimensional faces is a corner point. Thirdly, introducing a tensor product structure into

\mathbb C^4=\mathbb C^2\otimes\mathbb C^2

, one defines the separable joint numerical range - a subset of

W

useful in studies of quantum entanglement. The boundary of the separable numerical range is compared with that of

W