Usually, variational calculations for the second-order reduced density matrix are performed subject to the constraint that the `` $P$ '' and `` $Q$ '' matrices are positive semidefinite, which only constrains the lowest eigenvalue of these matrices. We characterize the highest eigenvalue of these matrices and discuss how the associated constraint (which is related to the ground-state energy of the Hamiltonians $H=\ensuremath{-}P$ and $H=\ensuremath{-}Q$) can be implemented in practical calculations. This necessary condition for $N$ -representability should help ensure that the second-order reduced density matrix is not ``overcorrelated.''Usually, variational calculations for the second-order reduced density matrix are performed subject to the constraint that the ``$P$'' and ``$Q$'' matrices are positive semidefinite, which only constrains the lowest eigenvalue of these matrices. We characterize the highest eigenvalue of these matrices and discuss how the associated constraint (which is related to the ground-state energy of the Hamiltonians $H=\ensuremath{-}P$ and $H=\ensuremath{-}Q$) can be implemented in practical calculations. This necessary condition for $N$-representability should help ensure that the second-order reduced density matrix is not ``overcorrelated.''

Physical Review AMar 5, 200713 citations

Geometry of reduced density matrices

Bei Zeng

Jan 1, 20150 citations

On the one-particle reduced density matrices of a pure three-qutrit quantum state

Atsushi Higuchi

We present a necessary and sufficient condition for three qutrit density matrices to be the one-particle reduced density matrices of a pure three-qutrit quantum state. The condition consists of seven classes of inequalities satisfied by the eigenvalues of these matrices. One of them is a generalization of a known inequality for the qubit case. Some of these inequalities are proved algebraically whereas the proof of the others uses the fact that a continuous function of the state must have a minimum. Construction of states satisfying these inequalities relies on a representation of the convex set of the allowed set of eigenvalues in terms of corner points. We also present a result for a more general quantum system concerning the nature of the boundary surface of the set of the allowed set of eigenvalues.We present a necessary and sufficient condition for three qutrit density matrices to be the one-particle reduced density matrices of a pure three-qutrit quantum state. The condition consists of seven classes of inequalities satisfied by the eigenvalues of these matrices. One of them is a generalization of a known inequality for the qubit case. Some of these inequalities are proved algebraically whereas the proof of the others uses the fact that a continuous function of the state must have a minimum. Construction of states satisfying these inequalities relies on a representation of the convex set of the allowed set of eigenvalues in terms of corner points. We also present a result for a more general quantum system concerning the nature of the boundary surface of the set of the allowed set of eigenvalues.

arXiv:quant-ph/0309186Sep 25, 200312 citations

How Random Are Ergodic Eigenstates of the Ultrametric Random Matrices and the Quantum Sun Model?

Tanay Pathak

We numerically study the extreme-value statistics of the Schmidt eigenvalues of reduced density matrices obtained from the ergodic eigenstates. We start by exploring the extreme value statistics of the ultrametric random matrices and then the related Quantum Sun Model, which is also a toy model of avalanche theory. It is expected that these ergodic eigenstates are purely random and thus possess random matrix theory-like features, and the corresponding eigenvalue density should follow the universal Marchenko-Pastur law. Nonetheless, we find deviations, specifically near the tail in both cases. Similarly, the distribution of maximum eigenvalue, after appropriate centering and scaling, should follow the Tracy-Widom distribution. However, our results show that, for both the ultrametric random matrix and the Quantum Sun model, it can be better described using the extreme value distribution. As the extreme value distribution is associated with uncorrelated or weakly correlated random variables, the results hence indicate that the Schmidt eigenvalues exhibit much weaker correlations compared to the strong correlations typically observed in Wishart matrices. Similar deviations are observed for the case of minimum Schmidt eigenvalues as well . Despite the spectral statistics, such as nearest neighbor spacing ratios, aligning with the random matrix theory predictions, our findings reveal that randomness is still not fully achieved. This suggests that deviations in extreme-value statistics offer a stringent test to probe the randomness of ergodic eigenstates and can provide deeper insights into the underlying structure and correlations in ergodic systems.We numerically study the extreme-value statistics of the Schmidt eigenvalues of reduced density matrices obtained from the ergodic eigenstates. We start by exploring the extreme value statistics of the ultrametric random matrices and then the related Quantum Sun Model, which is also a toy model of avalanche theory. It is expected that these ergodic eigenstates are purely random and thus possess random matrix theory-like features, and the corresponding eigenvalue density should follow the universal Marchenko-Pastur law. Nonetheless, we find deviations, specifically near the tail in both cases. Similarly, the distribution of maximum eigenvalue, after appropriate centering and scaling, should follow the Tracy-Widom distribution. However, our results show that, for both the ultrametric random matrix and the Quantum Sun model, it can be better described using the extreme value distribution. As the extreme value distribution is associated with uncorrelated or weakly correlated random variables, the results hence indicate that the Schmidt eigenvalues exhibit much weaker correlations compared to the strong correlations typically observed in Wishart matrices. Similar deviations are observed for the case of minimum Schmidt eigenvalues as well . Despite the spectral statistics, such as nearest neighbor spacing ratios, aligning with the random matrix theory predictions, our findings reveal that randomness is still not fully achieved. This suggests that deviations in extreme-value statistics offer a stringent test to probe the randomness of ergodic eigenstates and can provide deeper insights into the underlying structure and correlations in ergodic systems.

arXiv:2501.19244Jan 31, 20250 citations

Usually, variational calculations for the second-order reduced density matrix are performed subject to the constraint that the ``

P

'' and ``

Q

'' matrices are positive semidefinite, which only constrains the lowest eigenvalue of these matrices. We characterize the highest eigenvalue of these matrices and discuss how the associated constraint (which is related to the ground-state energy of the Hamiltonians $H=\ensuremath{-}P$ and $H=\ensuremath{-}Q$) can be implemented in practical calculations. This necessary condition for

N

-representability should help ensure that the second-order reduced density matrix is not ``overcorrelated.''