We investigate upper and lower bounds on the entropy of entanglement of a superposition of bipartite states as a function of the individual states in the superposition. In particular, we extend the results by Gour [Phys. Rev. A 76, 052320 (2007)] to superpositions of several states rather than just two. We then investigate the entanglement in a subspace as a function of its basis states: we find upper bounds for the largest entanglement in a subspace and demonstrate that no such lower bound for the smallest entanglement exists. Finally, we consider entanglement of superpositions using measures of entanglement other than the entropy of entanglement.
Abstract We prove that the Hilbert–Schmidt norm of k -particle reduced density matrices of N -body fermionic states is bounded from above by
CkNk/2 C k N k / 2 - matching the scaling behaviour of Slater determinant states. This generalises a recent result of Christiansen [3] on 2-particle reduced density matrices to higher order density matrices. Moreover, our estimate directly yields a lower bound on the von Neumann entropy and the 2-Rényi entropy of reduced density matrices, thereby providing further insight into conjectures of Carlen–Lieb–Reuvers [2, 8].
The least upper bound of the eigenvalues of second-order reduced density matrices for a system of fermions is proved to be
n for a system of
2n or
2n+1 identical fermions. It is also shown that this limiting state may be interpreted as a system of identical pairs behaving as quasibosons.
We discuss a general bipartitelike representation and Schmidt decomposition of an arbitrary pure state of
N indistinguishable fermions, based on states of
M<N and $(N\ensuremath{-}M)$ fermions. It is directly connected with the reduced
M- and $(N\ensuremath{-}M)$-body density matrices (DMs), which have the same spectrum in such states. The concept of
M-body entanglement emerges naturally in this scenario, generalizing that of one-body entanglement. Rigorous majorization relations satisfied by the normalized
M-body DM are then derived, which imply that the associated entropy will not increase, on average, under a class of operations which have these DMs as postmeasurement states. Moreover, such entropy is an upper bound to the bipartite entanglement entropy generated by a class of operations which map the original state to a bipartite state of
M and $N\ensuremath{-}M$ effectively distinguishable fermions. Analytic evaluation of the spectrum of
M-body DMs in some strongly correlated fermionic states is also provided.
Abstract We consider entanglement of first-quantized identical particles by adopting an algebraic approach. In particular, we investigate fermions whose wave functions are given by the Slater determinants, as for singlet sectors of one-matrix models. We show that the upper bounds of the general Rényi entropies are N log 2 for N particles or an N × N matrix. We compute the target space entanglement entropy and the mutual information in a free one-matrix model. We confirm the area law: the single-interval entropy for the ground state scales as
31 1 3 log N in the large N model. We obtain an analytical
O(N0) O N 0 expression of the mutual information for two intervals in the large N expansion.