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 frac{1}{3} log N in the large N model. We obtain an analytical mathcal{O}left({N}^0right) expression of the mutual information for two intervals in the large N expansion.
Highlights
We compute the target space entanglement entropy and the mutual information in a free one-matrix model
The notion of the base space is meaningless for the quantum mechanics, that is, (1 + 0)-dimensional QFTs
The singlet sector described by the eigenvalues can be mapped to a quantum mechanics of non-relativistic non-interacting fermions [27]
Summary
Let us begin with the quantum mechanics of a single particle on a manifold M. It is convenient to introduce the projection operators onto the region A and Aas. The total Hilbert space H(1) is decomposed into a direct sum as H(1) = ΠAH(1) ⊕ ΠAH(1). A(A) = L(HAk ) ⊗ 1Ak. Since we have specified the subalgebra A(A), we can define the reduced density matrix ρA from a given total density matrix ρ following the definition (2.9). The entanglement entropy of the state ρ associated with the subalgebra A(A), which we will represent by S(ρ; A), is given by. The first term is the classical part, and the second term −p1 trA1 ρ1 log ρ1 is the quantum part. Note that this quantum part vanishes if the original state ρ is pure.
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