Abstract
We study odd entanglement entropy (odd entropy in short), a candidate of measure for mixed states holographically dual to the entanglement wedge cross section, in two-dimensional free scalar field theories. Our study is restricted to Gaussian states of scale-invariant theories as well as their finite temperature generalizations, for which we show that odd entropy is a well-defined measure for mixed states. Motivated from holographic results, the difference between odd and von Neumann entropy is also studied. In particular, we show that large amounts of quantum correlations ensure the odd entropy to be larger than von Neumann entropy, which is qualitatively consistent with the holographic CFT. In general cases, we also find that this difference is not even a monotonic function with respect to size of (and distance between) subsystems.
Highlights
We study odd entanglement entropy, a candidate of measure for mixed states holographically dual to the entanglement wedge cross section, in two-dimensional free scalar field theories
We show that large amounts of quantum correlations ensure the odd entropy to be larger than von Neumann entropy, which is qualitatively consistent with the holographic CFT
We study the odd entropy for reduced density matrices of the vacuum in two dimensional free scalar fields
Summary
We review the odd (entanglement) entropy briefly. Let ρAB (AB ≡ A ∪ B) be a density matrix acting on bi-partite Hilbert space HA ⊗ HB. We are interested in the difference between odd entropy and von Neumann entropy, So(A : B) − S(AB) We expect this difference gives the area of the minimal cross section of the entanglement wedge, so-called entanglement wedge cross section [6, 7], So(A : B) − S(AB) = EW (A : B). Since it is related to an area of a certain surface, we should have So(A : B) ≥ S(AB) at the leading order of the large-N limit. One may expect this inequality always holds, we can find counterexamples of the inequality from mixed states in two qubit systems, for example (see appendix A.1).
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