We introduce a new method permitting the analytical determination of entanglement entropy (and related quantities) between configurations of a quantum field, which is either free or in interaction with a classical source, at two distinct spatial locations. We show how such a setup can be described by a bipartite, continuous Gaussian system. This allows us to derive explicit and exact formulas for the entanglement entropy, the mutual information and the quantum discord, solely in terms of the Fourier-space power spectra of the field. This contrasts with previous studies, which mostly rely on numerical considerations. As an illustration, we apply our formalism to massless fields in flat space, where exact expressions are derived that only involve the ratio between the size of the regions over which the field is coarse-grained, and the distance between these regions. In particular, we recover the well-known fact that mutual information decays as the fourth power of this ratio at large distances, as previously observed in numerical works. Our method leads to the first analytical derivation of this result, and to an exact formula that also applies to arbitrary distances. Finally, we determine the quantum discord and find that it identically vanishes (unless coarse-graining is performed over smeared spheres, in which case it obeys the same suppression at large distance as mutual information).
We introduce a new method permitting the analytical determination of entanglement entropy (and related quantities) between configurations of a quantum field, which is either free or in interaction with a classical source, at two distinct spatial locations. We show how such a setup can be described by a bipartite, continuous Gaussian system. This allows us to derive explicit and exact formulas for the entanglement entropy, the mutual information and the quantum discord, solely in terms of the Fourier-space power spectra of the field. This contrasts with previous studies, which mostly rely on numerical considerations. As an illustration, we apply our formalism to massless fields in flat space, where exact expressions are derived that only involve the ratio between the size of the regions over which the field is coarse-grained, and the distance between these regions. In particular, we recover the well-known fact that mutual information decays as the fourth power of this ratio at large distances, as previously observed in numerical works. Our method leads to the first analytical derivation of this result, and to an exact formula that also applies to arbitrary distances. Finally, we determine the quantum discord and find that it identically vanishes (unless coarse-graining is performed over smeared spheres, in which case it obeys the same suppression at large distance as mutual information).
We compute the entanglement entropy, mutual information and quantum discord of the Cosmic Microwave Background (CMB) fluctuations in real space. To that end, we first show that measurements of these fluctuations at two distinct spatial locations can be described by a bipartite, continuous Gaussian system. This leads to explicit formulas for the mutual information and the quantum discord in terms of the Fourier-space power spectra of the curvature perturbation. We then find that quantum entanglement, that builds up in Fourier space between opposite wave momenta as an effect of quantum squeezing, is transferred to real space. In particular, both the mutual information and quantum discord, which decay as the fourth power of the distance between the two measurements in flat space time, asymptotes a constant in cosmological backgrounds. At the scales probed in the CMB however, they are highly suppressed, while they can reach order-one values at much smaller scales, where primordial black holes could have formed.
Quantum discord, a kind of quantum correlation, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. It has been discussed so far for small systems with only a few independent parameters. We extend here to a much broader class of states when the second party is of arbitrary dimension d, so long as the first, measured, party is a qubit. We present two formulae to calculate quantum discord, the first relating to the original entropic definition and the second to a recently proposed geometric distance measure which leads to an analytical formulation. The tracing over the qubit in the entropic calculation is reduced to a very simple prescription. And, when the d-dimensional system is a so-called X state, the density matrix having non-zero elements only along the diagonal and anti-diagonal so as to appear visually like the letter X, the entropic calculation can be carried out analytically. Such states of the full bipartite qubit-qudit system may be named "extended X states", whose density matrix is built of four block matrices, each visually appearing as an X. The optimization involved in the entropic calculation is generally over two parameters, reducing to one for many cases, and avoided altogether for an overwhelmingly large set of density matrices as our numerical investigations demonstrate. Our results also apply to states of a N-qubit system, where "extended X states" consist of (2^(N+2) - 1) states, larger in number than the (2^(N+1) - 1) of X states of N qubits. While these are still smaller than the total number (2^(2N) - 1) of states of N qubits, the number of parameters involved is nevertheless large. In the case of N = 2, they encompass the entire 15-dimensional parameter space, that is, the extended X states for N = 2 represent the full qubit-qubit system.
First, we study the entanglement entropy S AB of a d + 1 dimensional massless free scalar field on two disjoint compact spatial regions A and B whose sizes are R 1 and R 2 , respectively, and the distance between them is r.The state of the massless free scalar field is the vacuum state.We consider the mutual information S A;B ≡ S A + S B -S AB where S A and S B are the entanglement entropy on the inside region of A and B, respectively.We develop the computational method based on that of Bombelli et al and obtain the result thatWhen A and B are spheres in d = 2 and 3, we obtain explicit resultsBy using the method used in the Minkowski spacetime case with some modifications, we study the entanglement entropy, S C , of the massless free scalar field on the outside region C of two black holes A and B whose radii are R 1 and R 2 and how it depends on the distance, r( R 1 , R 2 ) , between two black holes.If we can consider the entanglement entropy as thermodynamic entropy, we can see the entropic force (we call this entanglement entropic force) acting on the two black holes from the r dependence of S C .We consider the property of the entropic force.
Quantum discord, a kind of quantum correlation, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. In general, this correlation is different from entanglement, and quantum discord may be nonzero even for certain separable states. Even in the simple case of bipartite quantum systems, this different kind of quantum correlation has interesting and significant applications in quantum information processing. So far, quantum discord has been calculated explicitly only for a rather limited set of two-qubit quantum states and expressions for more general quantum states are not known. In this article, we derive explicit expressions for quantum discord for a larger class of two-qubit states, namely, a seven-parameter family of so called X states that have been of interest in a variety of contexts in the field. We also study the relation between quantum discord, classical correlation, and entanglement for a number of two-qubit states to demonstrate that they are independent measures of correlation with no simple relative ordering between them.
Quantum correlations characterized by quantum entanglement and quantum discord play important roles in many quantum information processing. We study the relations among the entanglement of formation, concurrence, tangle, linear entropy-based classical correlation and von Neumann entropy-based classical correlation . We present analytical formulae of linear entropy-based classical correlation for arbitrary d⊗2 quantum states and von Neumann entropy-based classical correlation for arbitrary 2⊗2 rank-2 quantum states. From the von Neumann entropy-based classical correlation, we derive an explicit formula of quantum discord for arbitrary rank-2 two-qubit quantum states.
A bstract We continue our study of reflected entropy, R ( A, B ), for Gaussian systems. In this paper we provide general formulas valid for free scalar fields in arbitrary dimensions. Similarly to the fermionic case, the resulting expressions are fully determined in terms of correlators of the fields, making them amenable to lattice calculations. We apply this to the case of a (1 + 1)-dimensional chiral scalar, whose reflected entropy we compute for two intervals as a function of the cross-ratio, comparing it with previous holographic and free-fermion results. For both types of free theories we find that reflected entropy satisfies the conjectural monotonicity property R ( A, BC ) ≥ R ( A, B ). Then, we move to (2 + 1) dimensions and evaluate it for square regions for free scalars, fermions and holography, determining the very-far and very-close regimes and comparing them with their mutual information counterparts. In all cases considered, both for (1 + 1)- and (2 + 1)-dimensional theories, we verify that the general inequality relating both quantities, R ( A, B ) ≥ I ( A, B ), is satisfied. Our results suggest that for general regions characterized by length-scales L A ∼ L B ∼ L and separated a distance ℓ , the reflected entropy in the large-separation regime ( x ≡ L / ℓ ≪ 1) behaves as R ( x ) ∼ −I ( x ) log x for general CFTs in arbitrary dimensions.