We develop a systematic method to compute the subsystem fidelity and Rényi fidelity in two-dimensional conformal field theories (CFTs) from the operator product expansion of twist operators, and apply it to the subsystem quantum Bhattacharyya distance (BD) and Rényi-Bhattacharyya distance (RBD). We validate the method against exact results in the free massless fermion and boson theories, and then use it to compute the BD and RBD in holographic CFTs between two heavy primary states, between two thermal states, and between a heavy primary state and a thermal state. When the two states differ in their classical, order-
c data, the distances are of order
c, and the states are distinguishable already at the level of classical gravity; when the leading data coincide, the distances are of order
1/c at every order of the short interval expansion, so that the distinguishability becomes a perturbative quantum gravity effect. For a heavy primary state and its thermal counterpart, the leading contribution to the BD appears at order
ℓ8 and is of order
c0, whereas the RBD is generically of order
c, in line with the earlier finding that a heavy primary state and a thermal state share the same order-
c entanglement entropy but have different order-
c Rényi entropies in the short interval expansion.
Abstract We report on a systematic replica approach to calculate the subsystem trace distance for a quantum field theory. This method has been recently introduced in [J. Zhang, P. Ruggiero and P. Calabrese, Phys. Rev. Lett.122(2019) 141602 ], of which this work is a completion. The trace distance between two reduced density matricesρAandσAis obtained from the moments tr(ρA− σA)nand taking the limitn →1 of the traces of the even powers. We focus here on the case of a subsystem consisting of a single interval of lengthℓembedded in the low lying eigenstates of a one-dimensional critical system of lengthL, a situation that can be studied exploiting the path integral form of the reduced density matrices of two-dimensional conformal field theories. The trace distance turns out to be ascale invariant universal functionofℓ/L. Here we complete our previous work by providing detailed derivations of all results and further new formulas for the distances between several low-lying states in two-dimensional free massless compact boson and fermion theories. Remarkably, for one special case in the bosonic theory and for another in the fermionic one, we obtain the exact trace distance, as well as the Schattenn-distance, for an interval of arbitrary length, while in generic case we have a general form for the first term in the expansion in powers ofℓ/L. The analytical predictions in conformal field theories are tested against exact numerical calculations in XX and Ising spin chains, finding perfect agreement. As a byproduct, new results in two-dimensional CFT are also obtained for other entanglement-related quantities, such as the relative entropy and the fidelity.
We develop a systematic method to calculate the trace distance between two reduced density matrices in 1+1 dimensional quantum field theories. The approach exploits the path integral representation of the reduced density matrices and an ad hoc replica trick. We then extensively apply this method to the calculation of the distance between reduced density matrices of one interval of length ℓ in eigenstates of conformal field theories. When the interval is short, using the operator product expansion of twist operators, we obtain a universal form for the leading order in ℓ of the trace distance. We compute the trace distances among the reduced density matrices of several low lying states in two-dimensional free massless boson and fermion theories. We compare our analytic conformal results with numerical calculations in XX and Ising spin chains finding perfect agreement.
We report on a systematic replica approach to calculate the subsystem trace distance for a generic quantum field theory. This method has been recently introduced in [J. Zhang, P. Ruggiero, P. Calabrese, Phys. Rev. Lett. 122, 141602 (2019)], of which this work is a completion. The trace distance between two reduced density matrices
ρA and
σA is obtained from the moments
tr(ρA−σA)n and taking the limit
n→1 of the traces of the even powers. We focus here on the case of a subsystem consisting of a single interval of length
ℓ embedded in the low lying eigenstates of a one-dimensional critical system of length
L, a situation that can be studied exploiting the path integral form of the reduced density matrices of two-dimensional conformal field theories. The trace distance turns out to be a {\it scale invariant universal function} of
ℓ/L. Here we complete our previous work by providing detailed derivations of all results and further new formulas for the distances between several low-lying states in two-dimensional free massless compact boson and fermion theories. Remarkably, for one special case in the bosonic theory and for another in the fermionic one, we obtain the exact trace distance, as well as the Schatten
n-distance, for an interval of arbitrary length, while in generic case we have a general form for the first term in the expansion in powers of
ℓ/L. The analytical predictions in conformal field theories are tested against exact numerical calculations in XX and Ising spin chains, finding perfect agreement. As a byproduct, new results are also obtained for other entanglement-related quantities, such as the relative entropy and the fidelity.
In this thesis we apply techniques from quantum information theory to study quantum gravity within the framework of the anti-de Sitter / conformal field theory correspondence (AdS/CFT). Through AdS/CFT, progress has been made in understanding the structure of entanglement in quantum field theories, and in how gravitational physics can emerge from these structures. However, this understanding is far from complete and will require the development of new tools to quantify correlations in CFT. This thesis presents refinements of a duality between operator product expansion (OPE) blocks in the CFT, giving the contribution of a conformal family to the OPE, and geodesic integrated fields in AdS which are diffeomorphism invariant quantities. This duality was originally discovered in the maximally symmetric setting of pure AdS dual to the CFT ground state. In less symmetric states the duality must be modified. Working with excited states within AdS
3/CFT
2, this thesis shows how the OPE block decomposes into more fine-grained CFT observables that are dual to AdS fields integrated over non-minimal geodesics. Additionally, this thesis contains results on the dynamics of entanglement measures for general quantum systems. Results are presented for the family of quantum Rényi entropies and entanglement negativity. Rényi entropies are studied for general dynamics by imposing special initial conditions. Around pure, separable initial states, all Rényi entropies grow with the same timescale at leading, and next-to-leading order. Mathematical tools are developed for the differentiation of non-analytic matrix functions with respect to constrained arguments and are used to construct analytic expressions for derivatives of negativity. We establish bounds on the rate of change of state distinguishability and the rate of entanglement growth for closed systems. Note: Abstract shortened.
The existence of a positive linear functional acting on the space of (differences between) conformal blocks has been shown to rule out regions in the parameter space of conformal field theories (CFTs). We argue that at the boundary of the allowed region the extremal functional contains, in principle, enough information to determine the dimensions and operator product expansion (OPE) coefficients of an infinite number of operators appearing in the correlator under analysis. Based on this idea we develop the extremal functional method (EFM), a numerical procedure for deriving the spectrum and OPE coefficients of CFTs lying on the boundary (of solution space). We test the EFM by using it to rederive the low lying spectrum and OPE coefficients of the two-dimensional Ising model based solely on the dimension of a single scalar quasiprimary--no Virasoro algebra required. Our work serves as a benchmark for applications to more interesting, less known CFTs in the near future.