Abstract
Finding pure states in an enlarged Hilbert space that encode the mixed state of a quantum field theory as a partial trace is necessarily a challenging task. Nevertheless, such purifications play the key role in characterizing quantum information-theoretic properties of mixed states via entanglement and complexity of purifications. In this article, we analyze these quantities for two intervals in the vacuum of free bosonic and Ising conformal field theories using, for the first time, the~most general Gaussian purifications. We provide a comprehensive comparison with existing results and identify universal properties. We further discuss important subtleties in our setup: the massless limit of the free bosonic theory and the corresponding behaviour of the mutual information, as well as the Hilbert space structure under the Jordan-Wigner mapping in the spin chain model of the Ising conformal field theory.
Highlights
Understanding quantum information theoretic properties of quantum field theories (QFTs) and, via holography, of quantum gravity has been an enormously fruitful research front in the past two decades.The main player in this endeavour has been the notion of entanglement and its entropy S
We find its estimated value to gradually decrease at large d/w as N is increased, though mutual information (MI) can be well approximated by a power law with b2 ≈ 0.2 in the range 1 < d/w < 100, consistent with earlier numerical studies in the d/w < 50 range [35]
Using the fact that this geometry is compatible with the group action, i.e., the Fubini-Study metric on the manifold of purifications is left invariant under the group action of GA, we do not need to recompute the metric at every step, but we can fix an orthonormal basis of Lie algebra generators equal to the dimension of the manifold
Summary
Understanding quantum information theoretic properties of quantum field theories (QFTs) and, via holography, of quantum gravity has been an enormously fruitful research front in the past two decades (as seen, for example, in Refs. [1–5]). We want to stress instead that the key motivating feature behind our work stems from both of these quantities involving in their definition scanning over purifications of mixed many-body states.2 Such purifications, i.e., embedding a mixed state in an enlarged Hilbert space in which it arises as a reduced density matrix, in the context we are interested in, i.e., QFT physics, are clearly challenging to operate with. Utilizing efficient Gaussian techniques allows us to minimize the two quantities of interest, EoP and CoP, for a judicious choice of a definition of pure state complexity [29] over general purifications to a given number of bosonic or fermionic modes. IV, we discuss briefly the mathematics of purifications of Gaussian states as seen by covariance matrices, which is the working horse behind most of the results reported in the present article We use this machinery to study EoP and CoP in the two-interval case of Fig. 1, respectively, in Secs. For the Ising model, we will discuss in detail how there are two distinct notions of locality associated to the spin and fermion formulation, respectively
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