Abstract
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 adhoc 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.
Highlights
During recent times, in several disconnected fields of physics emerged the necessity to characterize the properties of extended subsystems rather than of the entire system
When Λ is Hermitian, λi are just the absolute values of the eigenvalues of Λ. [The normalization 21=n ensures 0 ≤ Dnðρ; σÞ ≤ 1.] In finite dimensional Hilbert spaces, all norms are equivalent, but this ceases to be the case for infinite dimensional spaces and we are interested in quantum field theories (QFTs)
There are several reasons why the trace distance D1 is special; e.g., the difference of expectation values of an operator O in different states is bounded as jtrðρ − σÞOj ≤ D1ðρ; σÞjjOjj1, and the bound has no factor depending on the Hilbert space dimension, as it would be the case for other norms; see, e.g., Ref. [24]
Summary
We develop a systematic method to calculate the trace distance between two reduced density matrices in 1 þ 1 dimensional quantum field theories. In this Letter, we develop a systematic method to calculate the trace distance between two RDMs in quantum field theories This is based on an ad hoc replica trick: we first calculate the n distance (1) for a general even integer n, we analytically continue it to real n, and we take the limit n → 1 to get the trace distance. This is based on an ad hoc replica trick: we first calculate the n distance (1) for a general even integer n, we analytically continue it to real n, and we take the limit n → 1 to get the trace distance. (This trick is reminiscent of the one for entanglement negativity in QFT [35].) For even n, we express Dn as a correlation
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