Zhang Jiaju, Ruggiero Paola, Calabrese Pasquale
SISSA and INFN, Via Bonomea 265, 34136 Trieste, Italy.
International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy.
Phys Rev Lett. 2019 Apr 12;122(14):141602. doi: 10.1103/PhysRevLett.122.141602.
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.
我们开发了一种系统方法，用于计算1+1维量子场论中两个约化密度矩阵之间的迹距离。该方法利用了约化密度矩阵的路径积分表示和一个特殊的复制技巧。然后，我们广泛地将此方法应用于共形场论本征态中长度为ℓ的一个区间的约化密度矩阵之间距离的计算。当区间较短时，利用扭场算符的算符乘积展开，我们得到了迹距离ℓ的主导阶的通用形式。我们计算了二维无质量自由玻色子和费米子理论中几个低能态的约化密度矩阵之间的迹距离。我们将解析共形结果与XX和伊辛自旋链中的数值计算进行比较，发现结果完全一致。
