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The von Neumann entropy and unitary equivalence of quantum states. (English) Zbl 1321.15013
The paper starts with a short proof of a result due to K. He et al. [Appl. Math. Lett. 25, No. 8, 1153–1156 (2012; Zbl 1253.81022)]. Namely, a density operator \(\rho\) on an \(n\)–dimensional complex Hilbert space is determined up to unitary equivalence by the function \[ [0,1]\to\mathbb R,\qquad \lambda\mapsto S(\lambda\rho+(1-\lambda)I/n), \] where \(I\) is the identity operator and \(S\) stands for von Neumann entropy. Equivalently, a probability distribution \(x=(x_1, \dots, x_n)\) is determined up to permutation by the function \[ [0,1]\to\mathbb R,\qquad \lambda\mapsto \sum_{i=1}^nh\left(\lambda x_i+\frac{1-\lambda}n\right), \] where \(h(p)=p\log (1/p)\). Moreover, the restriction of the above function to any initial subinterval \([0,a)\) of \([0,1]\) already determines \(\rho\), resp. \(x\) up to unitary equivalence, resp. permutation.
The author then proves a stronger version of this result. Namely, the restriction of the above function to any given \(2n\)-element subset of \((0,1]\) already determines \(\rho\), resp. \(x\) up to unitary equivalence, resp. permutation. Note that the value of the function at \(\lambda=0\) is always \(\log n\).

MSC:
15A15 Determinants, permanents, traces, other special matrix functions
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[1] DOI: 10.1016/S0024-3795(03)00386-0 · Zbl 1027.15007 · doi:10.1016/S0024-3795(03)00386-0
[2] DOI: 10.1016/j.aml.2012.02.027 · Zbl 1253.81022 · doi:10.1016/j.aml.2012.02.027
[3] DOI: 10.1080/00207217.2012.669716 · doi:10.1080/00207217.2012.669716
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