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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
##### Keywords:
von Neumann entropy; quantum states; unitary equivalence
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##### References:
  DOI: 10.1016/S0024-3795(03)00386-0 · Zbl 1027.15007 · doi:10.1016/S0024-3795(03)00386-0  DOI: 10.1016/j.aml.2012.02.027 · Zbl 1253.81022 · doi:10.1016/j.aml.2012.02.027  DOI: 10.1080/00207217.2012.669716 · doi:10.1080/00207217.2012.669716
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