# Monotonicity of the unified quantum (r,s)-entropy and (r,s)-mutual information

Monotonicity of the unified quantum (r,s)-entropy and (r,s)-mutual information Monotonicity of the unified quantum (r, s)-entropy $$E_{r}^{s}(\rho )$$ E r s ( ρ ) and the unified quantum (r, s)-mutual information $$I_{r}^{s}(\rho )$$ I r s ( ρ ) is discussed in this paper. Some basic properties of them are explored, and the following conclusions are established. (1) For any $$0<r<1, E_{r}^{s}(\rho )$$ 0 < r < 1 , E r s ( ρ ) is increasing with respect to $$s\in (-\infty ,+\infty )$$ s ∈ ( - ∞ , + ∞ ) , and for any $$r\ge 1, E_{r}^{s}(\rho )$$ r ≥ 1 , E r s ( ρ ) is decreasing with respect to $$s\in (-\infty ,+\infty )$$ s ∈ ( - ∞ , + ∞ ) ; (2) for any $$s>0$$ s > 0 , $$E_{r}^{s}(\rho )$$ E r s ( ρ ) is decreasing with respect to $$r\in (0,+\infty )$$ r ∈ ( 0 , + ∞ ) ; (3) for any $$r>0, E_{r}^{s}(\rho )$$ r > 0 , E r s ( ρ ) is convex with respect to $$s\in (-\infty ,+\infty )$$ s ∈ ( - ∞ , + ∞ ) ; (4) for a product state $$\rho _{AB}$$ ρ A B , there are two real numbers a and b such that $$I_{r}^{s}(\rho _{AB})$$ I r s ( ρ A B ) is increasing with respect to $$s\in [0,a]$$ s ∈ [ 0 , a ] when $$r\ge 1$$ r ≥ 1 and it is decreasing with respect to $$s\in [b,0]$$ s ∈ [ b , 0 ] when $$0<r<1$$ 0 < r < 1 ; (5) for a product state $$\rho _{AB}$$ ρ A B , $$I_{r}^{s}(\rho _{AB})$$ I r s ( ρ A B ) is decreasing with respect to $$r\in [r_s,+\infty )$$ r ∈ [ r s , + ∞ ) for each $$s>0$$ s > 0 , where $$r_s={\mathrm {max}}\{a_s,b_s\}$$ r s = max { a s , b s } , $$m>2$$ m > 2 with $$m-2\ln m=1$$ m - 2 ln m = 1 and $${\mathrm {tr}}\rho _{A}^{a_s}={\mathrm {tr}}\rho _{B}^{b_s}=m^{-\frac{1}{s}}$$ tr ρ A a s = tr ρ B b s = m - 1 s . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

# Monotonicity of the unified quantum (r,s)-entropy and (r,s)-mutual information

, Volume 14 (12) – Sep 25, 2015
19 pages

/lp/springer_journal/monotonicity-of-the-unified-quantum-r-s-entropy-and-r-s-mutual-lpvRdpQpxS
Publisher
Springer US
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-015-1126-6
Publisher site
See Article on Publisher Site

### Abstract

Monotonicity of the unified quantum (r, s)-entropy $$E_{r}^{s}(\rho )$$ E r s ( ρ ) and the unified quantum (r, s)-mutual information $$I_{r}^{s}(\rho )$$ I r s ( ρ ) is discussed in this paper. Some basic properties of them are explored, and the following conclusions are established. (1) For any $$0<r<1, E_{r}^{s}(\rho )$$ 0 < r < 1 , E r s ( ρ ) is increasing with respect to $$s\in (-\infty ,+\infty )$$ s ∈ ( - ∞ , + ∞ ) , and for any $$r\ge 1, E_{r}^{s}(\rho )$$ r ≥ 1 , E r s ( ρ ) is decreasing with respect to $$s\in (-\infty ,+\infty )$$ s ∈ ( - ∞ , + ∞ ) ; (2) for any $$s>0$$ s > 0 , $$E_{r}^{s}(\rho )$$ E r s ( ρ ) is decreasing with respect to $$r\in (0,+\infty )$$ r ∈ ( 0 , + ∞ ) ; (3) for any $$r>0, E_{r}^{s}(\rho )$$ r > 0 , E r s ( ρ ) is convex with respect to $$s\in (-\infty ,+\infty )$$ s ∈ ( - ∞ , + ∞ ) ; (4) for a product state $$\rho _{AB}$$ ρ A B , there are two real numbers a and b such that $$I_{r}^{s}(\rho _{AB})$$ I r s ( ρ A B ) is increasing with respect to $$s\in [0,a]$$ s ∈ [ 0 , a ] when $$r\ge 1$$ r ≥ 1 and it is decreasing with respect to $$s\in [b,0]$$ s ∈ [ b , 0 ] when $$0<r<1$$ 0 < r < 1 ; (5) for a product state $$\rho _{AB}$$ ρ A B , $$I_{r}^{s}(\rho _{AB})$$ I r s ( ρ A B ) is decreasing with respect to $$r\in [r_s,+\infty )$$ r ∈ [ r s , + ∞ ) for each $$s>0$$ s > 0 , where $$r_s={\mathrm {max}}\{a_s,b_s\}$$ r s = max { a s , b s } , $$m>2$$ m > 2 with $$m-2\ln m=1$$ m - 2 ln m = 1 and $${\mathrm {tr}}\rho _{A}^{a_s}={\mathrm {tr}}\rho _{B}^{b_s}=m^{-\frac{1}{s}}$$ tr ρ A a s = tr ρ B b s = m - 1 s .

### Journal

Quantum Information ProcessingSpringer Journals

Published: Sep 25, 2015

### References

• Information content for quantum states
Brody, DC; Hughston, LP

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