• Corpus ID: 1453

# New Perspectives and some Celebrated Quantum Inequalities

@article{Effros2008NewPA,
title={New Perspectives and some Celebrated Quantum Inequalities},
author={Edward G. Effros},
journal={ArXiv},
year={2008},
volume={abs/0802.0006}
}
• E. Effros
• Published 31 January 2008
• Mathematics
• ArXiv
Some of the important inequalities associated with quantum entropy are immediate algebraic consequences of the Hansen-Pedersen-Jensen inequality. A general argument is given in terms of the matrix perspective of an operator convex function. A matrix analogue of Mar\'{e}chal's extended perspectives provides additional inequalities, including a $p+q\leq 1$ result of Lieb.
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## References

SHOWING 1-10 OF 16 REFERENCES

### A simple proof of the strong subadditivity inequality

• Mathematics
Quantum Inf. Comput.
• 2005
This short tutorial describes a simple proof of strong subadditivity due to Petz, and assumes only knowledge of elementary linear algebra and quantum mechanics.

### Jensen's Operator Inequality

• Mathematics
• 2003
Jensen's operator inequality and Jensen's trace inequality for real functions defined on an interval are established in what might be called their definitive versions. This is accomplished by the

### Multiplicativity of Completely Bounded p-Norms Implies a New Additivity Result

• Mathematics
• 2006
AbstractWe prove additivity of the minimal conditional entropy associated with a quantum channel Φ, represented by a completely positive (CP), trace-preserving map, when the infimum of S(γ12) − S(γ1)

### Proof of the strong subadditivity of quantum‐mechanical entropy

• Physics
• 1973
We prove several theorems about quantum‐mechanical entropy, in particular, that it is strongly subadditive.

### Convex Trace Functions and the Wigner-Yanase-Dyson Conjecture

• H.
• Mathematics
This paper is concerned with certain convex or concave mappings of linear operators on a Hilbert space into the reals. [f(A) is convex if f(ilA + (1 il)B) <; t..j(A) + (1 il)f(B) for 0 < il < 1 and

### On a Functional Operation Generating Convex Functions, Part 2: Algebraic Properties

Algebraic properties of the functional operation introduced in Part 1 of this paper (Ref. 1) are considered. In essence, the functional operation is shown to be associative, right distributive with

### Jensen's inequality for operators and Löwner's theorem

• Mathematics
• 1982
We shall say that a continuous, real function f on I satisfies Jensen's Operator Inequality if (JO) holds for any self-adjoint matrix x with spectrum in I and every matrix a with ][a[[< 1. Since we

### The non-commutative Legendre-Fenchel transform

We use the theory of matrix convex sets of Effros-Winkler to introduce a non-commutative version of convex functions, and we show how the Legendre-Fenchel transform generalizes to this situation