# Sandwiched Rényi divergence satisfies data processing inequality

@article{Beigi2013SandwichedRD, title={Sandwiched R{\'e}nyi divergence satisfies data processing inequality}, author={Salman Beigi}, journal={Journal of Mathematical Physics}, year={2013}, volume={54}, pages={122202-122202} }

Sandwiched (quantum) α-Renyi divergence has been recently defined in the independent works of Wilde et al. [“Strong converse for the classical capacity of entanglement-breaking channels,” preprint arXiv:1306.1586 (2013)] and Muller-Lennert et al. [“On quantum Renyi entropies: a new definition, some properties and several conjectures,” preprint arXiv:1306.3142v1 (2013)]. This new quantum divergence has already found applications in quantum information theory. Here we further investigate…

## 196 Citations

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Optimized Quantum F-Divergences

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## References

SHOWING 1-10 OF 37 REFERENCES

On quantum Rényi entropies: A new generalization and some properties

- Computer Science
- 2013

This work proposes a new quantum generalization of the family of Renyi entropies that contains the von Neumann entropy, min-entropy, collision entropy, and the max-entropies as special cases, thus encompassing most quantum entropie in use today.

On quantum R\'enyi entropies: a new definition, some properties and several conjectures

- Computer Science
- 2013

It is argued that previous quantum extensions are incompatible and thus unsatisfactory, and proposed a new quantum generalization that contains the min-entropy, collision entropy, von Neumann entropy as well as the max-entropies as special cases, thus encompassing most quantum entropies in use today.

Monotonicity of a relative Rényi entropy

- Computer Science, MathematicsArXiv
- 2013

We show that a recent definition of relative Renyi entropy is monotone
under completely positive, trace preserving maps. This proves a recent
conjecture of Muller-Lennert et al. [“On quantum Renyi…

Notes on super-operator norms induced by schatten norms

- MathematicsQuantum Inf. Comput.
- 2005

It is proved that if Φ is completely positive, the value of the supremum of the definition of ||Φ||q→p is achieved by a positive semidefinite operator X, answering a question recently posed by King and Ruskai.

On the Quantum Rényi Relative Entropies and Related Capacity Formulas

- Computer ScienceIEEE Transactions on Information Theory
- 2011

It is shown that various generalizations of the Holevo capacity, defined in terms of the α-relative entropies, coincide for the parameter range α ∈ (0,2], and an upper bound on the one-shot ε-capacity of a classical-quantum channel interms of these capacities is given.

Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Relative Entropy

- Computer Science
- 2013

It is shown that a strong converse theorem holds for the classical capacity of all entanglement-breaking channels and all Hadamard channels (the complementary channels of the former) that the probability of correctly decoding a classical message converges exponentially fast to zero in the limit of many channel uses if the rate of communication exceeds the classical Capacity.

Arimoto channel coding converse and Rényi divergence

- Computer Science2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton)
- 2010

A simple derivation of the Arimoto converse based on the data-processing inequality for Rényi divergence gives the simplest proof of the strong converse for the DMC with feedback and demonstrates that the sphere-packing bound is strictly tighter than Arimoto Converse for all channels, blocklengths and rates.

Asymptotic Error Rates in Quantum Hypothesis Testing

- Mathematics
- 2007

We consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies, and find a closed form expression for the asymptotic…

On general minimax theorems

- Mathematics
- 1958

There have been several generalizations of this theorem. J. Ville [9], A. Wald [11], and others [1] variously extended von Neumann's result to cases where M and N were allowed to be subsets of…