Quantum Entropy and Its Use

Ohya, Masanori; Petz, Dénes

doi:10.1007/978-3-642-57997-4

Cited by 877 publications

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“…As it was proved in [13] our relative entropy is larger than the LAU relative entropy, so that based on it quantum mutual information will give a larger entropy bound for quantum capacity.…”

Section: Introductionmentioning

confidence: 82%

“…if ̟ (E) = 0 for the maximal null-orthoprojector Eφ = 0 (otherwise the entropy is infinite by definition). As proved in [13], it gives a larger relative entropy than the expectation of ln ω − ln φ, and it has a positive value R (̟ : ϕ) ∈ [0, ∞] if the states are equally normalized, say (as usually) Trω = 1 = Trφ.…”

Section: Quantum Entropy Via Entanglementsmentioning

confidence: 99%

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On Entangled Information and Quantum Capacity

Belavkin

2001

Open Syst. Inf. Dyn.

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Abstract. The pure quantum entanglement is generalized to the case of mixed compound states on an operator algebra to include the classical and quantum encodings as particular cases. The true quantum entanglements are characterized by quantum couplings which are described as transpose-CP, but not Completely Positive (CP), trace-normalized linear positive maps of the algebra.The entangled (total) information is defined in this paper as a relative entropy of the conditional (the derivative of the compound state with respect to the input) and the unconditional output states. Thus defined the total information of the entangled states leads to two different types of the entropy for a given quantum state: the von Neumann entropy, or c-entropy, which is achieved as the supremum of the information over all c-entanglements and thus is semi-classical, and the true quantum entropy, or q-entropy, which is achieved at the standard entanglement. The q-capacity, defined as the supremum over all entanglements, coincides with the topological entropy. In the case of the simple algebra it doubles the c-capacity, coinciding with the rank-entropy. The conditional q-entropy based on the q-entropy, is positive, unlike the von Neumann conditional entropy, and the q-information of a quantum channel is proved to be additive.

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Section: Introductionmentioning

confidence: 82%

Section: Quantum Entropy Via Entanglementsmentioning

confidence: 99%

On Entangled Information and Quantum Capacity

Belavkin

2001

Open Syst. Inf. Dyn.

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“…Since Klein's inequality [14,16,20] says that H(P, Q) ≥ Tr P − Tr Q, it follows that H(P, Q) ≥ 0 when Tr P = Tr Q. It is easy to verify that H(xP, xQ) = xH(P, Q) for x > 0.…”

Section: A Appendixmentioning

confidence: 99%

“…The joint convexity of relative entropy implies the well-known fact [13,16,18,23] that it decreases under completely positive, trace-preserving (CPT) maps. These maps represent quantum channels.…”

Section: Monotonicity Of Relative Entropymentioning

confidence: 99%

Another short and elementary proof of strong subadditivity of quantum entropy

Ruskai

2007

Reports on Mathematical Physics

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A short and elementary proof of the joint convexity of relative entropy is presented, using nothing beyond linear algebra. The key ingredients are an easily verified integral representation and the strategy used to prove the Cauchy-Schwarz inequality in elementary courses. Several consequences are proved in a way which allow an elementary proof of strong subadditivity in a few more lines. Some expository material on Schwarz inequalities for operators and the Holevo bound for partial measurements is also included.

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“…The mathematical structure of the fluctuation algebra is discussed in many places in the literature, see e.g. [GVV1]- [GVV6] and [MSTV,BR2,Pe,OP,De2] for general results about CCR algebras. For notational and reference purposes we recall a few basic facts.…”

Section: Theorem 14mentioning

confidence: 99%

Central Limit Theorem for Locally Interacting Fermi Gas

Jakšić

Pautrat

Pillet

2008

Commun. Math. Phys.

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We consider a locally interacting Fermi gas in its natural non-equilibrium steady state and prove the Quantum Central Limit Theorem (QCLT) for a large class of observables. A special case of our results concerns finitely many free Fermi gas reservoirs coupled by local interactions. The QCLT for flux observables, together with the Green-Kubo formulas and the Onsager reciprocity relations previously established [JOP4], complete the proof of the Fluctuation-Dissipation Theorem and the development of linear response theory for this class of models.

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Quantum Entropy and Its Use

Cited by 877 publications

References 0 publications

On Entangled Information and Quantum Capacity

On Entangled Information and Quantum Capacity

Another short and elementary proof of strong subadditivity of quantum entropy

Central Limit Theorem for Locally Interacting Fermi Gas

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