Structure of Sufficient Quantum Coarse-Grainings

Mosonyi, Milán; Petz, Dénes

doi:10.1007/s11005-004-4072-2

Cited by 27 publications

(41 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One can extend this notion to that of a sufficient channel (or sufficient stochastic map), as discussed in [Pet86b,Pet88,MP04,Mos05]. A channel T ≡ T Y |X is sufficient for two input distributions p X and q X if there exists another channel (a recovery channel ) such that both these inputs can be recovered perfectly by sending the outputs of the channel T Y |X corresponding to them through it.…”

Section: I(x; Z|y ) ≡ H(xy ) + H(zy ) − H(y ) − H(xy Z)mentioning

confidence: 99%

Quantum Markov chains, sufficiency of quantum channels, and Rényi information measures

Datta¹,

Wilde²

2015

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

A short quantum Markov chain is a tripartite state ρ ABC such that system A can be recovered perfectly by acting on system C of the reduced state ρ BC . Such states have conditional mutual information I(A; B|C) equal to zero and are the only states with this property. A quantum channel N is sufficient for two states ρ and σ if there exists a recovery channel using which one can perfectly recover ρ from N (ρ) and σ from N (σ). The relative entropy difference D(ρ σ) − D(N (ρ) N (σ)) is equal to zero if and only if N is sufficient for ρ and σ. In this paper, we show that these properties extend to Rényi generalizations of these information measures which were proposed in [Berta et al., J. Math. Phys. 56, 022205, (2015) and Seshadreesan et al., J. Phys. A 48, 395303, 2015], thus providing an alternate characterization of short quantum Markov chains and sufficient quantum channels. These results give further support to these quantities as being legitimate Rényi generalizations of the conditional mutual information and the relative entropy difference. Along the way, we solve some open questions of Ruskai and Zhang, regarding the trace of particular matrices that arise in the study of monotonicity of relative entropy under quantum operations and strong subadditivity of the von Neumann entropy.

show abstract

Section: I(x; Z|y ) ≡ H(xy ) + H(zy ) − H(y ) − H(xy Z)mentioning

confidence: 99%

Quantum Markov chains, sufficiency of quantum channels, and Rényi information measures

Datta¹,

Wilde²

2015

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…The case of equality was studied in several papers recently but always restricted to finite dimensional Hilbert spaces [7,10]. Our aim now is to allow infinite dimensional spaces.…”

Section: Exponential Familiesmentioning

confidence: 99%

“…Equality in the strong subadditivity is equivalent to sufficiency of the subalgebra B(H A ⊗ H B ) ⊗ C1 C for (B(H), S) where S := {ω ABC , ω A ⊗ ω BC }, and the latter is equivalent to The structure (29) of the density matrix ω ABC is similar to the finite dimensional situation discussed in [7,10], however the direct sum decomposition may be infinite.…”

Section: Theoremmentioning

confidence: 99%

Sufficiency in Quantum Statistical Inference

Jenčová

Petz

2006

Commun. Math. Phys.

129

View full text Add to dashboard Cite

This paper attempts to develop a theory of sufficiency in the setting of non-commutative algebras parallel to the ideas in classical mathematical statistics. Sufficiency of a coarse-graining means that all information is extracted about the mutual relation of a given family of states. In the paper sufficient coarse-grainings are characterized in several equivalent ways and the non-commutative analogue of the factorization theorem is obtained. Among the applications the equality case for the strong subadditivity of the von Neumann entropy, the Imoto-Koashi theorem and exponential families are treated. The setting of the paper allows the underlying Hilbert space to be infinite dimensional.MSC: 46L53, 81R15, 62B05.

show abstract

“…4 It has been shown that the Markov property is tightly related to the sufficiency of conditional expectations through the strong subadditivity of von Neumann entropy: A state of a threecomposed tensor-product system is Markovian if and only if it takes the equality for the strong subadditivity inequality of entropy, which will be referred to as "the strong additivity of entropy." 16,6,12,7 We show that a similar equivalence relation of the Markov property and the strong additivity of entropy is valid for graded quantum systems. Its proof proceeds in much the same way as that for the tensor-product case following Ref.…”

Section: Introductionmentioning

confidence: 68%

Markov property and strong additivity of von Neumann entropy for graded quantum systems

Moriya

2006

Journal of Mathematical Physics

View full text Add to dashboard Cite

The quantum Markov property is equivalent to the strong additivity of von Neumann entropy for graded quantum systems. The additivity of von Neumann entropy for bipartite graded systems implies the statistical independence of states. However, the structure of Markov states for graded systems is different from that for tensorproduct systems which have trivial grading. For three-composed graded systems we have U͑1͒-gauge invariant Markov states whose restriction to the marginal pair of subsystems is nonseparable.

show abstract

Structure of Sufficient Quantum Coarse-Grainings

Cited by 27 publications

References 15 publications

Quantum Markov chains, sufficiency of quantum channels, and Rényi information measures

Quantum Markov chains, sufficiency of quantum channels, and Rényi information measures

Sufficiency in Quantum Statistical Inference

Markov property and strong additivity of von Neumann entropy for graded quantum systems

Contact Info

Product

Resources

About