Rényi generalizations of the conditional quantum mutual information

Berta, Mario; Seshadreesan, Kaushik P.; Wilde, Mark M.

doi:10.1063/1.4908102

Cited by 68 publications

(134 citation statements)

References 91 publications

(174 reference statements)

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“…As already stated in Section 1, the aforementioned monotonicity was proven in [BSW15a] for local quantum operations on one of the two systems A or B, and it was left as an open question to determine if the property holds for local quantum operations on the other system (with numerical evidence supporting a positive answer). Here, we assume the latter to hold.…”

Section: Overview Of Resultsmentioning

confidence: 95%

“…Complementary to this aim, we take the results here as further justification that [BSW15a] has pinned down a useful Rényi generalization of the conditional mutual information. Furthermore, our work suggests that the axiomatic approach to entanglement theory and discord does not uniquely single out the von Neumann entropy-based squashed entanglement or discord (however, further effort is needed to justify this claim in full).…”

Section: Overview Of Resultsmentioning

confidence: 99%

“…In particular, Rényi generalizations of the conditional quantum mutual information of (1.2) were proposed [BSW15a, Sections 5 and 6]. These generalizations retain most of the desired properties of the von Neumann entropy-based conditional quantum mutual information; e.g., they are non-negative, monotone with respect to local quantum operations on one of the two systems A or B, and obey a duality property for four-party pure states |ψ ABCD [BSW15a,Section 7]. There it was left as an open question to determine whether monotonicity also holds for local operations on the other system, with numerical evidence supporting a positive answer.…”

Section: Introductionmentioning

confidence: 99%

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Rényi squashed entanglement, discord, and relative entropy differences

Seshadreesan¹,

Berta²,

Wilde³

2015

J. Phys. A: Math. Theor.

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The squashed entanglement quantifies the amount of entanglement in a bipartite quantum state, and it satisfies all of the axioms desired for an entanglement measure. The quantum discord is a measure of quantum correlations that are different from those due to entanglement. What these two measures have in common is that they are both based upon the conditional quantum mutual information. In [Berta et al., J. Math. Phys. 56, 022205 (2015)], we recently proposed Rényi generalizations of the conditional quantum mutual information of a tripartite state on ABC (with C being the conditioning system), which were shown to satisfy some properties that hold for the original quantity, such as non-negativity, duality, and monotonicity with respect to local operations on the system B (with it being left open to show that the Rényi quantity is monotone with respect to local operations on system A). Here we define a Rényi squashed entanglement and a Rényi quantum discord based on a Rényi conditional quantum mutual information and investigate these quantities in detail. Taking as a conjecture that the Rényi conditional quantum mutual information is monotone with respect to local operations on both systems A and B, we prove that the Rényi squashed entanglement and the Rényi quantum discord satisfy many of the properties of the respective original von Neumann entropy based quantities. In our prior work [Berta et al., Phys. Rev. A 91, 022333 (2015)], we also detailed a procedure to obtain Rényi generalizations of any quantum information measure that is equal to a linear combination of von Neumann entropies with coefficients chosen from the set {−1, 0, 1}. Here, we extend this procedure to include differences of relative entropies. Using the extended procedure and a conjectured monotonicity of the Rényi generalizations in the Rényi parameter, we discuss potential remainder terms for well known inequalities such as monotonicity of the relative entropy, joint convexity of the relative entropy, and the Holevo bound.

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Section: Overview Of Resultsmentioning

confidence: 95%

Section: Overview Of Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Rényi squashed entanglement, discord, and relative entropy differences

Seshadreesan¹,

Berta²,

Wilde³

2015

J. Phys. A: Math. Theor.

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“…Note that the inequality (2.15) for the Rényi entropies originally appeared in [55]. The last assertion concerns the Rényi conditional mutual information I α (A; B|C) ρ , defined in [10] for a tripartite state ρ ABC and α > 0 as…”

Section: Further Properties Of Rényi Entropic Quantitiesmentioning

confidence: 99%

“…We can regard the expressions S β (R|B) ρ − S α (R|AB) ρ and I α (R; AB) ρ − I β (R; B) ρ appearing in the bounds on the fidelity in Lemma 3.1 and Theorem 3.2 as Rényi generalizations of the conditional mutual information I(A; R|B) ρ (see [10] for a detailed discussion of this concept).…”

Section: Rényi Generalizations Of the Conditional Mutual Informationmentioning

confidence: 99%

Strong converse theorems using Rényi entropies

Leditzky

Wilde

Datta

2016

Journal of Mathematical Physics

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We use a Rényi entropy method to prove strong converse theorems for certain information-theoretic tasks which involve local operations and quantum or classical communication between two parties. These include state redistribution, coherent state merging, quantum state splitting, measurement compression with quantum side information, randomness extraction against quantum side information, and data compression with quantum side information. The method we employ in proving these results extends ideas developed by Sharma [41], which he used to give a new proof of the strong converse theorem for state merging. For state redistribution, we prove the strong converse property for the boundary of the entire achievable rate region in the (e, q)-plane, where e and q denote the entanglement cost and quantum communication cost, respectively. In the case of measurement compression with quantum side information, we prove a strong converse theorem for the classical communication cost, which is a new result extending the previously known weak converse. For the remaining tasks, we provide new proofs for strong converse theorems previously established using smooth entropies. For each task, we obtain the strong converse theorem from explicit bounds on the figure of merit of the task in terms of a Rényi generalization of the optimal rate. Hence, we identify candidates for the strong converse exponents for each task discussed in this paper. To prove our results, we establish various new entropic inequalities, which might be of independent interest. These involve conditional entropies and mutual information derived from the sandwiched Rényi divergence. In particular, we obtain novel bounds relating these quantities, as well as the Rényi conditional mutual information, to the fidelity of two quantum states.

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