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arXiv:2208.14438 · Annales Henri Poincaré

Interpolating Between Rényi Entanglement Entropies for Arbitrary Bipartitions via Operator Geometric Means

Dávid Bugár, Péter Vrana

Published Sep 14, 2024• 3 citations
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Abstract

Abstract The asymptotic restriction problem for tensors can be reduced to finding all parameters that are normalized, monotone under restrictions, additive under direct sums and multiplicative under tensor products, the simplest of which are the flattening ranks. Over the complex numbers, a refinement of this problem, originating in the theory of quantum entanglement, is to find the optimal rate of entanglement transformations as a function of the error exponent. This trade-off can also be characterized in terms of the set of normalized, additive, multiplicative functionals that are monotone in a suitable sense, which includes the restriction-monotones as well. For example, the flattening ranks generalize to the (exponentiated) Rényi entanglement entropies of order $$\alpha \in [0,1]$$ α ∈ [ 0 , 1 ] . More complicated parameters of this type are known, which interpolate between the flattening ranks or Rényi entropies for special bipartitions, with one of the parts being a single tensor factor. We introduce a new construction of subadditive and submultiplicative monotones in terms of a regularized Rényi divergence between many copies of the pure state represented by the tensor and a suitable sequence of positive operators. We give explicit families of operators that correspond to the flattening-based functionals, and show that they can be combined in a nontrivial way using weighted operator geometric means. This leads to a new characterization of the previously known additive and multiplicative monotones, and gives new submultiplicative and subadditive monotones that interpolate between the Rényi entropies for all bipartitions. We show that for each such monotone there exist pointwise smaller multiplicative and additive ones as well. In addition, we find lower bounds on the new functionals that are superadditive and supermultiplicative.

DOI
10.1007/s00023-024-01486-3

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Abstract The asymptotic restriction problem for tensors can be reduced to finding all parameters that are normalized, monotone under restrictions, additive under direct sums and multiplicative under tensor products, the simplest of which are the flattening ranks. Over the complex numbers, a refinement of this problem, originating in the theory of quantum entanglement, is to find the optimal rate of entanglement transformations as a function of the error exponent. This trade-off can also be characterized in terms of the set of normalized, additive, multiplicative functionals that are monotone in a suitable sense, which includes the restriction-monotones as well. For example, the flattening ranks generalize to the (exponentiated) Rényi entanglement entropies of order α∈[0,1]\alpha \in [0,1]α∈[0,1] α ∈ [ 0 , 1 ] . More complicated parameters of this type are known, which interpolate between the flattening ranks or Rényi entropies for special bipartitions, with one of the parts being a single tensor factor. We introduce a new construction of subadditive and submultiplicative monotones in terms of a regularized Rényi divergence between many copies of the pure state represented by the tensor and a suitable sequence of positive operators. We give explicit families of operators that correspond to the flattening-based functionals, and show that they can be combined in a nontrivial way using weighted operator geometric means. This leads to a new characterization of the previously known additive and multiplicative monotones, and gives new submultiplicative and subadditive monotones that interpolate between the Rényi entropies for all bipartitions. We show that for each such monotone there exist pointwise smaller multiplicative and additive ones as well. In addition, we find lower bounds on the new functionals that are superadditive and supermultiplicative.
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Historically, entanglement and coherence distillation focused on the distillable rate under full knowledge of the initial state. In this paper, we develop a unified finite-blocklength and asymptotic theory for universal distillation from convex compact sequences of possible states stable under tensor products. We first resolve an open question by showing that two regularized variants of the generalized quantum Hoeffding divergence need not coincide, and establish fundamental properties of the Hoeffding divergence and anti-divergence. For universal entanglement distillation, we connect finite-blocklength performance under non-entangling, PPT-preserving, and dually free operations to composite correlated hypothesis testing, yielding reliability functions for non-entangling and PPT-preserving operations characterized by regularized quantum Hoeffding divergences and strong converse bounds via the anti-divergence. We recover the entanglement concentration error exponent, zero-rate exponents under a continuity condition, and distillable entanglements, and characterize the dually free cases by regularized relative entropies under restricted measurements. Additionally, five concrete noise models justify the black-box assumptions for both resources. For coherence distillation under maximally incoherent and dephasing-covariant incoherent operations, we extend the one-shot semidefinite program characterization to universal inputs; relating a relaxed optimization to hypothesis testing yields the high-rate reliability function. A coherent-seed approach extends the single-state formula to the entire direct regime when the zero-error rate is positive. Finally, we completely characterize the states with positive zero-error coherence distillation rate, show that the rate equals the order-zero Petz Rényi relative entropy of coherence, and derive dimension-independent one-shot bounds ......
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