Pith. sign in

REVIEW 2 major objections 4 minor 43 references

Relative entropy for von Neumann subalgebras

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The Pimsner-Popa index equals the maximal sandwiched Renyi relative entropy to a subalgebra.

desk verdict Solid index-entropy core with a false decoherence theorem as printed; the exponential bound in the abstract is right and fixes it. read the letter →

arxiv 1909.01906 v3 pith:FT66XEG4 submitted 2019-09-04 math.OA

Li Gao , Marius Junge , Nicholas LaRacuente This is my paper · ORCID
classification math.OA MSC 46L5346L6046L3746L51
keywords Pimsner-PopaindexsandwichedRenyirelativeentropysubfactorsconditionalexpectationsquantumMarkovsemigroupsdecoherencetimeamalgamatedLpspacescompletelybounded
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that for an inclusion $N\subset M$ of II$_1$ factors or finite-dimensional von Neumann algebras, the Pimsner-Popa index is an information-theoretic quantity: $\log\lambda(M:N)$ equals the maximal sandwiched Renyi relative entropy from a state of $M$ to the subalgebra $N$, for every Renyi parameter $1/2\le p\le\infty$. The equality is $-\log\lambda(M:N)=\sup_\rho D_p(\rho\|N)=\sup_\rho D_p(\rho\|E(\rho))$, with $E$ the trace-preserving conditional expectation. If correct, the paper unifies the subfactor index with quantum information measures such as Renyi conditional entropy, coherence, and asymmetry, and shows the index does not depend on $p$. It then defines completely bounded versions of this relative entropy, proves additivity, and derives a dimension-independent decoherence-time estimate for self-adjoint quantum Markov semigroups from a spectral-gap condition.

What carries the argument

The load-bearing identity is the Pimsner-Popa structural equality $\lambda(M:N)=\inf\{\|E(e)\|_\infty : e\text{ projection in }M\}$, where $E$ is the trace-preserving conditional expectation and $\lambda(M:N)=\max\{\lambda:\lambda\rho\le E(\rho)\ \forall\rho\ge 0\}$ is the Pimsner-Popa index. The proof of Theorem 3.1 applies this equality only at the $p=1/2$ step: a Holder estimate gives $D_{1/2}(\rho\|N)\ge -\log\|E(\mathrm{supp}\,\rho)\|_\infty$, so the supremum reaches $-\log\lambda(M:N)$, while monotonicity in $p$ and the $p=\infty$ identity $-\log\lambda(M:N)=\sup_\rho D_\infty(\rho\|E(\rho))$ provide the reverse direction. An extremal projection $e_0$ with $E(e_0)$ a scalar multiple of a projection certifies that the maximum is attained. For the decoherence application, the additional mechanism is an amalgamated $L_p$-norm estimate imported from [16] that transfers an $L^2$ spectral-gap contraction to the conditional space $L^2_1(N\subset M)$.

What would settle it

Compute the supremum of $D_{1/2}(\rho\|N)$ for the explicit finite-dimensional inclusion $M_6$ with $N=M_2$ embedded with multiplicity 2; the paper predicts the value $\log 6$. A numerical optimization that returns any strictly smaller supremum would refute Theorem 3.1, since monotonicity in $p$ makes $p=1/2$ the most sensitive endpoint.

Watch

Extended reading notes

Core claim

The central discovery, stated as Theorem 3.1, is that for $N\subset M$ an inclusion of II$_1$ factors or finite-dimensional von Neumann algebras, the supremum over densities of the sandwiched Renyi relative entropy to the subalgebra is exactly $-\log\lambda(M:N)$, and the same value is obtained when the second argument is $E(\rho)$ instead of the infimum over $\sigma\in N$. Because sandwiched entropies are monotone in $p$, it suffices to prove the lower bound at $p=1/2$, where the divergence is fidelity-based; the upper bound follows from the identity $-\log\lambda(M:N)=\sup_\rho D_\infty(\rho\|E(\rho))$. The paper also introduces $D_p(M\|N)=\sup_\rho D_p(\rho\|N)$ and its completely bounded version $D_{p,cb}(M\|N)$, and shows $D_{p,cb}(M\|N)=\log[M:N]$ for finite subfactors. For self-adjoint quantum Markov semigroups with spectral gap $\lambda$, it proves $D(\mathrm{id}\otimes T_t(\rho)\|M_n(N))\le 2e^{-\lambda t}+D_2(\rho\|M_n(N))/2$, giving the uniform decoherence-time bound $t\ge \lambda^{-1}(2\log(2/\epsilon)+D_{2,cb}(M\|N)/2)$.

Load-bearing premise

The proof leans on a structural fact about the Pimsner-Popa index, that it is determined by the worst projection in $M$ rather than by all positive operators; this is known only for II$_1$ factors and finite-dimensional algebras, and if it fails, only the inequality $D_p(M\|N)\le -\log\lambda(M:N)$ remains.

Editorial extensions

If this is right

  • For finite subfactors, the completely bounded relative entropy recovers exactly the subfactor index: $D_{p,cb}(M\|N)=\log[M:N]$ for every $1/2\le p\le\infty$, so the index is a divergence maximum and not merely a coupling constant.
  • The completely bounded relative entropy is additive: $D_{cb}(M_1\otimes M_2\|N_1\otimes N_2)=D_{cb}(M_1\|N_1)+D_{cb}(M_2\|N_2)$, so composite systems behave like independent information reservoirs.
  • Every finite-dimensional self-adjoint quantum Markov semigroup whose decoherence-free subalgebra is commutative has an $\epsilon$-separability time that is independent of the dimension of any auxiliary system entangled with $M$.
  • For Schur-multiplier dephasing semigroups on $M_m$, the uniform decoherence time is at most $\lambda^{-1}(2\log(2/\epsilon)+m/2)$, with $\lambda$ the smallest nonzero decoherence rate.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • I infer that if the Pimsner-Popa structural equality were established for a wider class of inclusions, Theorem 3.1 would extend verbatim, making the subfactor index an information-theoretic invariant for general finite von Neumann algebras.
  • The additivity of $D_{cb}$ suggests using $e^{D_{p,cb}}$ as a tensor-stable, computable proxy for the index in inclusions where the index is not defined, potentially connecting entropy inequalities to classification questions.
  • The decoherence bound is likely not tight: for concrete dephasing channels one can numerically compare the true worst-case trace distance with the paper's exponential bound and test whether the $D_{2,cb}$ term overestimates the transient time.
  • A testable extension is to replace the spectral-gap assumption by a complete logarithmic Sobolev inequality for the same semigroup; the paper's framework predicts the same form of estimate with $D_{2,cb}$ replaced by $D_{1,cb}$.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies the Pimsner-Popa index λ(M:N) for inclusions N⊂M of finite von Neumann algebras and connects it to sandwiched Rényi relative entropies. The central result, Theorem 3.1, states that for inclusions of II₁ factors or finite-dimensional von Neumann algebras, and for every 1/2≤p≤∞, one has −log λ(M:N) = sup_ρ D_p(ρ||E(ρ)) = sup_ρ D_p(ρ||N), where E is the trace-preserving conditional expectation. The authors introduce D_p(M||N) and a completely bounded version D_{p,cb}(M||N), prove that the latter equals log[M:N] for finite subfactors and satisfies an additivity property, and apply these quantities to bound the decoherence time of self-adjoint quantum Markov semigroups in terms of the spectral gap and D_{2,cb}. The paper is written so that the main structural result is accessible to a quantum information audience, with the more technical operator-space material placed in an appendix.

Significance. If correct, Theorem 3.1 is a substantial conceptual contribution: it identifies the Pimsner-Popa index with a maximal Rényi-type divergence from the algebra to a subalgebra, uniformly in p, and gives an information-theoretic interpretation of the index. The completely bounded relative entropy D_{p,cb}, with its additivity and its equality to log[M:N] for finite subfactors, is a useful new invariant. The paper is also commendably explicit about its dependency on the Pimsner-Popa structural equality (12), whose converse direction is noted to be open in general, and it provides a concrete finite-dimensional formula (14). The decoherence-time application is interesting and potentially important. However, the main application theorem as printed in Section 4 contains a false inequality, so the paper needs a substantive correction before acceptance.

major comments (2)
  1. [Section 4, Theorem 4.1, Eq. (24)] The asserted bound D(T_t(ρ)||N) ≤ 2e^{-λt} + D_2(ρ||N)/2 is false. The proof reduces the last step to 2 log(1+ab) ≤ 2a + log b with a=e^{-λt}∈[0,1] and b=‖ρ‖_{L^2_1(N⊂M)}≥1; this inequality is not true, e.g. a=1, b=100 gives 2log(101)≈9.21 > 6.61. A concrete state counterexample is M=M_n with normalized trace, N=C1, ρ=n|ψ><ψ|, and T_t=e^{-t}(id−E)+E, which is a self-adjoint quantum Markov semigroup with spectral gap 1. At t=0 one has D(ρ||N)=D_2(ρ||N)=log n, so (24) would require log n ≤ 2 + log n/2, which fails for every n>e^4≈54.6. The abstract's exponential bound, D(T_tρ||N) ≤ 2exp(−λt + D_2(ρ||N)/2), is the correct statement and follows by replacing the false step with 2log(1+ab)≤2ab. Corollary 4.2's displayed inequality (25) is false for the same reason, although the decoherence-time estimate (26) can still be recovered from the corrected exponential bound. This is a load-bearing error in the application section and must be fixed.
  2. [Section 4, proof of Theorem 4.1] The proof uses the estimate ‖T_t−E : L^2_1(N⊂M)→L^2_1(N⊂M)‖ ≤ e^{-λt}, imported from Lemma 3.12 of the authors' own preprint [16]. This lemma is central to both Theorem 4.1 and Theorem 4.6, but it is neither stated nor proved in the present paper. Since the current theorem statement must be revised in any case, the authors should make the dependence on this result explicit and either state the lemma in full or provide a proof in an appendix, so that the corrected decoherence bound is self-contained.
minor comments (4)
  1. [Abstract and Introduction] The phrase 'a new notation of relative entropy' should be 'a new notion of relative entropy', and 'egordic theory' should be 'ergodic theory'.
  2. [Section 3, Eq. (11)] The displayed derivation of Eq. (11) contains erroneous reciprocal and exponent placement (e.g. '(sup ... )^{-1}' and the final '−1'); the stated conclusion is correct, but the chain of equalities as written does not parse and should be rewritten.
  3. [Section 4, Theorem 4.6] In the statement of Theorem 4.6, the map on M_n(M) should be id⊗T^k and the target subalgebra should be M_n(N); as written, 'D(T^k(ρ)||M_n(N))' applies T^k, defined on M, to a density in M_n(M).
  4. [Throughout] There are several typographical slips, including 'gourp von Neumann algebra', 'quantum Makrov map', and 'correpsonds'; these should be corrected in a final revision.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular step: Theorem A is proven from the external Pimsner-Popa projection equality plus monotonicity; self-citations are not load-bearing in the circular sense.

full rationale

Walking the derivation chain, the central equality (Theorem 3.1) is assembled as follows. For p=∞, equation (11) rewrites the definition of λ(M:N) in terms of D∞; this is an explicit observation, not a fitted or predicted output. The nontrivial direction is sup_ρ D_{1/2}(ρ||N) ≥ −log λ. The proof bounds D_{1/2}(ρ||N) below by −log ||E(e)||∞ and then invokes [31, Theorem 2.2 & Corollary 5.6] for inf_e ||E(e)||∞ = λ(M:N). That equality is external to the present authors and does not contain the claimed D_p identity. Monotonicity of D_p in p and data processing are cited from standard literature. The finite-dimensional index formula (14) is re-derived in the text by maximizing H(E(ρ)) − H(ρ), and Corollary 3.8 uses Jones' multiplicativity of the subfactor index, again external. The self-citations [15] and [16] do not smuggle the target conclusion: [15] only introduces the notation D_p(ρ||N), and [16, Lemma 3.12] supplies an independent norm identity for N-bimodule maps on amalgamated L^2 spaces. Even if Theorem 4.1's printed additive bound is mathematically questionable (the final numerical inequality 2log(1+ab) ≤ 2a + log b fails in general), that is a correctness defect, not a circular reduction: the bound is not obtained by assuming the conclusion. No parameter is fitted and no "prediction" is defined in terms of the claimed index value. Thus there is no circular step; the score 2 reflects the presence of author self-citations in the derivation chain, which are not load-bearing in the circular sense.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The central identity is assembled from independently defined objects (λ from Pimsner-Popa, D_p from quantum information) plus imported structural facts. The p=∞ component is a direct rewriting of λ's definition; the p<∞ extension uses the Pimsner-Popa equality (12) and the established monotonicity and data-processing calculus of sandwiched Rényi divergences. No numbers are fitted: the finite-dimensional index formula (14) is re-derived by entropy maximization, and the subfactor identification D_{p,cb} = log[M:N] follows from Theorem 3.1 plus Jones's multiplicativity. The invented objects (relative entropy to a subalgebra, cb-version) have independent handles: they reduce to the Jones index on subfactors, to log Σ l_k² on finite-dimensional inclusions, and they satisfy additivity.

assumptions (6)
  • domain assumption Pimsner-Popa equality (12): inf{||E(e)||_∞ : e projection in M} = λ(M:N), with an extremal projection e₀ such that E(e₀) is a scalar multiple of a projection, for II₁ factors and finite-dimensional algebras.
    Invoked at the decisive step of Theorem 3.1: the p=1/2 lower bound yields sup D_{1/2} ≥ −log inf_e ||E(e)||, and equality with −log λ requires (12). The paper explicitly notes the converse direction is open in general, which confines Theorem A to these two classes. Cited to [31, Theorem 2.2 and Corollary 5.6].
  • domain assumption Sandwiched Rényi divergence D_p is monotone in p on [1/2,∞], satisfies the data processing inequality and joint convexity, and limits to Umegaki entropy at p=1, on finite von Neumann algebras.
    Used to sandwich D_p between D_{1/2} and D_∞ in Theorem 3.1 and throughout Sections 2-3; cited to [28,41,6,20,21,18]. For p < 1 on general von Neumann algebras these are recent results (Berta-Scholz-Tomamichel, Jenčová).
  • domain assumption [16, Lemma 3.12]: for a self-adjoint semigroup T_t with fixed-point algebra N, the amalgamated norm satisfies ||T_t−E : L²₁(N⊂M)→L²₁(N⊂M)|| = ||T_t−E : L²(M)→L²(M)|| ≤ e^{−λt} under λ-spectral gap.
    The central estimate in the decoherence application (Theorems 4.1 and 4.6). Cited to the authors' own preprint arXiv:1807.08838; the proof is not reproduced in this paper.
  • standard math Duality L^{p'}_∞(N⊂M) ⊂ (L^p_1(N⊂M))_* (w*-dense) and complex interpolation for amalgamated and conditional Lp spaces from [23, Junge-Parcet].
    Used in Section 2.3 and in the proof of Theorem 3.9 (cb-norm formula (i), additivity (iii)); from the published Junge-Parcet monograph [23].
  • domain assumption Index multiplicativity for subfactors: [M_n(M):M_n(N)] = [M:N] and [M_1⊗M_2 : N_1⊗N_2] = [M_1:N_1][M_2:N_2].
    Used in Corollary 3.8 iii) to conclude D_{p,cb}(M||N) = log[M:N] for subfactors; cited to [22, Proposition 2.1.15].
  • standard math Quantum Pinsker inequality: ||ρ−σ||₁² ≤ 2D(ρ||σ) for Umegaki relative entropy.
    Converts the relative-entropy decay (Theorem 4.1) into the trace-distance decoherence time bound; cited to [40, Theorem 5.38].
invented entities (2)
  • Relative entropy to a subalgebra D_p(ρ||N) = inf_{σ∈S(N)} D_p(ρ||σ) independent evidence
    purpose: Measures how far a state is from the subalgebra; unifies conditional Rényi entropy, coherence and asymmetry measures; its sup over states equals the Pimsner-Popa index.
    First appeared in the authors' [15] (grounded in Marvian-Spekkens asymmetry [26]); here the novel content is the sup = index equality, not the object itself. Reduces to known quantities in standard cases (coherence for diagonal subalgebras, conditional entropy for tensor factors).
  • Completely bounded relative entropy D_{p,cb}(M||N) = sup_n D_p(M_n(M)||M_n(N)) independent evidence
    purpose: Extends the subfactor index to general finite inclusions; its finiteness is the condition in the complete decoherence time estimate.
    New in this paper. Independent handles: equals log[M:N] on subfactors (Cor 3.8), equals log Σ l_k² on finite-dimensional inclusions (Example 3.10, verified by direct computation), and is additive (Theorem 3.9 iii). These are checkable predictions, not fitted constants.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Relative entropy for von Neumann subalgebras." pith.science (2026). https://pith.science/paper/FT66XEG4

@misc{pith2026190901906,
  author       = {Pith},
  title        = {Pith review of: Relative entropy for von Neumann subalgebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FT66XEG4}},
  note         = {Machine review of arXiv:1909.01906}
}
abstract

We revisit the connection between index and relative entropy for an inclusion of finite von Neumann algebras. We observe that the Pimsner-Popa index connects to sandwiched Renyi $p$-relative entropy for all $1/2\le p\le \infty$, including Umegaki's relative entropy at $p=1$. Based on that, we introduce a new notation of relative entropy to a subalgebra which generalizes subfactors index. This relative entropy has application in estimating decoherence time of quantum Markov semigroups.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 40 canonical work pages

  1. [16]

    Fisher Information and Logarithmic Sobolev Inequality for Matrix Valued Functions

    Li Gao, Marius Junge, and Nicolas LaRacuente. Fisher informat ion and logarithmic sobolev inequality for matrix valued functions. arXiv preprint arXiv:1807.08838 , 2018

    work page Pith review arXiv 2018

  2. [1]

    Relative entropy of states of von neumann algebr as

    Huzihiro Araki. Relative entropy of states of von neumann algebr as. Publications of the Research Institute for Mathematical Sciences , 11(3):809–833, 1976

    work page 1976

  3. [2]

    Analysis and geometry of Markov diffusion op- erators, volume 348

    Dominique Bakry, Ivan Gentil, and Michel Ledoux. Analysis and geometry of Markov diffusion op- erators, volume 348. Springer Science & Business Media, 2013

    work page 2013

  4. [3]

    Estimating the decoherence time using non-commutative Functional Inequalities

    Ivan Bardet. Estimating the decoherence time using non-commu tative functional inequalities. arXiv preprint arXiv:1710.01039, 2017

    work page Pith review arXiv 2017

  5. [4]

    Hypercontractivity and logarithmic Sobolev Inequality for non-primitive quantum Markov semigroups and estimation of decoherence rates

    Ivan Bardet and Cambyse Rouz´ e. Hypercontractivity and loga rithmic sobolev inequality for non-primitive quantum markov semigroups and estimation of decohe rence rates. arXiv preprint arXiv:1803.05379, 2018

    work page Pith review arXiv 2018

  6. [5]

    Quantifying coherence

    T Baumgratz, M Cramer, and MB Plenio. Quantifying coherence. Physical review letters , 113(14):140401, 2014

    work page 2014

  7. [6]

    R´ enyi dive rgences as weighted non- commutative vector-valued lp-spaces

    Mario Berta, Volkher B Scholz, and Marco Tomamichel. R´ enyi dive rgences as weighted non- commutative vector-valued lp-spaces. In Annales Henri Poincar´ e, volume 19, pages 1843–1867. Springer, 2018. RELATIVE ENTROPY FOR VON NEUMANN SUBALGEBRAS 33

    work page 2018

  8. [7]

    Gradient flow structure and exponential decay of the sandwiched r´ enyi divergence for primitive lindblad equations with gns-detailed b alance

    Yu Cao, Jianfeng Lu, and Yulong Lu. Gradient flow structure and exponential decay of the sandwiched r´ enyi divergence for primitive lindblad equations with gns-detailed b alance. Journal of Mathematical Physics, 60(5):052202, 2019

    work page 2019

Show all 43 references
  1. [8]

    Gradient flow and entropy inequalities f or quantum markov semigroups with detailed balance

    Eric A Carlen and Jan Maas. Gradient flow and entropy inequalities f or quantum markov semigroups with detailed balance. Journal of Functional Analysis , 273(5):1810–1869, 2017

    2017

  2. [9]

    Non-commutative calculus, optimal tr ansport and functional inequalities in dissipative quantum systems

    Eric A Carlen and Jan Maas. Non-commutative calculus, optimal tr ansport and functional inequalities in dissipative quantum systems. arXiv preprint arXiv:1811.04572 , 2018

    2018 arXiv

  3. [10]

    Noncommutative de leeuw theo- rems

    Martijn Caspers, Javier Parcet, Mathilde Perrin, and ´Eric Ricard. Noncommutative de leeuw theo- rems. In Forum of Mathematics, Sigma , volume 3. Cambridge University Press, 2015

    2015

  4. [11]

    A brief history o f the gkls equation

    Dariusz Chru´ sci´ nski and Saverio Pascazio. A brief history o f the gkls equation. arXiv preprint arXiv:1710.05993, 2017

    2017 arXiv

  5. [12]

    Entropy for automorphisms of ii 1 von neumann algebras

    Alain Connes and Erling Størmer. Entropy for automorphisms of ii 1 von neumann algebras. Acta Mathematica, 134(1):289–306, 1975

    1975

  6. [13]

    Operator spaces

    Edward G Effros and Ruan Zhong Jin. Operator spaces. Clarendon Press, 2000

    2000

  7. [14]

    Generalized s-numbers of τ-measurable operators

    Thierry Fack and Hideki Kosaki. Generalized s-numbers of τ-measurable operators. Pacific Journal of Mathematics , 123(2):269–300, 1986

    1986

  8. [15]

    Unifying Entan glement with Uncertainty via Sym- metries of Observable Algebras

    Li Gao, Marius Junge, and Nicholas LaRacuente. Unifying Entan glement with Uncertainty via Sym- metries of Observable Algebras. arXiv:1710.10038 [quant-ph] , October 2017. arXiv: 1710.10038

    2017 arXiv

  9. [17]

    Measuring th e quality of a quantum reference frame: The relative entropy of frameness

    Gilad Gour, Iman Marvian, and Robert W Spekkens. Measuring th e quality of a quantum reference frame: The relative entropy of frameness. Physical Review A , 80(1):012307, 2009

    2009

  10. [18]

    Interpolation of quasi no ncommutative lp-spaces

    Juan Gu, Zhi Yin, and Haonan Zhang. Interpolation of quasi no ncommutative lp-spaces. arXiv preprint arXiv:1905.08491, 2019

    1905 arXiv

  11. [19]

    Multiplicativity of completely bound ed p-norms implies a strong converse for entanglement-assisted capacity

    Manish K Gupta and Mark M Wilde. Multiplicativity of completely bound ed p-norms implies a strong converse for entanglement-assisted capacity. Communications in Mathematical Physics , 334(2):867– 887, 2015

    2015

  12. [20]

    R´ enyi relative entropies and noncommutative lp-spaces

    Anna Jenˇ cov´ a. R´ enyi relative entropies and noncommutative lp-spaces. In Annales Henri Poincar´ e, volume 19, pages 2513–2542. Springer, 2018

    2018

  13. [21]

    R\’enyi relative entropies and noncommutative Lp-spaces II

    Anna Jenˇ cov´ a. R\’enyi relative entropies and noncommutative Lp-spaces II. arXiv e-prints , page arXiv:1707.00047, Jun 2017

    2017 arXiv

  14. [22]

    Index for subfactors

    Vaughan FR Jones. Index for subfactors. Inventiones mathematicae, 72(1):1–25, 1983

    1983

  15. [23]

    Mixed-norm inequalities and operator space L p embedding theory

    Marius Junge and Javier Parcet. Mixed-norm inequalities and operator space L p embedding theory. Number 952-956. American Mathematical Soc., 2010

    2010

  16. [24]

    Applications of the complex interpolation method t o a von neumann algebra: non- commutative lp-spaces

    Hideki Kosaki. Applications of the complex interpolation method t o a von neumann algebra: non- commutative lp-spaces. Journal of functional analysis , 56(1):29–78, 1984

    1984

  17. [25]

    Strong conver se theorems using r´ enyi entropies

    Felix Leditzky, Mark M Wilde, and Nilanjana Datta. Strong conver se theorems using r´ enyi entropies. Journal of Mathematical Physics , 57(8):082202, 2016

    2016

  18. [26]

    Extending noethers the orem by quantifying the asymmetry of quantum states

    Iman Marvian and Robert W Spekkens. Extending noethers the orem by quantifying the asymmetry of quantum states. Nature communications, 5:3821, 2014

    2014

  19. [27]

    Sandwiched r´ enyi convergence for quantum evo- lutions

    Alexander M¨ uller-Hermes and Daniel Stilck Franca. Sandwiched r´ enyi convergence for quantum evo- lutions. Quantum, 2:55, 2018. 34 LI GAO, MARIUS JUNGE ∗, AND NICHOLAS LARACUENTE

    2018

  20. [28]

    On quan- tum r´ enyi entropies: A new generalization and some properties

    Martin M¨ uller-Lennert, Fr´ ed´ eric Dupuis, Oleg Szehr, SergeFehr, and Marco Tomamichel. On quan- tum r´ enyi entropies: A new generalization and some properties. Journal of Mathematical Physics , 54(12):122203, 2013

    2013

  21. [29]

    Strong converse and st ein’s lemma in quantum hypothesis testing

    Tomohiro Ogawa and Hiroshi Nagaoka. Strong converse and st ein’s lemma in quantum hypothesis testing. In Asymptotic Theory Of Quantum Statistical Inference: Selec ted Papers, pages 28–42. World Scientific, 2005

    2005

  22. [30]

    Completely bounded maps and operator algebras , volume 78

    Vern Paulsen. Completely bounded maps and operator algebras , volume 78. Cambridge University Press, 2002

    2002

  23. [31]

    Entropy and index for subfactor s

    Mihai Pimsner and Sorin Popa. Entropy and index for subfactor s. In Annales scientifiques de l’Ecole normale sup´ erieure, volume 19, pages 57–106, 1986

    1986

  24. [32]

    Relating relative entropy , optimal transport and fisher infor- mation: A quantum hwi inequality

    Cambyse Rouz´ e and Nilanjana Datta. Relating relative entropy , optimal transport and fisher infor- mation: A quantum hwi inequality. arXiv preprint arXiv:1709.07437 , 2017

    2017 arXiv

  25. [33]

    Metric spaces and positive definite funct ions

    Isaac J Schoenberg. Metric spaces and positive definite funct ions. Transactions of the American Mathematical Society, 44(3):522–536, 1938

    1938

  26. [34]

    Relative entropy for finite von neumann algebras

    Yuki Seo. Relative entropy for finite von neumann algebras. Journal of Operator Theory , pages 373– 387, 1993

    1993

  27. [35]

    Alexander Streltsov, Gerardo Adesso, and Martin B. Plenio. Co lloquium: Quantum coherence as a resource. Reviews of Modern Physics , 89(4):041003, October 2017

    2017

  28. [36]

    Conditional expectation in an operator algeb ra, iv (entropy and information)

    Hisaharu Umegaki. Conditional expectation in an operator algeb ra, iv (entropy and information). In Kodai Mathematical Seminar Reports , volume 14, pages 59–85. Department of Mathematics, Tokyo Institute of Technology, 1962

    1962

  29. [37]

    Tradeoff between ex- tractable mechanical work, accessible entanglement, and ability to act as a reference system, under arbitrary superselection rules

    Joan Alfina Vaccaro, F Anselmi, Howard Mark Wiseman, and Kurt J acobs. Tradeoff between ex- tractable mechanical work, accessible entanglement, and ability to act as a reference system, under arbitrary superselection rules. Physical Review A , 77(3):032114, 2008

    2008

  30. [38]

    The role of relative entropy in quantum informatio n theory

    Vlatko Vedral. The role of relative entropy in quantum informatio n theory. Reviews of Modern Physics, 74(1):197, 2002

    2002

  31. [39]

    Resource theory of asymmetric disting uishability

    Xin Wang and Mark M Wilde. Resource theory of asymmetric disting uishability. arXiv preprint arXiv:1905.11629, 2019

    1905 arXiv

  32. [40]

    The theory of quantum information

    John Watrous. The theory of quantum information . Cambridge University Press, 2018

    2018

  33. [41]

    Strong converse for the classical capacity of entanglement-breaking and hadamard channels via a sandwiched r´ enyi relative entropy

    Mark M Wilde, Andreas Winter, and Dong Yang. Strong converse for the classical capacity of entanglement-breaking and hadamard channels via a sandwiched r´ enyi relative entropy. Commu- nications in Mathematical Physics , 331(2):593–622, 2014

    2014

  34. [42]

    Tight uniform continuity bounds for quantum e ntropies: conditional entropy, rela- tive entropy distance and energy constraints

    Andreas Winter. Tight uniform continuity bounds for quantum e ntropies: conditional entropy, rela- tive entropy distance and energy constraints. Communications in Mathematical Physics , 347(1):291– 313, 2016

    2016

  35. [43]

    Operational resource theory of coherence

    Andreas Winter and Dong Yang. Operational resource theory of coherence. Physical review letters , 116(12):120404, 2016. RELATIVE ENTROPY FOR VON NEUMANN SUBALGEBRAS 35 Department of Mathematics, Texas A&M University, College S tation, TX 77840, USA E-mail address, Li Gao: liga...

    2016

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.