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Multivariate Trace Inequalities

David Sutter, Mario Berta, Marco Tomamichel

arXiv:1604.03023v3math-phcs.ITmath.ITmath.MPquant-ph

TL;DR

Trace inequalities can be difficult to prove for non-commuting matrices, and extensions of Golden–Thompson and Araki–Lieb–Thirring to many matrices are needed. The paper uses asymptotic spectral pinching and complex interpolation to prove multivariate extensions. It reports a strengthened triple-matrix inequality and recoverability-based entropy remainder terms, including bounds tight in the commutative case.

  • Problem

    Trace inequalities can be difficult to prove for non-commuting matrices, motivating extensions of the Golden–Thompson and Araki–Lieb–Thirring inequalities to many matrices.

    Source: §1 · p. 1

  • Method

    The proofs use asymptotic spectral pinching and complex interpolation theory to derive multivariate trace inequalities.

    Source: §2.4 · p. 5, §3.1 · p. 6, §5 · p. 12

  • Results

    The extensions include a strengthened triple-matrix inequality and recoverability-based entropy remainder terms that are explicit and tight in the commutative case.

    Source: §1 · pp. 1–2

  • Takeaways & Limitations

    The authors present pinching and interpolation as transparent methods for multivariate trace inequalities and apply the four-matrix Golden–Thompson extension to entropy inequalities.

    Source: §1 · p. 2, §5 · p. 12

    Corollary 4.2 is no longer valid if measured relative entropy is replaced with relative entropy.

    Source: §4 · p. 11

Abstract

from arXiv · show

We prove several trace inequalities that extend the Golden-Thompson and the Araki-Lieb-Thirring inequality to arbitrarily many matrices. In particular, we strengthen Lieb's triple matrix inequality. As an example application of our four matrix extension of the Golden-Thompson inequality, we prove remainder terms for the monotonicity of the quantum relative entropy and strong sub-additivity of the von Neumann entropy in terms of recoverability. We find the first explicit remainder terms that are tight in the commutative case. Our proofs rely on complex interpolation theory as well as asymptotic spectral pinching, providing a transparent approach to treat generic multivariate trace inequalities.

1 Introduction

Trace inequalities are often straightforward when matrices commute but difficult for non-commuting matrices; this work extends the Golden–Thompson and Araki–Lieb–Thirring inequalities to arbitrarily many matrices. It also applies the extensions to entropy inequalities, with remainder bounds in terms of recoverability.

1 Introduction

  • The motivating difficulty is that trace inequalities can be straightforward equalities for commuting matrices yet difficult to prove for non-commuting ones.Straightforward three-matrix extensions of Golden–Thompson are incorrect.Source: §1 · p. 1
  • For n matrices, the Schatten p-norm extension of Golden–Thompson simplifies to the standard inequality when n=2 and p=2, and strengthens Lieb’s triple matrix inequality for n=3 and p=2.The integrand is independent of t when n=2.Source: §1 · p. 1
  • The paper extends the Araki–Lieb–Thirring inequality to n positive semi-definite matrices, with the Golden–Thompson extension arising in the limit r → 0.The authors also extend both inequalities to general square matrices.Source: §1 · p. 2
  • The four-matrix Golden–Thompson extension yields remainder terms for monotonicity of quantum relative entropy and strong sub-additivity in terms of recoverability.The bound uses measured relative entropy and an explicit universal rotated Petz recovery map independent of ρ; the authors describe it as the first explicit lower bound tight in the commutative case.Source: §1 · p. 2
  • Asymptotic spectral pinching and complex interpolation theory provide the proof methods, which the authors present as transparent approaches to multivariate trace inequalities.The authors suggest these methods may apply beyond the GT and ALT extensions studied here.Source: §1 · p. 2

2 Trace inequalities via asymptotic spectral pinching

This section develops asymptotic spectral pinching as a transparent route to multivariate trace inequalities. It uses the method to extend ALT and GT to arbitrarily many matrices.

2 Trace inequalities via asymptotic spectral pinching

  • The pinching method yields results for p > 0 but does not give explicit forms for the distributions β0 and βr.Source: §2 · p. 2

2.1 The asymptotic spectral pinching method

  • Pinching removes off-diagonal blocks, makes the pinched matrix commute with A, and preserves tr PA[X]A = tr XA; tensor powers have only polynomially many distinct eigenvalues.The polynomial growth of the distinct-eigenvalue count under tensor powers is used in the asymptotic method.Source: §2.1 · p. 3

2.2 Illustrative example: intuitive proof of Golden-Thompson inequality

  • The illustrative two-matrix GT proof pinches B^⊗m against A^⊗m, uses the resulting commutation to apply GT with equality, and then takes m → ∞.The proof's intuition is that tensor-power pinching enforces commutation without destroying the operator too much in the limit.Source: §2.2 · pp. 3–4

2.3 A convexity result for Schatten quasi-norms

  • For 0 < p < 1, despite loss of ordinary convexity, tensor-product Schatten quasi-norms satisfy an asymptotic convexity property.The section combines that property with ordinary convexity for p ≥ 1 to establish quasi-convexity for all p > 0.Source: §2.3 · p. 4

2.4 Main results and proofs via pinching

  • Spectral pinching yields arbitrary-matrix extensions of ALT and GT; the multivariate Lie–Trotter formula obtains the GT extension from ALT as r→0.The results apply to Schatten (quasi-)norms with p > 0.Source: §2.4 · pp. 4, 6
  • For n = 2, the GT extension's right-hand side is independent of t and recovers the original GT inequality at p = 2.Source: §2.4 · p. 6

3 Trace inequalities via interpolation theory

The section uses complex interpolation to extend Golden–Thompson and Araki–Lieb–Thirring inequalities to arbitrarily many matrices. It also extends the scope of these results to general square matrices and relates the multivariate Golden–Thompson result to Lieb’s triple matrix inequality.

3.1 The complex interpolation method

  • The proofs use a complex interpolation result for Schatten norms attributed to Stein and based on Hirschman’s improvement of the Hadamard three-lines theorem [60, 33].The section presents this as the main ingredient for most of its proofs.Source: §3.1 · p. 6

3.2 Main results and proofs via interpolation theory

  • For n = 2, the extended Araki–Lieb–Thirring result recovers the original inequality only for q ≥1.The right-hand side becomes independent of t in this two-matrix case.Source: §3.2 · p. 8
  • The multivariate Lie–Trotter product formula yields an extension of the Golden–Thompson inequality to arbitrarily many matrices, obtained by taking the limit r→0 of (36).Source: §3.2 · p. 8
  • The Corollary 3.3 representation generalizes Lieb’s triple matrix inequality [44] to arbitrarily many matrices.The authors note that Lieb’s triple matrix inequality had not been extended beyond three matrices.Source: §3.2 · p. 8
  • Theorem 3.5 extends the result to general square matrices, and the section views its related inequality as an Araki–Lieb–Thirring inequality for such matrices.The general-square-matrix extension uses the same techniques as Corollary 3.3.Source: §3.2 · pp. 8–9

4 An application: entropy inequalities

The section applies the four-matrix Golden–Thompson extension to entropy inequalities, deriving recoverability remainder terms for strengthened monotonicity and strong sub-additivity. Its relative-entropy bound uses an explicit universal rotated Petz recovery map and is tight in the commutative case.

4 An application: entropy inequalities

  • Theorem 4.1 gives a recoverability remainder for monotonicity of relative entropy under partial trace, using the explicit universal rotated Petz recovery map and measured relative entropy.The map is independent of ρ, and the lower bound is tight in the commutative case.Source: §4 · p. 10
  • The four-matrix Golden–Thompson inequality is the only inequality used in the proof of Theorem 4.1.The derivation uses the variational formula for measured relative entropy at its final step.Source: §4 · pp. 10–11
  • Applying Theorem 4.1 to suitable marginals yields a recoverability remainder for conditional quantum mutual information, and the result extends to trace-preserving completely positive maps.The map extension uses Stinespring dilation and invariance of relative entropy under isometries.Source: §4 · p. 11
  • Corollary 4.2 is invalid if measured relative entropy is replaced by quantum relative entropy in its remainder term.The authors infer that the stated bound may not admit further improvement.Source: §4 · p. 11
  • The universal measured-relative-entropy remainder can also be transformed into bounds for conditional-entropy concavity and joint convexity of relative entropy [8].The authors refer to [35, Section 5] for further discussion of these bounds.Source: §4 · p. 11
  • The authors’ alternative derivation uses the four-matrix Golden–Thompson extension, whereas Lieb and Ruskai’s original proof uses a three-matrix extension.The authors state that this leads to a stronger statement.Source: §4 · p. 11

5 Discussion

The discussion presents asymptotic pinching and complex interpolation as direct proof techniques for multivariate Golden–Thompson and Araki–Lieb–Thirring extensions. It identifies further applications and extensions as open questions.

5 Discussion

  • Asymptotic pinching and complex interpolation provide transparent, direct proofs of the multivariate Golden–Thompson and Araki–Lieb–Thirring extensions.The authors suggest these techniques may apply to other trace inequalities.Source: §5 · p. 12
  • Whether the techniques can yield multivariate extensions of complementary Golden–Thompson and Araki–Lieb–Thirring inequalities remains for future research.The discussion frames this as an open question about matrix means.Source: §5 · p. 12
  • The authors identify unifying Hansen’s multivariate Golden–Thompson extension with their result, and extending the resulting concavity applications, as open questions [25] [27].Corollary 3.3 generalizes Lieb’s triple matrix inequality, which is equivalent to several concavity results.Source: §5 · p. 12
  • Whether the multivariate Golden–Thompson extension can sharpen random-matrix tail bounds remains an open question.Earlier work applied the two-matrix Golden–Thompson inequality sequentially, while Lieb’s concavity theorem yielded sharper bounds [67].Source: §5 · p. 12

A About the probability distribution in Corollary 3.3

The examples constrain matrix-independent probability distributions used to bound κ by averages of γ(t): they cannot be too narrow around zero or too flat. They also show that a three-matrix Golden–Thompson extension without phases can fail.

A About the probability distribution in Corollary 3.3

  • The direct three-matrix trace-exponential bound and a more symmetric conjecture both fail in general.The passage gives the direct bound as tr exp(H1 + H2 + H3) ≤ tr exp(H1) exp(H2) exp(H3).Source: A About the probability distribution… · p. 12
  • The two examples show that a matrix-independent measure making the bound κ ≤ ∫γ(t) dμ(t) hold cannot be too narrow around zero or too flat.The distribution in Corollary 3.3 is β0; the examples motivate examining alternatives independent of A1, A2, and A3.Source: A About the probability distribution… · p. 13
  • For the matrices in (70), κ can exceed γ(0), so a three-matrix Golden–Thompson extension without phases does not hold in general.Figure 2 compares κ with γ(t) for the matrices in (70) and (71).Source: A About the probability distribution… · p. 13

B Proof of Lemma 2.1

The proof constructs a probability distribution for averaging conjugations: its Fourier transform must vanish at differences between logarithms of distinct eigenvalues, and a triangular function provides the required construction.

B Proof of Lemma 2.1

  • Averaging conjugations by A^{it} uses a measure whose Fourier transform vanishes at differences between logarithms of distinct eigenvalues of A.The spectral decomposition of A reduces the requirement to the eigenvalue differences in logarithmic coordinates.Source: B Proof of Lemma 2.1 · pp. 13–14
  • A triangular function is chosen to satisfy the Fourier-transform requirements, and its inverse Fourier transform gives the candidate measure.The construction uses a fixed τ > 0.Source: B Proof of Lemma 2.1 · p. 14
  • The measure defined in (77) is a probability distribution, completing the proof of Lemma 2.1.The passage verifies nonnegativity and unit integral.Source: B Proof of Lemma 2.1 · p. 14

C Proof of Lemma 2.2

The proof uses purifications and symmetric-subspace dimension bounds to obtain a discrete measure with polynomial support, then applies Schatten quasi-norm estimates to complete the inequalities.

C Proof of Lemma 2.2

  • Purifying each matrix as a vector makes the tensor-power projectors lie in a symmetric subspace whose dimension grows polynomially in m.The symmetric-subspace dimension is polynomial in m for fixed matrix dimension d.Source: C Proof of Lemma 2.2 · p. 14
  • Carathéodory’s theorem yields a discrete probability measure supported on a set of polynomial size.The passage states that the support set I has |I| = poly(m).Source: C Proof of Lemma 2.2 · p. 14
  • For p in (0, 1), the proof uses the weakened Schatten p-norm triangle inequality, convexity, and polynomial support size to obtain the inequality chain.The final step relies on |I| = poly(m).Source: C Proof of Lemma 2.2 · pp. 14–15

D Proof of Theorem 3.1

The proof applies Hirschman’s strengthening of the three-line theorem to a scalar holomorphic function built from the operator family. Hölder bounds on its boundary values then yield the desired result.

D Proof of Theorem 3.1

  • The interpolation argument uses Hirschman’s strengthening of Hadamard’s three-line theorem, including a relaxed growth condition for the holomorphic function.The proof identifies the strip as 0 ≤ Re(z) ≤ 1.Source: D Proof of Theorem 3.1 · p. 15
  • For x in [0, 1], the proof pairs conjugate exponents px and qx, then uses the polar decomposition of G(θ) to construct a holomorphic bounded scalar function g(z).The function is defined as the Hilbert–Schmidt inner product tr X(z)†G(z).Source: D Proof of Theorem 3.1 · pp. 15–16
  • Hölder’s inequality bounds g on both strip boundaries; substituting those bounds into Hirschman’s lemma gives the desired result.The proof first verifies the endpoint bounds at it and 1 + it.Source: D Proof of Theorem 3.1 · p. 16

E Proof of Lemma 3.4

The proof uses the known integral-representation expression for the derivative, then reduces the claim in an eigenbasis to a scalar condition for positive x and y.

E Proof of Lemma 3.4

  • The derivative expression in (41) is well known and can be derived from integral representations of the operator logarithm.The passage cites as an example.Source: E Proof of Lemma 3.4 · p. 16
  • In an orthonormal eigenbasis of A, the claim is expressed using eigenvalues and matrix elements of H.The displayed expression is given in (95).Source: E Proof of Lemma 3.4 · p. 16
  • The proof reduces the remaining step to a scalar condition involving an integral for all positive x and y.This scalar condition is identified in (96).Source: E Proof of Lemma 3.4 · p. 16

F Additional recoverability bounds

This section presents two additional entropy inequalities derived from the multivariate Golden–Thompson extension, including a displayed relative-entropy bound and an extension to trace-preserving completely positive maps.

F Additional recoverability bounds

  • D2(ρ∥σ) is defined as log tr ρ^2σ^-1 and identified as Petz’s Rényi relative entropy of order 2.Source: F Additional recoverability bounds · p. 16
  • The Peierls–Bogoliubov inequality used in the derivation follows by applying Klein’s inequality with the exponential function.The passage gives the choice of Hermitian matrices used to obtain it.Source: F Additional recoverability bounds · p. 17
  • In the proof of (99), the penultimate step uses the extension of the Golden–Thompson inequality from Corollary 3.3 with n = 4 and p = 1.The proof assumes positive semidefinite ρAB and σAB with ρAB ≪ σAB and tr ρAB = 1.Source: F Additional recoverability bounds · p. 17
  • D(ρAB∥σAB) − D(ρA∥σA) ≤ log tr exp(2 log ρAB + log σA − log ρA − log σAB) is obtained using Corollary 3.3 with n = 4 and p = 2.The passage states that this is the second inequality in the derivation.Source: F Additional recoverability bounds · p. 17
  • The argument for (99) extends to arbitrary trace-preserving completely positive maps and reproduces a result in [35, Section 3].Source: F Additional recoverability bounds · p. 17

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