Sutter, Berta, and Tomamichel asked whether the averaging density \(\beta_0\) in their three-matrix trace inequality could be replaced by another probability distribution independent of the matrices. We show that it cannot. In every fixed dimension \(d\geq2\), if a Borel probability measure \(\mu\) satisfies the averaged inequality for all \(d\times d\) Hermitian triples, then \(\mathrm d\mu(t)=\beta_0(t)\,\mathrm d t\). The full rigidity is already detected in dimension two. No moment or regularity assumption is needed. The proof combines an exact, dimension-free Fourier representation of the averaged term with a second-order perturbation of the commuting configuration \(A=C=-B\), which determines the characteristic function of \(\mu\) at every frequency.
Sutter, Berta, and Tomamichel asked whether the averaging density (\beta_0) in their three-matrix trace inequality could be replaced by another probability distribution independent of the matrices. We show that it cannot. In every fixed dimension (d\geq2), if a Borel probability measure (\mu) satisfies the averaged inequality for all (d\times d) Hermitian triples, then (\mathrm d\mu(t)=\beta_0(t),\mathrm d t). The full rigidity is already detected in dimension two. No moment or regularity assumption is needed. The proof combines an exact, dimension-free Fourier representation of the averaged term with a second-order perturbation of the commuting configuration (A=C=-B), which determines the characteristic function of (\mu) at every frequency.
We consider a family of multivariate trace inequalities recently derived by Sutter, Berta, and Tomamichel. These inequalities generalize the Golden-Thompson inequality and Lieb’s triple matrix inequality to an arbitrary number of matrices in a way that features complex matrix powers (i.e., certain unitaries). We show that their inequalities can be rewritten as an n-matrix generalization of Lieb’s original triple matrix inequality. The complex matrix powers are replaced by resolvents and appropriate maximally entangled states. We expect that the technically advantageous properties of resolvents, in particular for perturbation theory, can be of use in applications of the n-matrix inequalities, e.g., for analyzing the performance of the rotated Petz recovery map in quantum information theory and for removing the unitaries altogether.
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.
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.
We study continuous-time quantum walks on connected strongly regular graphs that are not complete, observed at a random time drawn from a freely chosen probability law. Uniform average mixing (UAM) asks for a law under which every averaged transition probability equals 1/n, where n is the number of vertices. On a strongly regular graph this is equivalent to two affine constraints on three cosine moments. We construct a bounded, compactly supported time density for every strongly regular graph with nonintegral eigenvalues. For integral spectra we give an exact finite Toeplitz criterion and its Hankel form. Every averaged mixing matrix of such a graph is realized by at most two observation times. Three elementary inequalities on the moment line, which also give a short proof of Chan's classification of complex Hadamard matrices in the Bose-Mesner algebra, lead to a determination of all strongly regular graphs that admit UAM. Apart from the conference graphs of nonsquare order and the graphs with instantaneous uniform mixing, these are the members of two infinite families of parameter sets and their complements, and for them we give explicit laws with two observation times. The Petersen graph and its complement are the smallest members. A strongly regular graph with UAM admits a bounded time density exactly when it has no instantaneous uniform mixing. We also correct the classification of instantaneous uniform mixing on strongly regular graphs by Godsil, Mullin and Roy. Its sign condition excludes the halved 5-cube, which mixes uniformly at time π/4. With the order 4θ2 read literally, its parity condition also excludes the Clebsch graph and includes the parameters (36,14,4,6), for which no time law gives uniform average mixing.
We investigate the uniqueness of local quadratic probability densities associated with complex-valued smooth functions of compact support on Rd. Consider finite-order local Hermitian quadratic densities constructed from the function and its derivatives. Under assumptions of pointwise non-negativity, finite differential order, locality, and universal normalization with respect to the standard L2 norm, we establish a uniqueness result identifying the density with ∣ψ(x)∣2 almost everywhere. The argument combines positivity of the coefficient matrix, high-frequency localized test functions, and elimination of derivative-dependent terms. We also discuss the role of the assumptions and the limitations of the result. The theorem concerns uniqueness within the specified class of local quadratic densities; it does not by itself derive the Born rule from more general physical principles or provide a theory of quantum measurement dynamics.
Standard finite quantum de Finetti theorems approximate the k-system marginals of permutation-invariant states of n-systems by mixtures of independent and identically distributed (iid) states, usually in trace distance. We prove both standard and Renner's exponential de Finetti theorems in the stronger form of operator inequalities, implying bounds in every Schatten norm and for every quantum Rényi divergence satisfying data processing. In the standard case, for fixed local dimension, our k/n error bound in max-relative entropy improves on the previously best known k2/n scaling, even in the classical setting. The operator-inequality approach is particularly suited to study channel de Finetti representations because operator order between Choi states is equivalent to completely positive (CP) order between the underlying channels. For permutation-covariant channels N(n):A⊗n→B⊗n, where dA=dimA, we prove that the k-system reduced channel is CP-dominated by a mixture of tensor-power channels with error O(k/n) and polynomial dependence on the local dimensions, addressing a question raised by Berta et al. [Math. Program. 194, 781-829 (2022)]. Under the no-signalling condition, we also prove an exponential channel de Finetti theorem where the approximating mixture consists of Choi-almost-iid channels, whose normalized Choi states are almost-iid in the sense of Mazzola-Sutter-Renner. In the case of r defects, the representation error is at most poly(n)(2dA4k3/(nr2))(r+1)/2 and decays exponentially in n for a suitable choice of parameters r and k.