Let $H$ be a positive semi-definite matrix partitioned in $β\times β$ Hermitian blocks, $H=[A_{s,t}]$ , $1\le s,t,\le β$ . Then, for all symmetric norms, {equation*} \| H \| \le \| \sum_{s=1}^β A_{s,s} \|. {equation*} The proof uses a nice decomposition for positive matrices and unitary congruences with the generators of a Clifford algebra. A few corollaries are given, in particular the partial trace operation increases norms of separable states on a real Hilbert space, leading to a conjecture for usual complex Hilbert spaces.Let $H$ be a positive semi-definite matrix partitioned in $β\times β$ Hermitian blocks, $H=[A_{s,t}]$, $1\le s,t,\le β$. Then, for all symmetric norms, {equation*} \| H \| \le \| \sum_{s=1}^β A_{s,s} \|. {equation*} The proof uses a nice decomposition for positive matrices and unitary congruences with the generators of a Clifford algebra. A few corollaries are given, in particular the partial trace operation increases norms of separable states on a real Hilbert space, leading to a conjecture for usual complex Hilbert spaces.

arXiv:1208.6494Aug 31, 20121 citation

Partial traces and entropy inequalities

Rajendra Bhatia

Linear Algebra and its ApplicationsMar 25, 200321 citations

Approximate pushforward designs and image bounds on approximations

Jakub Czartowski, Adam Sawicki, Karol Życzkowski

We extend the framework of quantum pushforward designs to the approximate setting, in which moment operators agree only up to finite precision. We first formulate a general transfer theorem for maps that act linearly at the level of moments. This separates direct applications, controlled by standard Schatten-norm estimates, from refinements that use additional tensorial structure. Dephasing is shown to be contractive and therefore sends an approximate projective design to a simplex design without increasing its error. Ordinary partial-trace estimates give immediate bounds for mixed-state and channel designs. We then prove a sharp Schatten-norm bound for the partial trace restricted to the totally symmetric subspace. This replaces the full-environment norm coefficient by one governed by a symmetric-subspace dimension and yields asymptotically tighter estimates for both mixed-state and channel designs. Numerical simulations for induced mixed-state designs are consistent with the resulting hierarchy of bounds.We extend the framework of quantum pushforward designs to the approximate setting, in which moment operators agree only up to finite precision. We first formulate a general transfer theorem for maps that act linearly at the level of moments. This separates direct applications, controlled by standard Schatten-norm estimates, from refinements that use additional tensorial structure. Dephasing is shown to be contractive and therefore sends an approximate projective design to a simplex design without increasing its error. Ordinary partial-trace estimates give immediate bounds for mixed-state and channel designs. We then prove a sharp Schatten-norm bound for the partial trace restricted to the totally symmetric subspace. This replaces the full-environment norm coefficient by one governed by a symmetric-subspace dimension and yields asymptotically tighter estimates for both mixed-state and channel designs. Numerical simulations for induced mixed-state designs are consistent with the resulting hierarchy of bounds.

quant-pharXiv:2512.01858Dec 2, 20250 citations

Partial trace relations beyond normal matrices

Pablo Costa Rico, Michael M. Wolf

We investigate the relationship between partial traces and their dilations for general complex matrices, focusing on two main aspects: the existence of (joint) dilations and norm inequalities relating partial traces and their dilations. Throughout our analysis, we pay particular attention to rank constraints. We find that every pair of matrices of equal size and trace admits dilations of any rank larger than one. We generalize Audenaert’s subadditivity inequality to encompass general matrices, multiple tensor factors, and different norms. A central ingredient for this is a novel majorization relation for Kronecker sums. As an application, we extend the interval of Werner states in which they are provably two-undistillable in any dimension d ≥ 4. We also prove new Schmidt-number witnesses and k-positive maps.We investigate the relationship between partial traces and their dilations for general complex matrices, focusing on two main aspects: the existence of (joint) dilations and norm inequalities relating partial traces and their dilations. Throughout our analysis, we pay particular attention to rank constraints. We find that every pair of matrices of equal size and trace admits dilations of any rank larger than one. We generalize Audenaert’s subadditivity inequality to encompass general matrices, multiple tensor factors, and different norms. A central ingredient for this is a novel majorization relation for Kronecker sums. As an application, we extend the interval of Werner states in which they are provably two-undistillable in any dimension d ≥ 4. We also prove new Schmidt-number witnesses and k-positive maps.

arXiv:2507.18278Journal of Mathematical PhysicsAug 1, 20260 citations

A Relation Between Completely Bounded Norms and Conjugate Channels

Anna Jenčová

Communications in Mathematical PhysicsMay 8, 200615 citations

Let

H

be a positive semi-definite matrix partitioned in

β\times β

Hermitian blocks,

H=[A_{s,t}]

1\le s,t,\le β

. Then, for all symmetric norms, {equation*} \| H \| \le \| \sum_{s=1}^β A_{s,s} \|. {equation*} The proof uses a nice decomposition for positive matrices and unitary congruences with the generators of a Clifford algebra. A few corollaries are given, in particular the partial trace operation increases norms of separable states on a real Hilbert space, leading to a conjecture for usual complex Hilbert spaces.