SIAM Journal on Matrix Analysis and Applications

A Lower Bound on the C-Numerical Radius of Nilpotent Matrices Appearing in Coherent Spectroscopy

Chi-Kwong Li, Hugo J. Woerdeman

Published Jan 1, 2005• 9 citations

Abstract

We provide a lower bound for the efficiency of polarization or coherence transfer between quantized states under unitary transformations. Mathematically the problem is the determination of the C-numerical radius of A for certain nilpotent matrices C and A. The presented lower bound is conjectured to be exact as it coincides with numerical data provided in [U. Helmke et al., J. Global Optim., 23 (2002), pp. 283-308].

DOI: 10.1137/040613330

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SIAM Journal on Matrix Analysis and Applications

A Lower Bound on the C-Numerical Radius of Nilpotent Matrices Appearing in Coherent Spectroscopy

Chi-Kwong Li, Hugo J. Woerdeman

Published Jan 1, 2005• 9 citations

Publisher

Abstract

DOI: 10.1137/040613330

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We investigate the geometric structure of the set of spectra of bipartite absolute separable states (

\mathrm{ASEP}_{m,n}

) and absolute positive partial transpose states (

\mathrm{APPT}_{m,n}

), \emph{i.e.}, bipartite quantum states that remain separable or PPT, respectively, under all global unitary transformations. First, we establish general geometric properties of absolute convex sets of matrices, their spectra and extreme points. Regarding absolute separability, we present a permutation-symmetric reformulation of the absolute PPT criterion and use it to demonstrate that

\mathrm{APPT}_{m,n}

is a spectrahedron for all

m\leq n

: in particular, all its faces are exposed. While

\mathrm{ASEP}_{2,n}

is a spectrahedron, too, for

\mathrm{ASEP}_{m,n}

we can in general only show that it is a semialgebraic set for all

m,n\geq 3

. Furthermore, we provide a complete characterization of the faces and extreme points of

\mathrm{APPT}_{m,n}

and demonstrate that the dimension of a face is determined by the rank of a certain matrix, with maximal proper faces having dimension

mn-m-1

. In the quantitative setting, we provide a rigorous lower bound on the maximal attainable purity of

\mathrm{APPT}_{m,n}

via an inscribed polytope

\mathcal{P}_{m,n}

and conjecture that the maximal purity of

\mathrm{APPT}_{m,n}

(along with its spectra) coincides with the polytope for arbitrary dimensions except when

m=n=2

. Additionally, we also provide a rigorous upper bound on the minimal von Neumann entropy of

\mathrm{APPT}_{m,n}

and demonstrate numerically that the minimum entropy eventually coincides with the polytope

\mathcal{P}_{m,n}

as the local system dimension

n

increases. Finally, we show that the relative spectral volume of

\mathrm{APPT}_{m,n}

decays exponentially in

n

by a constant multiplicative factor of the relative volume of the inscribed polytope

\mathcal{P}_{m,n}

We propose candidate lower bounds on the real Grothendieck constants

K_{\mathrm G}(d)

and their infinite-dimensional limit

K_{\mathrm G}

using rotationally invariant cubic kernels. For the continuous construction, we identify an explicit threshold above which hemispherical binary strategies are unstable under cubic boundary deformations. Supported by numerical evidence, we conjecture that hemispherical strategies are globally optimal at this threshold. This conjecture would imply

K_{\mathrm G}\geq9π/16\simeq1.767146

, improving the rigorous lower bound

6π/11\simeq1.713596

. We also construct finite symmetric coefficient matrices from unit vectors in dimensions up to

24

. Large-scale see-saw optimization supports the continuous predictions and suggests improved lower bounds on

K_{\mathrm G}(d)

in the studied dimensions from

4

24

, conditional on the conjectured binary optima being exact. Selected matrices also yield finite-setting candidate dimension witnesses when interpreted as bipartite correlation Bell functionals. If the corresponding binary upper bounds hold, these witnesses would certify a local Hilbert-space dimension of at least five for each party.