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arXiv:1107.1047 · Journal of Physics A Mathematical and Theoretical

Induced metric and matrix inequalities on unitary matrices

H. F. Chau, Chi-Kwong Li, Yiu‐Tung Poon, Nung-Sing Sze

Published Feb 16, 2012• 2 citations
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Abstract

Recently, Chau [Quant. Inform. & Comp. 11, 721 (2011)] showed that one can define certain metrics and pseudo-metrics on U(n), the group of all $n\times n$ unitary matrices, based on the arguments of the eigenvalues of the unitary matrices. More importantly, these metrics and pseudo-metrics have quantum information theoretical meanings. So it is instructive to study this kind of metrics and pseudo-metrics on U(n). Here we show that any symmetric norm on ${\mathbb R}^n$ induces a metric on U(n). Furthermore, using the same technique, we prove an inequality concerning the eigenvalues of a product of two unitary matrices which generalizes a few inequalities obtained earlier by Chau [arXiv:1006.3614v1].

DOI
10.1088/1751-8113/45/9/095201

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Recently, Chau [Quant. Inform. & Comp. 11, 721 (2011)] showed that one can define certain metrics and pseudo-metrics on U(n), the group of all n×nn\times nn×n unitary matrices, based on the arguments of the eigenvalues of the unitary matrices. More importantly, these metrics and pseudo-metrics have quantum information theoretical meanings. So it is instructive to study this kind of metrics and pseudo-metrics on U(n). Here we show that any symmetric norm on Rn{\mathbb R}^nRn induces a metric on U(n). Furthermore, using the same technique, we prove an inequality concerning the eigenvalues of a product of two unitary matrices which generalizes a few inequalities obtained earlier by Chau [arXiv:1006.3614v1].
Recently, Chau [Quant. Inform. & Comp. 11, 721 (2011)] found a family of metrics and pseudo-metrics on nnn-dimensional unitary operators that can be interpreted as the minimum resources (given by certain tight quantum speed limit bounds) needed to transform one unitary operator to another. This result is closely related to the weighted ℓ1\ell^1ℓ1-norm on Rn{\mathbb R}^nRn. Here we generalize this finding by showing that every weighted ℓp\ell^pℓp-norm on Rn{\mathbb R}^nRn with $1\le p \le \limitingp$ induces a metric and a pseudo-metric on nnn-dimensional unitary operators with quantum information-theoretic meanings related to certain tight quantum speed limit bounds. Besides, we investigate how far the correspondence between the existence of metrics and pseudo-metrics of this type and the quantum speed limits can go.
A general strategy is provided to identify the most general metric for diagonalizable pseudo-Hermitian and anti-pseudo-Hermitian Hamilton operators represented by two-dimensional matrices. It is investigated how a permutation of the eigen-values of the Hamilton operator in the process of its diagonalization influences the metric and how this permutation equivalence affects energy eigen-values. We try to understand on one hand, how the metric depends on the normalization of the chosen left and right eigen-basis of the matrix representing the diagonalizable pseudo-Hermitian or anti-pseudo-Hermitian Hamilton operator, on the other hand, whether there has to exist a positive semi-definite metric required to set up a meaningful Quantum Theory even for non-Hermitian Hamilton operators of this type. Using our general strategy we determine the metric with respect to the two elements of the two-dimensional permutation group for various topical examples of matrices representing two-dimensional Hamilton operators found in the literature assuming on one hand pseudo-Hermiticity, on the other hand anti-pseudo-Hermiticity. The (unnecessary) constraint inferred by C. M. Bender and collegues that the C{\cal C}C-operator of PT{\cal PT}PT-symmetric Quantum Theory should be an involution (C2=1{\cal C}^2=1C2=1) is shown - in the unbroken phase of PT{\cal PT}PT-symmetry - to require the Hamilton operator to be symmetric. This inconvenient restriction had been already - with hesitation - noted by M. Znojil and H. B. Geyer in 2006 (arXiv:quant-ph/0607104). A Hamilton operator proposed by T. D. Lee and C. G. Wick is used to outline implications of the formalism to higher dimensional Hamilton operators.
The aim of the manuscript is to characterize monotone `metric' in the space of Markov map. Here, `metric' means the square of the norm defined on the tangent space, and not necessarily induced from an inner product (this property hereafter will be called inner-product-assumption), different from usual metric used in differential geometry. As for metrics in So far, there have been plenty of literatures on the metric in the space of probability distributions and quantum states. Among them, Cencov proved the monotone metric in probability distribution space is unique up to constant multiple, and identical to Fisher information metric. Petz characterized all the monotone metrics in the quantum state space using operator mean. As for channels, however, only a little had been known. In this paper, we impose monotonicity by concatenation of channels before and after the given channel families, and invariance by tensoring identity channels. (Notably, we do not use the inner-product-assumption.) To obtain this result, `resource conversion' technique, which is widely used in quantum information, is used. We consider distillation from and formation to a family of channels. Under these axioms, we identify the largest and the smallest `metrics'. Interestingly, they are not induced from any inner product, i.e., not a metric. Indeed, one can prove that any `metric' satisfying our axioms can not be a metric. This result has some impact on the axiomatic study of the monotone metric in the space of classical and quantum states, since both conventional theory relies on the inner-product-assumption. Also, we compute the lower and the upper bound for some concrete examples.