[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

Cornell hotel and restaurant administration quarterly

a ) p. 131 The discussion between eqs. (5.14) and (5.15) is incorrect (dA should be made as large as possible!). b ) p. 256 In the figure, the numbers 6) and 7) occur twice. c ) p. 292 At the end of section 12.5, it should be the space of isospectral 0Hermitian matrices. d ) p. 306 A ”Tr” is missing in eq. (13.43). e ) p. 327, Eq. (14.64b) is 〈Trρ〉B = N(14N+10) (5N+1)(N+3) should be 〈Trρ〉B = 8N+7 (N+2)(N+4)

Geometry of Quantum States: Frontmatter

https://chaos.if.uj.edu.pl/~karol/erratum10.pdf

Geometry of Quantum States

Geometry of quantum states: an introduction to quantum entanglement by Ingemar Bengtsson and Karol Zyczkowski

Let A=(A1, . . ., Am) be an m-tuple of n × n Hermitian matrices. For $1 \le k \le n$, the $k${\rm th} joint numerical range of A is defined by $$W_k(A) = \{ ({\rm \tr}(X^*A_1X), \dots, {\rm \tr}(X^*A_mX) ): X \in {\bf C}^{n\times k}, X^*X = I_k \}.$$ We consider linearly independent families of Hermitian matrices {A1, . . . , Am} so that Wk(A) is convex. It is shown that m can reach the upper bound 2k(n-k)+1. A key idea in our study is relating the convexity of Wk(A) to the problem of constructing rank k orthogonal projections under linear constraints determined by A. The techniques are extended to study the convexity of other generalized numerical ranges and the corresponding matrix construction problems.

/pdf/convexity-of-the-joint-numerical-range-33sk0ug53w.pdf

Convexity of the Joint Numerical Range

We characterize the relationship between the singular values of a Hermitian (resp., real symmetric, complex symmetric) matrix and the singular values of its off-diagonal block. We also characterize the eigenvalues of a Hermitian (or real symmetric) matrix C = A + B in terms of the combined list of eigenvalues of A and B . The answers are given by Horn-type linear inequalities. The proofs depend on a new inequality among Littlewood-Richardson coefficients.

Eigenvalues, singular values, and Littlewood-Richardson coefficients

Let be the set of all n × n unitary matrices and A, C two n × n complex matrices with C hermitian. In this note, the author gives another proof of the following result: the set is convex.

Another proof of a result of Westwick

It is shown that the rank-k numerical range of every n-by-n complex matrix is non-empty if k < n/3 + 1. The proof is based on a recent characterization of the rank-k numerical range by Li and Sze, the Helly's theorem on compact convex sets, and some eigenvalue inequalities. In particular, the result implies that rank-2 numerical range is non-empty if n ≥ 4. This confirms a conjecture of Choi et al. If k ≥ n/3 + 1, an n-by-n complex matrix is given for which the rank-k numerical range is empty. An extension of the result of bounded linear operators acting on an infinite dimensional Hilbert space is also discussed.

Condition for the higher rank numerical range to be non-empty

We show that the linear group of automorphism of Hermitian matrices which preserves the set of separable states is generated by natural automorphisms: change of an orthonormal basis in each tensor factor, partial transpose in each tensor factor, and interchanging two tensor factors of the same dimension. We apply our results to the preservers of the product numerical range.

/pdf/the-automorphism-group-of-separable-states-in-quantum-4mjmxsfyx6.pdf

Yiu-Tung Poon

Papers

Convexity of the Joint Numerical Range

Eigenvalues, singular values, and Littlewood-Richardson coefficients

Another proof of a result of Westwick

Condition for the higher rank numerical range to be non-empty

The automorphism group of separable states in quantum information theory