Higher-rank numerical ranges of unitary and normal matrices

2014

2010

Numerical range of a Hermitian operator X is defined as the set of all possible expectation values of this observable among a normalized quantum state. We analyze a modification of this definition in which the expectation value is taken among a certain subset of the set of all quantum states. One considers for instance the set of real states, the set of product states, separable states, or the set of maximally entangled states. We show exemplary applications of these algebraic tools in the theory of quantum information: analysis of k-positive maps and entanglement witnesses, as well as study of the minimal output entropy of a quantum channel.

Journal of Mathematical Analysis and Applications, 2008

For a positive integer k, the rank-k numerical range Λ k (A) of an operator A acting on a Hilbert space H of dimension at least k is the set of scalars λ such that P AP = λP for some rank k orthogonal projection P . In this paper, a close connection between low rank perturbation of an operator A and Λ k (A) is established. In particular, for 1 ≤ r < k it is shown that Λ k (A) ⊆ Λ k−r (A + F ) for any operator F with rank (F ) ≤ r. In quantum computing, this result implies that a quantum channel with a k-dimensional error correcting code under a perturbation of rank ≤ r will still have a (k − r)dimensional error correcting code. Moreover, it is shown that if A is normal or if the dimension of A is finite, then Λ k (A) can be obtained as the intersection of Λ k−r (A + F ) for a collection of rank r operators F . Examples are given to show that the result fails if A is a general operator. The closure and the interior of the convex set Λ k (A) are completely determined. Analogous results are obtained for Λ∞(A) defined as the set of scalars λ such that P AP = λP for an infinite rank orthogonal projection P . It is shown that Λ∞(A) is the intersection of all Λ k (A) for k = 1, 2, . . . . If A − µI is not compact for any µ ∈ C, then the closure and the interior of Λ∞(A) coincide with those of the essential numerical range of A. The situation for the special case when A − µI is compact for some µ ∈ C is also studied.

Linear Algebra and its Applications, 2013

We associate with a k-tuple of hermitian N × N matrices a probability measure on R k supported on their joint numerical range: The joint numerical shadow of these matrices. When k = 2 we recover the numerical range and the numerical shadow of the complex matrix corresponding to a pair of hermitian matrices. We apply this material to the theory of quantum information. Thus, we show that quantum maps on the set of quantum states defined by Kraus operators satisfying the identity resolution assumption shrink joint numerical ranges.

Annals of Functional Analysis, 2015

In this paper, the notion of rank−k numerical range of rectangular complex matrix polynomials are introduced. Some algebraic and geometrical properties are investigated. Moreover, for ϵ > 0, the notion of Birkhoff-James approximate orthogonality sets for ϵ−higher rank numerical ranges of rectangular matrix polynomials is also introduced and studied. The proposed definitions yield a natural generalization of the standard higher rank numerical ranges.

Linear Algebra and its Applications, 2006

We consider higher-rank versions of the standard numerical range for matrices. A central motivation for this investigation comes from quantum error correction. We develop the basic structure theory for the higher-rank numerical ranges, and give a complete description in the Hermitian case. We also consider associated projection compression problems.

2007

Results on matrix canonical forms are used to give a complete description of the higher rank numerical range of matrices arising from the study of quantum error correction. It is shown that the set can be obtained as the intersection of closed half planes (of complex numbers). As a result, it is always a convex set in C. Moreover, the higher rank numerical range of a normal matrix is a convex polygon determined by the eigenvalues. These two consequences confirm the conjectures of Choi et al. on the subject. In addition, the results are used to derive a formula for the optimal upper bound for the dimension of a totally isotropic subspace of a square matrix, and verify the solvability of certain matrix equations.

Reports on Mathematical Physics, 2006

We solve the fundamental quantum error correction problem for bi-unitary channels on two-qubit Hilbert space. By solving an algebraic compression problem, we construct qubit codes for such channels on arbitrary dimension Hilbert space, and identify correctable codes for Pauli-error models not obtained by the stabilizer formalism. This is accomplished through an application of a new tool for error correction in quantum computing called the "higher-rank numerical range". We describe its basic properties and discuss possible further applications.

Linear and Multilinear Algebra, 1996

Let A, C be n × n complex matrices. We prove in the affirmative the conjecture that the C-numerical range of A, defined by W C (A) = {tr (CU * AU) : U is unitary} , is always star-shaped with respect to star-center (tr A)(tr C)/n. This result is equivalent to that the image of the unitary orbit {U * AU : U is unitary} of A under any complex linear functional is always star-shaped.

submitted to Springer- …, 2004

arXiv (Cornell University), 2023

We establish rigorous connections between quantum circuit complexity and approximate quantum error correction (AQEC) capability, two properties of fundamental importance to the physics and practical use of quantum many-body systems, covering both all-to-all and geometric scenarios like lattice systems in finite spatial dimensions. To this end, we introduce a type of code parameter that we call subsystem variance, which is closely related to the optimal AQEC precision. Our key finding is that if the subsystem variance is below an O(k/n) threshold then any state in the code subspace must obey certain circuit complexity lower bounds, which identify nontrivial "phases" of codes. Based on our results, we propose O(k/n) as a boundary between subspaces that should and should not count as AQEC codes. This theory of AQEC provides a versatile framework for understanding the quantum complexity and order of many-body quantum systems, offering new insights for wide-ranging physical scenarios, in particular topological order and critical quantum systems which are of outstanding importance in many-body and high energy physics. We observe from various different perspectives that roughly O(1/n) represents a common, physically significant "scaling threshold" of subsystem variance for features associated with nontrivial quantum order.

World Academy of Science, Engineering and Technology, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, 2015

In this paper, the notion of rank−k numerical range of rectangular complex matrix polynomials are introduced. Some algebraic and geometrical properties are investigated. Moreover, for &gt; 0, the notion of Birkhoff-James approximate orthogonality sets for −higher rank numerical ranges of rectangular matrix polynomials is also introduced and studied. The proposed definitions yield a natural generalization of the standard higher rank numerical ranges. Keywords—Rank−k numerical range, isometry, numerical range, rectangular matrix polynomials.

Electronic Journal of Linear Algebra, 2018

The maximal numerical range W0(A) of a matrix A is the (regular) numerical range W (B) of its compression B onto the eigenspace L of A*A corresponding to its maximal eigenvalue. So, always W0(A) ⊆ W (A). Conditions under which W0(A) has a non-empty intersection with the boundary of W (A) are established, in particular, when W0(A) = W(A). The set W0(A) is also described explicitly for matrices unitarily similar to direct sums of 2-by-2 blocks, and some insight into the behavior of W0(A) is provided when L has codimension one.

2021

We describe here the higher rank numerical range, as defined by Choi, Kribs and Życzkowski, of a normal operator on an infinite dimensional Hilbert space in terms of its spectral measure. This generalizes a result of Avendaño for self-adjoint operators. An analogous description of the numerical range of a normal operator by Durszt is derived for the higher rank numerical range as an immediate consequence. It has several interesting applications. We show using Durszt’s example that there exists a normal contraction T for which the intersection of the higher rank numerical ranges of all unitary dilations of T contains the higher rank numerical range of T as a proper subset. Finally, we strengthen and generalize a result of Wu by providing a necessary and sufficient condition for the higher rank numerical range of a normal contraction being equal to the intersection of the higher rank numerical ranges of all possible unitary dilations of it.

IEEE Transactions on Information Theory, 2000

In Part I of this paper we formulated the problem of error detection with quantum codes on the completely depolarized channel and gave an expression for the probability of undetected error via the weight enumerators of the code. In this part we show that there exist quantum codes whose probability of undetected error falls exponentially with the length of the code and derive bounds on this exponent. The lower (existence) bound is proved for stabilizer codes by the counting argument for classical self-orthogonal quaternary codes. Upper bounds are proved by linear programming. First we formulate two linear programming problems that are convenient for the analysis of specific short codes. Next we give a relaxed formulation of the problem in terms of optimization on the cone of polynomials in the Krawtchouk basis. We present two general solutions of the problem. Together they give an upper bound on the exponent of undetected error. The upper and lower asymptotic bounds coincide for a certain interval of code rates close to 1.