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arXiv:2601.10943

The Hilbert-Schmidt norms of quantum channels and matrix integrals over the unit sphere

Yuan Li, Zhengli Chen, Zhihua Guo, Yongfeng Pang

quant-phmath-phSubmitted Jan 16, 2026• Announced Jan 19, 2026• 0 citations
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Abstract

The dynamics of quantum systems are generally described by a family of quantum channels (linear, completely positive and trace preserving maps). In this note, we mainly study the range of all possible values of $\|\mathcal{E}\|_2^2+\|\widetilde{\mathcal{E}}\|_2^2$ for quantum channels $\mathcal{E}$ and give the equivalent characterizations for quantum channels that achieve these maximum and minimum values, respectively, where $\|\mathcal{E}\|_2$ is the Hilbert-Schmidt norm of $\mathcal{E}$ and $\widetilde{\mathcal{E}}$ is a complementary channel of $\mathcal{E}.$ Also, we get a concrete description of completely positive maps on infinite dimensional systems preserving pure states. Moreover, the equivalency of several matrix integrals over the unit sphere is demonstrated and some extensions of these matrix integrals are obtained.

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•Jul 1, 2019•7 citations
The dynamics of quantum systems are generally described by a family of quantum channels (linear, completely positive and trace preserving maps). In this note, we mainly study the range of all possible values of ∥E∥22+∥E~∥22\|\mathcal{E}\|_2^2+\|\widetilde{\mathcal{E}}\|_2^2∥E∥22​+∥E∥22​ for quantum channels E\mathcal{E}E and give the equivalent characterizations for quantum channels that achieve these maximum and minimum values, respectively, where ∥E∥2\|\mathcal{E}\|_2∥E∥2​ is the Hilbert-Schmidt norm of E\mathcal{E}E and E~\widetilde{\mathcal{E}}E is a complementary channel of E.\mathcal{E}.E. Also, we get a concrete description of completely positive maps on infinite dimensional systems preserving pure states. Moreover, the equivalency of several matrix integrals over the unit sphere is demonstrated and some extensions of these matrix integrals are obtained.
There are various notions of positivity for matrices and linear matrix-valued maps that play important roles in quantum information theory. The cones of positive semidefinite matrices and completely positive linear maps, which represent quantum states and quantum channels respectively, are the most ubiquitous positive cones. There are also many natural cones that can been regarded as "more" or "less" positive than these standard examples. In particular, entanglement theory deals with the cones of separable operators and entanglement witnesses, which satisfy very strong and weak positivity properties respectively. Rather complementary to the various cones that arise in entanglement theory are norms. The trace and operator norms for operators and the diamond and completely bounded norms for superoperators are the typical norms that are seen throughout quantum information theory. In this work our main goal is to develop a family of norms that play a role analogous to the cone of entanglement witnesses. We investigate the basic mathematical properties of these norms, including their relationships with other well-known norms, their isometry groups, and their dual norms. We also make the place of these norms in entanglement theory rigorous by showing that entanglement witnesses arise from minimal operator systems, and analogously our norms arise from minimal operator spaces. Finally, we connect the various cones and norms considered here to several seemingly unrelated problems from other areas. We characterize the problem of whether or not non-positive partial transpose bound entangled states exist in terms of one of our norms, and provide evidence in favour of their existence. We also characterize the minimum gate fidelity of a quantum channel, the maximum output purity and its completely bounded counterpart, and the geometric measure of entanglement in terms of these norms.
In this paper we study quantum dynamical mappings called also quantum processes.The set of values of such mapping is a one-parameter family of completely positive trace-preserving linear operators defined on a finite-dimensional Hilbert space.In quantum information theory such operators are referred to as quantum channels.An important concept for quantum dynamical mappings is their divisibility.There are different types of this concept.The present paper deals with so-called completely positive divisible quantum processes.For two such processes, which are bijective and satisfy a commutativity condition, we construct a compound quantum process.It is shown that this compound quantum process is also completely positive divisible.Endowing a set of quantum channels with the norm topology, we consider continuous quantum processes and continuous completely positive evolutions.The latter are defined as two-parameter families of quantum channels satisfying additional properties.We prove that a continuous bijective completely positive divisible quantum process generates a continuous completely positive evolution.In order to illustrate the considered concepts and the results on them, we provide examples of quantum dynamical mappings with values in the set of qubit channels.In particular, a completely positive divisible compound quantum process is constructed for two bijective commuting quantum processes.Geometric and physical interpretations of this compound quantum process are given.
Abstract In this paper, we study the multiplicative behaviour of quantum channels, mathematically described by trace preserving, completely positive maps on matrix algebras. It turns out that the multiplicative domain of a unital quantum channel has a close connection to its spectral properties. A structure theorem (theorem 2.5), which reveals the automorphic property of an arbitrary unital quantum channel on a subalgebra, is presented. Various classes of quantum channels (irreducible, primitive, etc) are then analysed in terms of this stabilising subalgebra. The notion of the multiplicative index of a unital quantum channel is introduced, which measures the number of times a unital channel needs to be composed with itself for the multiplicative algebra to stabilise. We show that the maps that have trivial multiplicative domains are dense in completely bounded norm topology in the set of all unital completely positive maps. Some applications in quantum information theory are discussed.
A linear map LLL from Cn×n{\mathbb C}^{n \times n}Cn×n into Cn×n{\mathbb C}^{n \times n}Cn×n is called a quantum channel if it is completely positive and trace preserving. The set Ln{\cal L}_nLn​ of all such quantum channels is known to be a compact convex set. While the extreme points of Ln{\cal L}_nLn​ can be characterized, not much is known about the structure of its higher dimensional faces. Using the so called Choi matrix Z(L)Z(L)Z(L) associated with the quantum channel LLL, we compute the maximum dimension of a proper face of Ln{\cal L}_nLn​, and in addition the possible dimensions of faces generated by LLL when $rank \ Z(L)=2 $.
As we knew, a mathematical formalism of a quantum mechanics says that any quantum system is identified with some finite- or infinite-dimensional Hilbert space; pure states correspond to vectors of norm 1; observables are self-adjoint operators on the space of states. Thus the set of all pure states corresponds to the unit sphere in the Hilbert space [1-2]. It is of interest to describe all linear or nonlinear operators which preserve the pure states of the system. In the linear case, it is nothing more than isometries of Hilbert spaces [1]. In the nonlinear case, this problem was open. In this paper we shall describe all quadratic plus linear operators which preserve pure states of the quantum system.!
In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi’s characterization for completely positive maps between pairs of linear operators on finite dimensional Hilbert spaces. We apply our conditions to a completely positive map between two trace class operators on separable Hilbert spaces. A completely positive map μ is called a quantum channel, if it is trace preserving, and μ is called a quantum subchannel if it decreases the trace of a positive operator.We give simple neccesary and sufficient condtions for μ to be a quantum subchannel.We show that μ is a quantum subchannel if and only if it hasHellwig-Kraus representation. The last result extends the classical results of Kraus and the recent result of Holevo for characterization of a quantum channel.