With techniques borrowed from quantum information theory, we develop a method to systematically obtain operator inequalities and identities in several matrix variables. These take the form of trace polynomials: polynomial-like expressions that involve matrix monomials $X_{α_1} \cdots X_{α_r}$ and their traces $\operatorname{tr}(X_{α_1} \cdots X_{α_r})$. Our method rests on translating the action of the symmetric group on tensor product spaces into that of matrix multiplication. As a result, we extend the polarized Cayley-Hamilton identity to an operator inequality on the positive cone, characterize the set of multilinear equivariant positive maps in terms of Werner state witnesses, and construct permutation polynomials and tensor polynomial identities on tensor product spaces. We give connections to concepts in quantum information theory and invariant theory.
Quantum monotone metric was introduced by Petz,and it was proved that quantum monotone metrics on the set of quantum states with trace one were characterized by operator monotone functions. Later, these were extended to monotone metrics on the set of positive operators whose traces are not always one based on completely positive, trace preserving (CPTP) maps. It was shown that these extended monotone metrics were characterized by operator monotone functions continuously parameterized by traces of positive operators,and did not have some ideal properties such as monotonicity and convexity with …
We review single-qubit quantum process tomography for trace-preserving and nontrace-preserving processes, and derive explicit forms of the general constraints for fitting experimental data. These forms provide additional insight into the structure of the process matrix as well as reveal a tighter bound on the trace of a nontrace-preserving process than has been previously stated. We also describe, for completeness, how to incorporate measured imperfect input states.
We prove the linearity and injectivity of two maps $ϕ_1$ and $ϕ_2$ on certain subsets of $M_n$ that satisfy $\operatorname{tr}(ϕ_1(A)ϕ_2(B))=\operatorname{tr}(AB)$. We apply it to characterize maps $ϕ_i:\mathcal{S}\to \mathcal{S}$ ($i=1, \ldots, m$) satisfying $$\operatorname{tr} (ϕ_1(A_1)\cdots ϕ_m(A_m))=\operatorname{tr} (A_1\cdots A_m)$$ in which $\mathcal{S}$ is the set of $n$-by-$n$ general, Hermitian, or symmetric matrices for $m\ge 3$, or positive definite or diagonal matrices for $m\ge 2$. The real versions are also given.
We analyze the geometrical properties of trace-preserving Pauli maps. Using the Choi-Jamio\lkowski isomorphism, we express the Hilbert-Schmidt line and volume elements in terms of the eigenvalues of the Pauli map. We analytically compute the volumes of positive, trace-preserving Pauli maps and Pauli channels. In particular, we find the relative volumes of the entanglement breaking Pauli channels, as well as the channels that can be generated by a time-local generator. Finally, we show what part of the Pauli channels are P and CP-divisible, which is related to the notion of Markovianity.
We introduce and study two new classes of unital quantum channels. The first class describes a 2-parameter family of channels given by completely positive (CP) maps $M_3({\bf C}) \mapsto M_3({\bf C})$ which are both unital and trace-preserving. Almost every member of this family is factorizable and extreme in the set of CP maps which are both unital and trace-preserving, but is not extreme in either the set of unital CP maps or the set of trace-preserving CP maps. We also study a large class of maps which generalize the Werner-Holevo channel for $d = 3$ in the sense that they are defined in te…
Trace decreasing quantum operations naturally emerge in experiments involving postselection. However, the experiments usually focus on dynamics of the conditional output states as if the dynamics were trace preserving. Here we show that this approach leads to incorrect conclusions about the dynamics divisibility, namely, one can observe an increase in the trace distance or the system-ancilla entanglement although the trace decreasing dynamics is completely positive divisible. We propose solutions to that problem and introduce proper indicators of the information backflow and the indivisibility…
We consider open quantum systems with factorized initial states, providing the structure of the reduced system dynamics, in terms of environment cumulants. We show that such completely positive (CP) and trace preserving (TP) maps can be unraveled by linear stochastic Schrödinger equations (SSEs) characterized by sets of colored stochastic processes (with n-th order cumulants). We obtain both the conditions such that the SSEs provide CPTP dynamics, and those for unraveling an open system dynamics. We then focus on Gaussian non-Markovian unravellings, whose known structure displays a functional …
Efficient implementation of arbitrary Hermitian-preserving and trace-preserving maps
University of Science and Technology of China（中国科学技术大学）；QICI Quantum Information and Computation Initiative, School of Computing and Data Science, The University of Hong Kong（香港大学计算与数据科学学院量子信息与计算倡议中心）；The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；Hefei National Laboratory（合肥国家实验室）；Tsinghua University（清华大学）
On the relevance of Hermitian-preserving trace-preserving linear maps in asymptotic information retrieval in quantum communication
Indian Institute of Technology Bhubaneswar（印度技术研究所布巴内斯瓦尔分校）；Korea Research Institute of Standards and Science（韩国标准科学研究院）；The Institute of Mathematical Sciences（数学科学研究院）；Homi Bhabha National Institute（霍米·巴巴国立研究所）
We derive the equations governing static, spherically symmetric vacuum solutions to the Einstein equations, as modified by the frame-dependent effective action arising from trace dynamics. We give analytic and numerical results for the solutions of these equations, first in polar coordinates, and then in isotropic coordinates. General features of the static case are that (i) there is no horizon, since $g_{00}$ is non-vanishing for finite values of the polar radius, and only vanishes (in isotropic coordinates) at the internal singularity, (ii) the Ricci scalar $R$ vanishes identically, and (iii…
Alternatively to the full reconstruction of an unknown quantum process, the so-called selective and efficient quantum process tomography (SEQPT) allows estimating, individually and up to the required accuracy, a given element of the matrix that describes such an operation with a polynomial amount of resources. The implementation of this protocol has been carried out with success to characterize the evolution of a quantum system that is well described by a trace preserving quantum map. Here, we deal with a more general type of quantum process that does not preserve the trace of the input quantu…
We define an entanglement witness in a composite quantum system as an observable having nonnegative expectation value in every separable state. Then a state is entangled if and only if it has a negative expectation value of some entanglement witness. Equivalent representations of entanglement witnesses are as nonnegative biquadratic forms or as positive linear maps of Hermitian matrices. As reported elsewhere, we have studied extremal entanglement witnesses in dimension $3\times 3$ by constructing numerical examples of generic extremal nonnegative forms. These are so complicated that we do not…
