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arXiv:1709.03805 · Physical Review A

Convex approximations of quantum channels

Massimiliano Federico Sacchi, Tito Sacchi

Published Sep 8, 2017• 13 citations
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Abstract

We address the problem of optimally approximating the action of a desired and unavailable quantum channel $\mathrm{\ensuremath{\Phi}}$ having at our disposal a single use of a given set of other channels ${{\mathrm{\ensuremath{\Psi}}}_{i}}$. The problem is recast to look for the least distinguishable channel from $\mathrm{\ensuremath{\Phi}}$ among the convex set ${\ensuremath{\sum}}_{i}{p}_{i}{\mathrm{\ensuremath{\Psi}}}_{i}$, and the corresponding optimal weights ${{p}_{i}}$ provide the optimal convex mixing of the available channels ${{\mathrm{\ensuremath{\Psi}}}_{i}}$. For single-qubit channels we study specifically cases where the available convex set corresponds to covariant channels or to Pauli channels, and the desired target map is an arbitrary unitary transformation or a generalized damping channel.

DOI
10.1103/physreva.96.032311

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Comment on “Optimal convex approximations of quantum states”

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In a recent paper, M. F. Sacchi [Phys. Rev. A 96, 042325 (2017)] addressed the general problem of approximating an unavailable quantum state by the convex mixing of different available states. For the case of qubit mixed states, we show that the analytical solutions in some cases are invalid. In this Comment, we present complete analytical solutions for the optimal convex approximation. Our solutions can be viewed as correcting and supplementing the results in the aforementioned paper.

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We address the problem of optimally approximating the action of a desired and unavailable quantum channel \ensuremathΦ\mathrm{\ensuremath{\Phi}}\ensuremathΦ having at our disposal a single use of a given set of other channels \ensuremathΨi{{\mathrm{\ensuremath{\Psi}}}_{i}}\ensuremathΨi​. The problem is recast to look for the least distinguishable channel from \ensuremathΦ\mathrm{\ensuremath{\Phi}}\ensuremathΦ among the convex set \ensuremath∑ipi\ensuremathΨi{\ensuremath{\sum}}_{i}{p}_{i}{\mathrm{\ensuremath{\Psi}}}_{i}\ensuremath∑i​pi​\ensuremathΨi​, and the corresponding optimal weights pi{{p}_{i}}pi​ provide the optimal convex mixing of the available channels \ensuremathΨi{{\mathrm{\ensuremath{\Psi}}}_{i}}\ensuremathΨi​. For single-qubit channels we study specifically cases where the available convex set corresponds to covariant channels or to Pauli channels, and the desired target map is an arbitrary unitary transformation or a generalized damping channel.
We consider the problem of optimally approximating an unavailable quantum state $\ensuremath{\rho}$ by the convex mixing of states drawn from a set of available states ${{\ensuremath{\nu}}_{i}}$. The problem is recast to look for the least distinguishable state from $\ensuremath{\rho}$ among the convex set ${\ensuremath{\sum}}_{i}{p}_{i}{\ensuremath{\nu}}_{i}$, and the corresponding optimal weights pi{{p}_{i}}pi​ provide the optimal convex mixing. We present the complete solution for the optimal convex approximation of a qubit mixed state when the set of available states comprises the three bases of the Pauli matrices.
Optimal convex approximation of quantum channels based on ααα-affinity
Determining the minimal distance between a target channel and a convex hull of predefined set of implementable channels is a fundamental problem in quantum resource theory, and provides key guidance for experimental implementations. In this work, we develop a unified analytical framework for optimal convex approximation of quantum channels based on the quantum ααα-affinity measure. We construct a channel distance metric induced by the α-affinity and the ChoiJamiolkowski isomorphism, which satisfies the required properties of a well-defined channel distance. Subsequently, we present an optimization framework for the convex approximation of quantum channels, and derive analytical solutions for the optimal convex approximation of single-qubit unitary channels over both the SU(2)-covariant and Pauli channel families, obtaining closed-form expressions for the optimal parameters and the minimal approximation distance. This framework is further applied to the amplitude-damping channel, yielding the explicit form of its optimal approximation and the associated minimal α-affinity distance. In contrast to conventional approaches based on the diamond norm, our framework provides a systematic and analytically tractable approach to quantum channel approximation under realistic constraints.
The problem of an optimal mapping between Hilbert spaces IN and OUT, based on a series of density matrix mapping measurements ${\ensuremath{\rho}}^{(l)}\ensuremath{\rightarrow}{\ensuremath{\varrho}}^{(l)}$, $l=1\ensuremath{\cdots}M$, is formulated as an optimization problem maximizing the total fidelity $\mathcal{F}={\ensuremath{\sum}}_{l=1}^{M}{\ensuremath{\omega}}^{(l)}F({\ensuremath{\varrho}}^{(l)},{\ensuremath{\sum}}_{s}{B}_{s}{\ensuremath{\rho}}^{(l)}{B}_{s}^{\ifmmode\dagger\else\textdagger\fi{}})$ subject to probability preservation constraints on Kraus operators Bs{B}_{s}Bs​. For $F(\ensuremath{\varrho},\ensuremath{\sigma})$ in the form that total fidelity can be represented as a quadratic form with superoperator $\mathcal{F}={\ensuremath{\sum}}_{s}\ensuremath{\langle}{B}_{s}|S|{B}_{s}\ensuremath{\rangle}$ (either exactly or as an approximation) an iterative algorithm is developed. The work introduces two important generalizations of unitary learning: (1) IN/OUT states are represented as density matrices; (2) the mapping itself is formulated as a mixed unitary quantum channel ${A}^{OUT}={\ensuremath{\sum}}_{s}{|{w}_{s}|}^{2}{\mathcal{U}}_{s}{A}^{IN}{\mathcal{U}}_{s}^{\ifmmode\dagger\else\textdagger\fi{}}$ (no general quantum channel yet). This marks a crucial advancement from the commonly studied unitary mapping of pure states ${\ensuremath{\phi}}_{l}=\mathcal{U}{\ensuremath{\psi}}_{l}$ to a quantum channel, which allows us to distinguish probabilistic mixture of states and their superposition. An application of the approach is demonstrated on unitary learning of density matrix mapping ${\ensuremath{\varrho}}^{(l)}=\mathcal{U}{\ensuremath{\rho}}^{(l)}{\mathcal{U}}^{\ifmmode\dagger\else\textdagger\fi{}}$, in this case a quadratic on U\mathcal{U}U fidelity can be constructed by considering $\sqrt{{\ensuremath{\rho}}^{(l)}}\ensuremath{\rightarrow}\sqrt{{\ensuremath{\varrho}}^{(l)}}$ mapping, and on a quantum channel, where quadratic on Bs{B}_{s}Bs​ fidelity is an approximation---a quantum channel is then obtained as a hierarchy of unitary mappings, a mixed unitary channel. The approach can be applied to studying quantum inverse problems, variational quantum algorithms, quantum tomography, and more. A software product implementing the algorithm is available from the authors.
In this technical note, we study the problem of parameter estimation for quantum channels, which are (completely positive) maps acting on quantum states. We focus on an important subclass of channels called Pauli channels which are characterized by a certain set of vectors (directions) and a number of scalar parameters. For the case where the directions are known, a special parametrization turns the parameter estimation problem into a convex optimization problem. For the case of unknown directions we give a simple algorithm to estimate the directions for qubit Pauli channels. These results assume that the identification experiment configuration is given, namely, a set of quantum states are given on which the channel acts, and a set of positive measurement (operator)s are fixed from which information is gathered. In the second part of the technical note we consider the problem of determining the experiment design, namely, determining states and measurements for optimal parameter identification of the channels. We formalize this problem as a maximization problem for Fisher information and, assuming known channel directions, prove that this problem is convex. We also prove that the optimal states to be used in experiments are pure and the optimal measurements are extremal. For qubit Pauli channels we prove that both the optimal pure input states and projective measurements should be directed towards the channel directions. We illustrate the results of the technical note with numerical examples and simulations.
Abstract We investigate the optimal convex approximation, optimally approximating a desired and unavailable qubit state by the convex mixing of a given set of available states. When the available states are the eigenvectors of three Pauli matrices, we present the complete exact solution for the optimal convex approximation of an arbitrary qubit state based on the fidelity distance. By the comparison of optimal states based on fidelity and trace norm, the advantages and disadvantages of the optimal convex approximation are identified. Several examples are provided to support this. We also analyze the geometrical properties of the target states which can be completely represented by a set of available states. Using the feature of convex combination, we derive the maximum range of completely represented states and clearly illustrate that any qubit state can be optimally prepared by at most three available states in known useable states.