We study the problem of approximating a quantum channel by one with as few Kraus operators as possible (in the sense that, for any input state, the output states of the two channels should be close to one another). Our main result is that any quantum channel mapping states on some input Hilbert space A to states on some output Hilbert space B can be compressed into one with order dlogd Kraus operators, where d=max(|A|,|B|) , hence much less than |A||B| . In the case where the channel's outputs are all very mixed, this can be improved to order d . We discuss the optimality of this result as well as some consequences.
We study the problem of approximating a quantum channel by one with as few Kraus operators as possible (in the sense that, for any input state, the output states of the two channels should be close to one another). Our main result is that any quantum channel mapping states on some input Hilbert space A to states on some output Hilbert space B can be compressed into one with order dlogd Kraus operators, where d=max(|A|,|B|) , hence much less than |A||B| . In the case where the channel's outputs are all very mixed, this can be improved to order d . We discuss the optimality of this result as well as some consequences.
In general quantum operations, or quantum channels cannot be inverted by physical operations, i.e., by completely positive trace-preserving maps. An arbitrary state passing through a quantum channel loses its fidelity with the input. Given a quantum channel E, we discuss the concept of its quasi-inverse as a completely positive trace-preserving map Eqi which when composed with E increases its average input-output fidelity in an optimal way. The channel Eqi comes as close as possible to the inverse of a quantum channel. We give a complete classification of such maps for qubit channels and provide quite a few illustrative examples.
Earlier, we proved a duality between two optimizations problems [Phys. Rev. Lett. 104, 120501 (2010)]. The primary one is, given two quantum channels M and N, to find a quantum channel R such that $\mathcal{R}\ensuremath{\circ}\mathcal{N}$ is optimally close to M as measured by the worst-case entanglement fidelity. The dual problem involves the information obtained by the environment through the so-called complementary channels $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{M}}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{N}}$, and consists in finding a quantum channel ${\mathcal{R}}^{\ensuremath{'}}$ such that ${\mathcal{R}}^{\ensuremath{'}}\ensuremath{\circ}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{M}}$ is optimally close to $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{N}}$. It turns out to be easier to find an approximate solution to the dual problem in certain important situations, notably when M is the identity channel---the problem of quantum error correction---yielding a good near-optimal worst-case entanglement fidelity as well as the corresponding near-optimal correcting channel. Here we provide more detailed proofs of these results. In addition, we generalize the main theorem to the case where there are certain constraints on the implementation of R, namely, on the number of Kraus operators. We also offer a simple algebraic form for the near-optimal correction channel in the case M=id. For approximate error correction, we show that any $\ensuremath{\varepsilon}$-correctable channel is, up to appending an ancilla, $\ensuremath{\varepsilon}$-close to an exactly correctable one. We also demonstrate an application of our theorem to the problem of minimax state discrimination.
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. 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 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 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.
For large d, we study quantum channels on C d obtained by selecting ran- domly N independent Kraus operators according to a probability measure µ on the unitary group U(d). When µ is the Haar measure, we show that for N d/e 2 , such a channel is e-randomizing with high probability, which means that it maps every state within distance e/d (in operator norm) of the maximally mixed state. This slightly improves on a result by Hayden, Leung, Shor and Winter by optimizing their discreti- zation argument. Moreover, for general µ, we obtain an e-randomizing channel provided N d (log d) 6 /e 2 .F ord = 2 k (k qubits), this includes Kraus operators obtained by ten- soring k random Pauli matrices. This leads to more efficient constructions of almost randomizing channels. The proof uses recent results on empirical processes in Banach spaces.
Quantum process tomography is the task of reconstructing unknown quantum channels from measured data. In this work, we introduce compressed sensing-based methods that facilitate the reconstruction of quantum channels of low Kraus rank. Our main contribution is the analysis of a natural measurement model for this task: We assume that data is obtained by sending pure states into the channel and measuring expectation values on the output. Neither ancillary systems nor coherent operations across multiple channel uses are required. Most previous results on compressed process reconstruction reduce the problem to quantum state tomography on the channel's Choi matrix. While this ansatz yields recovery guarantees from an essentially minimal number of measurements, physical implementations of such schemes would typically involve ancillary systems. A priori, it is unclear whether a measurement model tailored directly to quantum process tomography might require more measurements. We establish that this is not the case.Technically, we prove recovery guarantees for three different reconstruction algorithms. The reconstructions are based on a trace, diamond, and ℓ2 -norm minimization, respectively. Our recovery guarantees are uniform in the sense that with one random choice of measurement settings all quantum channels can be recovered equally well. Moreover, stability against arbitrary measurement noise and robustness against violations of the low-rank assumption is guaranteed. Numerical studies demonstrate the feasibility of the approach.
Abstract In this chapter we visit the ‘church of the larger Hilbert space’: we take a closer look at the relationship of pure and mixed quantum states via purification, and discuss a similar connection between completely positive and trace-preserving (CPTP) maps and unitary dynamics on a larger Hilbert space. We show that such CPTP maps, which represent quantum channels, are equivalent to operations admitting a Kraus decomposition, as well as to unitary dynamics on a larger Hilbert space with factoring initial condition, the famous Stinespring dilation of quantum operations. In this context we introduce the Choi-Jamiołkowski isomorphism, which expresses a duality between maps between two Hilbert spaces and states on tensor products of these Hilbert spaces, and we review several important examples of quantum channels such as the dephasing channel, amplitude-damping channel, and depolarizing channel. Finally, we examine operations that are not possible in quantum theory, focusing on the no-cloning theorem.