Coherifying quantum channels

Is it always possible to explain random stochastic transitions between states of a finite-dimensional system as arising from the deterministic quantum evolution of the system? If not, then what is the minimal amount of randomness required by quantum theory to explain a given stochastic process? Here, we address this problem by studying possible coherifications of a quantum channel $\Phi$, i.e., we look for channels $\Phi^{\mathcal{C}}$ that induce the same classical transitions $T$, but are "more coherent". To quantify the coherence of a channel $\Phi$ we measure the coherence of the corresponding Jamio{\l}kowski state $J_{\Phi}$. We show that the classical transition matrix $T$ can be coherified to reversible unitary dynamics if and only if $T$ is unistochastic. Otherwise the Jamio{\l}kowski state $J_\Phi^{\mathcal{C}}$ of the optimally coherified channel is mixed, and the dynamics must necessarily be irreversible. To assess the extent to which an optimal process $\Phi^{\mathcal{C}}$ is indeterministic we find explicit bounds on the entropy and purity of $J_\Phi^{\mathcal{C}}$, and relate the latter to the unitarity of $\Phi^{\mathcal{C}}$. We also find optimal coherifications for several classes of channels, including all one-qubit channels. Finally, we provide a non-optimal coherification procedure that works for an arbitrary channel $\Phi$ and reduces its rank (the minimal number of required Kraus operators) from $d^2$ to $d$.Comment: 20 pages, 8 figures. Published versio

Characterization of non-perturbative qubit channel induced by a quantum field

In this work we provide some characterization of the quantum channel induced by non-perturbative interaction between a single qubit with a quantized massless scalar field in arbitrary globally hyperbolic curved spacetimes. The qubit interacts with the field via Unruh-DeWitt detector model and we consider two non-perturbative regimes: (i) when the interaction is very rapid, effectively at a single instant in time (\textit{delta-coupled detector}); and (ii) when the qubit has degenerate energy level (\textit{gapless detector}). We organize the results in terms of quantum channels and Weyl algebras of observables in the algebraic quantum field theory (AQFT). We collect various quantum information-theoretic results pertaining to these channels, such as entropy production of the field and the qubit, recoverability of the qubit channels, and causal propagation of noise due to the interactions. We show that by treating the displacement and squeezing operations as elements of the Weyl algebra, we can generalize existing non-perturbative calculations involving the qubit channels to non-quasifree Gaussian states in curved spacetimes with little extra effort and provide transparent interpretation of these unitaries in real space. We also generalize the existing result about cohering and decohering power of a quantum channel induced by the quantum field to curved spacetimes in a very compact manner.Comment: 22 pages + 3 pages of references; 3 figures, RevTeX4-2; v3: fixed citation

Coherence as a Resource for Shor’s Algorithm

Shor’s factoring algorithm provides a superpolynomial speedup over all known classical factoring algorithms. Here, we address the question of which quantum properties fuel this advantage. We investigate a sequential variant of Shor’s algorithm with a fixed overall structure and identify the role of coherence for this algorithm quantitatively. We analyze this protocol in the framework of dynamical resource theories, which capture the resource character of operations that can create and detect coherence. This allows us to derive a lower and an upper bound on the success probability of the protocol, which depends on rigorously defined measures of coherence as a dynamical resource. We compare these bounds with the classical limit of the protocol and conclude that within the fixed structure that we consider, coherence is the quantum resource that determines its performance by bounding the success probability from below and above. Therefore, we shine new light on the fundamental role of coherence in quantum computation.Universität UlmInstituto de Física La PlataInstitute for Quantum Science and Technology, University of CalgaryInstitute of Theoretical Physics, Technical University Dresde

Localizable quantum coherence

Coherence is a fundamental notion in quantum mechanics, defined relative to a reference basis. As such, it does not necessarily reveal the locality of interactions nor takes into account the accessible operations in a composite quantum system. In this paper, we put forward a notion of localizable coherence as the coherence that can be stored in a particular subsystem, either by measuring or just by disregarding the rest. We examine its spreading, its average properties in the Hilbert space and show that it can be applied to reveal the real-space structure of states of interest in quantum many-body theory, for example, localized or topological states.Comment: Close to published versio