Measurement sharpness and disturbance trade-off

Obtaining information from a quantum system through a measurement typically disturbs its state. The post-measurement states for a given measurement, however, are not unique and highly rely on the chosen measurement model, complicating the puzzle of information-disturbance. Two distinct questions are then in order. Firstly, what is the minimum disturbance a measurement may induce? Secondly, when a fixed disturbance occurs, how informative is the possible measurement in the best-case scenario? Here, we propose various approaches to tackle these questions and provide explicit solutions for the set of unbiased binary qubit measurements and post-measurement state spaces that are equivalent to the image of a unital qubit channel. In particular, we show there are different trade-off relations between the sharpness of this measurement and the average fidelity of the pre-measurement and post-measurement state spaces as well as the sharpness and quantum resources preserved in the post-measurement states in terms of coherence and discord-like correlation once the measurement is applied locally.Comment: 10 pages, 2 figures, 1 tabl

Quantum-embeddable stochastic matrices

The classical embeddability problem asks whether a given stochastic matrix $T$, describing transition probabilities of a $d$-level system, can arise from the underlying homogeneous continuous-time Markov process. Here, we investigate the quantum version of this problem, asking of the existence of a Markovian quantum channel generating state transitions described by a given $T$. More precisely, we aim at characterising the set of quantum-embeddable stochastic matrices that arise from memoryless continuous-time quantum evolution. To this end, we derive both upper and lower bounds on that set, providing new families of stochastic matrices that are quantum-embeddable but not classically-embeddable, as well as families of stochastic matrices that are not quantum-embeddable. As a result, we demonstrate that a larger set of transition matrices can be explained by memoryless models if the dynamics is allowed to be quantum, but we also identify a non-zero measure set of random processes that cannot be explained by either classical or quantum memoryless dynamics. Finally, we fully characterise extreme stochastic matrices (with entries given only by zeros and ones) that are quantum-embeddable.Comment: 14 pages, 3 figures, comments welcom

Log-convex set of Lindblad semigroups acting on N-level system

We analyze the set ${\cal A}_N^Q$ of mixed unitary channels represented in the Weyl basis and accessible by a Lindblad semigroup acting on an $N$-level quantum system. General necessary and sufficient conditions for a mixed Weyl quantum channel of an arbitrary dimension to be accessible by a semigroup are established. The set ${\cal A}_N^Q$ is shown to be log--convex and star-shaped with respect to the completely depolarizing channel. A decoherence supermap acting in the space of Lindblad operators transforms them into the space of Kolmogorov generators of classical semigroups. We show that for mixed Weyl channels the hyper-decoherence commutes with the dynamics, so that decohering a quantum accessible channel we obtain a bistochastic matrix form the set ${\cal A}_N^C$ of classical maps accessible by a semigroup. Focusing on $3$-level systems we investigate the geometry of the sets of quantum accessible maps, its classical counterpart and the support of their spectra. We demonstrate that the set ${\cal A}_3^Q$ is not included in the set ${\cal U}^Q_3$ of quantum unistochastic channels, although an analogous relation holds for $N=2$. The set of transition matrices obtained by hyper-decoherence of unistochastic channels of order $N\ge 3$ is shown to be larger than the set of unistochastic matrices of this order, and yields a motivation to introduce the larger sets of $k$-unistochastic matrices.Comment: 33 pages, 7 figure