Semidefinite programming converse bounds for quantum communication

We derive several efficiently computable converse bounds for quantum communication over quantum channels in both the one-shot and asymptotic regime. First, we derive one-shot semidefinite programming (SDP) converse bounds on the amount of quantum information that can be transmitted over a single use of a quantum channel, which improve the previous bound from [Tomamichel/Berta/Renes, Nat. Commun. 7, 2016]. As applications, we study quantum communication over depolarizing channels and amplitude damping channels with finite resources. Second, we find an SDP strong converse bound for the quantum capacity of an arbitrary quantum channel, which means the fidelity of any sequence of codes with a rate exceeding this bound will vanish exponentially fast as the number of channel uses increases. Furthermore, we prove that the SDP strong converse bound improves the partial transposition bound introduced by Holevo and Werner. Third, we prove that this SDP strong converse bound is equal to the so-called max-Rains information, which is an analog to the Rains information introduced in [Tomamichel/Wilde/Winter, IEEE Trans. Inf. Theory 63:715, 2017]. Our SDP strong converse bound is weaker than the Rains information, but it is efficiently computable for general quantum channels.Comment: 17 pages, extended version of arXiv:1601.06888. v3 is closed to the published version, IEEE Transactions on Information Theory, 201

Distillation and simulation in quantum information

University of Technology Sydney. Faculty of Engineering and Information Technology.We use the techniques of convex optimization, especially semidefinite programming, to study two kinds of fundamental tasks, i.e., distillation and simulation in quantum information theory. We investigate these tasks in a unified framework of resource theory and focus on their computation and characterization with finite resources. Particularly we study the tradeoff among relevant parameters such as the number of resource copies, resource transformation rate, error tolerance and success probability. In the first part, we study the task of distillation for two different resources, maximally entangled state and maximally coherent state, representing nonlocal and local “quantumness” respectively. For entanglement distillation, we derive an efficiently computable second-order estimation of the distillation rate for general quantum states, which are tight for quantum states of practical interest. Our study overcomes the limitation of conventional research either focusing on the asymptotic rate or ignoring the computability. For the coherence distillation, we perform finite analysis for both deterministic and probabilistic scenarios. Our results unveil several new features of coherence from a resource theoretic viewpoint and contribute to an increased understanding of the fundamental properties of different sets of free operations. In the second part, we investigate the resource cost of simulating a quantum channel via quantum coherence or another quantum channel. We introduce the channel’s analogs of max-relative entropy, logarithmic robustness and max-information of quantum states, providing their operational interpretation with the channel simulation cost via different resources. Particularly, we establish the asymptotic equipartition property of the channel’s max-information, that is, it converges to the quantum mutual information of the channel in the independent and identically distributed asymptotic limit. As applications, this asymptotic equipartition property implies the quantum reverse Shannon theorem in the presence of non-signalling correlations. From the perspective of resource theory, the worth of a resource can usually be characterized by the minimum distance to a set of useless resources under a proper distance measure. We give such characterization for all the tasks studied in this thesis, and find that the distance measure for the distillation and simulation process naturally corresponds to the quantum hypothesis testing relative entropy and the max-relative entropy, respectively

Constraining the slow-diffusion zone size and electron injection spectral index for the Geminga pulsar halo

Measuring the electron diffusion coefficient is the most straightforward task in the study of gamma-ray pulsar halos. The updated measurements of the spatial morphology and spectrum of the Geminga halo by the HAWC experiment enable us to constrain parameters beyond the diffusion coefficient, including the size of the slow-diffusion zone and the electron injection spectrum from the pulsar wind nebulae (PWN). Based on the two-zone diffusion model, we find that the slow-diffusion zone size ($r_*$) around Geminga is within the range of $30-70$pc. The lower boundary of this range is determined by the goodness of fit of the model to the one-dimensional morphology of the Geminga halo. The upper limit is derived from fitting the gamma-ray spectrum of the Geminga halo, along with the expectations for the power-law index of the injection spectrum based on simulations and PWN observations, i.e., $p\gtrsim1$. With $r_*$ set at its lower limit of $30$~pc, we obtain the maximum $p$ permitted by the HAWC spectrum measurement, with an upper limit of $2.17$ at a $3\sigma$ significance. Moreover, we find that when $r_*=30$pc and $p=2.17$, the predicted positron spectrum generated by Geminga at Earth coincides with the AMS-02 measurement in the $50-500$GeV range.Comment: 14 pages, 5 figure