Discriminating quantum states: the multiple Chernoff distance

We consider the problem of testing multiple quantum hypotheses $\{\rho_1^{\otimes n},\ldots,\rho_r^{\otimes n}\}$, where an arbitrary prior distribution is given and each of the $r$ hypotheses is $n$ copies of a quantum state. It is known that the average error probability $P_e$ decays exponentially to zero, that is, $P_e=\exp\{-\xi n+o(n)\}$. However, this error exponent $\xi$ is generally unknown, except for the case that $r=2$. In this paper, we solve the long-standing open problem of identifying the above error exponent, by proving Nussbaum and Szko\l a's conjecture that $\xi=\min_{i\neq j}C(\rho_i,\rho_j)$. The right-hand side of this equality is called the multiple quantum Chernoff distance, and $C(\rho_i,\rho_j):=\max_{0\leq s\leq 1}\{-\log\operatorname{Tr}\rho_i^s\rho_j^{1-s}\}$ has been previously identified as the optimal error exponent for testing two hypotheses, $\rho_i^{\otimes n}$ versus $\rho_j^{\otimes n}$. The main ingredient of our proof is a new upper bound for the average error probability, for testing an ensemble of finite-dimensional, but otherwise general, quantum states. This upper bound, up to a states-dependent factor, matches the multiple-state generalization of Nussbaum and Szko\l a's lower bound. Specialized to the case $r=2$, we give an alternative proof to the achievability of the binary-hypothesis Chernoff distance, which was originally proved by Audenaert et al.Comment: v2: minor change

Many Hard Examples in Exact Phase Transitions with Application to Generating Hard Satisfiable Instances

This paper first analyzes the resolution complexity of two random CSP models (i.e. Model RB/RD) for which we can establish the existence of phase transitions and identify the threshold points exactly. By encoding CSPs into CNF formulas, it is proved that almost all instances of Model RB/RD have no tree-like resolution proofs of less than exponential size. Thus, we not only introduce new families of CNF formulas hard for resolution, which is a central task of Proof-Complexity theory, but also propose models with both many hard instances and exact phase transitions. Then, the implications of such models are addressed. It is shown both theoretically and experimentally that an application of Model RB/RD might be in the generation of hard satisfiable instances, which is not only of practical importance but also related to some open problems in cryptography such as generating one-way functions. Subsequently, a further theoretical support for the generation method is shown by establishing exponential lower bounds on the complexity of solving random satisfiable and forced satisfiable instances of RB/RD near the threshold. Finally, conclusions are presented, as well as a detailed comparison of Model RB/RD with the Hamiltonian cycle problem and random 3-SAT, which, respectively, exhibit three different kinds of phase transition behavior in NP-complete problems.Comment: 19 pages, corrected mistakes in Theorems 5 and

Estimating decay rate of X±(5568)→Bsπ±X^{\pm}(5568)\to B_s\pi^{\pm} while assuming them to be molecular states

Discovery of $X(5568)$ brings up a tremendous interest because it is very special, i.e. made of four different flavors. The D0 collaboration claimed that they observed this resonance through portal $X(5568)\to B_s\pi$, but unfortunately, later the LHCb, CMS, CDF and ATLAS collaborations' reports indicate that no such state was found. Almost on the Eve of 2017, the D0 collaboration reconfirmed existence of $X(5568)$ via the semileptonic decay of $B_s$. To further reveal the discrepancy, supposing $X(5568)$ as a molecular state, we calculate the decay rate of $X(5568)\rightarrow B_s\pi^+$ in an extended light front model. Numerically, the theoretically predicted decay width of $\Gamma(X(5568)\rightarrow B_s\pi^+)$ is $20.28$ MeV which is consistent with the result of the D0 collaboration ($\Gamma=18.6^{+7.9}_{-6.1}(stat)^{+3.5}_{-3.8}(syst)$ MeV). Since the resonance is narrow, signals might be drowned in a messy background. In analog, two open-charm molecular states $DK$ and $BD$ named as $X_a$ and $X_b$, could be in the same situation. The rates of $X_a\to D_s\pi^0$ and $X_b\to B_c\pi^0$ are estimated as about 30 MeV and 20 MeV respectively. We suggest the experimental collaborations round the world to search for these two modes and accurate measurements may provide us with valuable information.Comment: 13 pages and 4 figures, accepted by EPJ

Iterative Instance Segmentation

Existing methods for pixel-wise labelling tasks generally disregard the underlying structure of labellings, often leading to predictions that are visually implausible. While incorporating structure into the model should improve prediction quality, doing so is challenging - manually specifying the form of structural constraints may be impractical and inference often becomes intractable even if structural constraints are given. We sidestep this problem by reducing structured prediction to a sequence of unconstrained prediction problems and demonstrate that this approach is capable of automatically discovering priors on shape, contiguity of region predictions and smoothness of region contours from data without any a priori specification. On the instance segmentation task, this method outperforms the state-of-the-art, achieving a mean $\mathrm{AP}^{r}$ of 63.6% at 50% overlap and 43.3% at 70% overlap.Comment: 13 pages, 10 figures; IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 201