- Quantum hypothesis testing is one of the most basic tasks in quantum information theory and has fundamental links with quantum communication and estimation theory. In this paper, we establish a formula that characterizes the decay rate of the minimal Type-II error probability in a quantum hypothesis test of two Gaussian states given a fixed constraint on the Type-I error probability. This formula is a direct function of the mean vectors and covariance matrices of the quantum Gaussian states in question. We give an application to quantum illumination, which is the task of determining whether there is a low-reflectivity object embedded in a target region with a bright thermal-noise bath. For the asymmetric-error setting, we find that a quantum illumination transmitter can achieve an error probability exponent much stronger than a coherent-state transmitter of the same mean photon number, and furthermore, that it requires far fewer trials to do so. This occurs when the background thermal noise is either low or bright, which means that a quantum advantage is even easier to witness than in the symmetric-error setting because it occurs for a larger range of parameters. Going forward from here, we expect our formula to have applications in settings well beyond those considered in this paper, especially to quantum communication tasks involving quantum Gaussian channels.
- Aug 26 2016 math.CO arXiv:1608.07028v1A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph $K_n$ has a rainbow Hamiltonian path. Although this conjecture turned out to be false, it was widely believed that such a colouring always contains a rainbow cycle of length almost $n$. In this paper, improving on several earlier results, we confirm this by proving that every properly edge-coloured $K_n$ has a rainbow cycle of length $n-O(n^{3/4})$. One of the main ingredients of our proof, which is of independent interest, shows that a random subgraph of a properly edge-coloured $K_n$ formed by the edges of a random set of colours has a similar edge distribution as a truly random graph with the same edge density. In particular it has very good expansion properties.
- In this paper we develop an operational formulation of General Relativity similar in spirit to existing operational formulations of Quantum Theory. To do this we introduce an operational space (or op-space) built out of scalar fields. A point in op-space corresponds to some nominated set of scalar fields taking some given values in coincidence. We assert that op-space is the space in which we observe the world. We introduce also a notion of agency (this corresponds to the ability to set knob settings just like in Operational Quantum Theory). The effects of agents' actions should only be felt to the future so we introduce also a time direction field. Agency and time direction can be understood as effective notions. We show how to formulate General Relativity as a possibilistic theory and as a probabilistic theory. In the possibilistic case we provide a compositional framework for calculating whether some operationally described situation is possible or not. In the probabilistic version we introduce probabilities and provide a compositional framework for calculating the probability of some operationally described situation. Finally we look at the quantum case. We review the operator tensor formulation of Quantum Theory and use it to set up an approach to Quantum Field Theory that is both operational and compositional. Then we consider strategies for solving the problem of Quantum Gravity. By referring only to operational quantities we are able to provide formulations for the possibilistic, probabilistic, and (the nascent) quantum cases that are manifestly invariant under diffeomorphisms.
- Aug 26 2016 astro-ph.EP astro-ph.SR arXiv:1608.07278v1
- Aug 26 2016 hep-ph arXiv:1608.07274v1
- Aug 26 2016 physics.plasm-ph physics.comp-ph arXiv:1608.07273v1
- Aug 26 2016 hep-ph arXiv:1608.07271v1
- Aug 26 2016 math.MG arXiv:1608.07270v1
- Aug 26 2016 math.NA arXiv:1608.07268v1
- Aug 26 2016 physics.ins-det hep-ex arXiv:1608.07267v1
- Aug 26 2016 cond-mat.soft arXiv:1608.07266v1
- Aug 26 2016 hep-th cond-mat.quant-gas arXiv:1608.07262v1
- Aug 26 2016 cs.PL arXiv:1608.07261v1
- Aug 26 2016 physics.hist-ph arXiv:1608.07257v1
- Aug 26 2016 cond-mat.soft arXiv:1608.07254v1
- Aug 26 2016 astro-ph.IM arXiv:1608.07252v1
- Aug 26 2016 cond-mat.soft cond-mat.stat-mech arXiv:1608.07250v1
- Aug 26 2016 math.CO arXiv:1608.07247v1
- Aug 26 2016 q-bio.NC arXiv:1608.07244v1
- Aug 26 2016 cs.CV arXiv:1608.07242v1
- Aug 26 2016 stat.ML arXiv:1608.07241v1
- Aug 26 2016 math.NT arXiv:1608.07236v1
- Aug 26 2016 math.NT arXiv:1608.07234v1
- Aug 26 2016 physics.comp-ph cond-mat.dis-nn arXiv:1608.07231v1
- Aug 26 2016 astro-ph.GA arXiv:1608.07230v1
- Aug 26 2016 math.MG arXiv:1608.07229v1
- Aug 26 2016 q-fin.MF arXiv:1608.07226v1