Higher-order interference and single-system postulates characterizing quantum theory

We present a new characterization of quantum theory in terms of simple physical principles that is different from previous ones in two important respects: first, it only refers to properties of single systems without any assumptions on the composition of many systems; and second, it is closer to experiment by having absence of higher-order interference as a postulate, which is currently the subject of experimental investigation. We give three postulates -- no higher-order interference, classical decomposability of states, and strong symmetry -- and prove that the only non-classical operational probabilistic theories satisfying them are real, complex, and quaternionic quantum theory, together with 3-level octonionic quantum theory and ball state spaces of arbitrary dimension. Then we show that adding observability of energy as a fourth postulate yields complex quantum theory as the unique solution, relating the emergence of the complex numbers to the possibility of Hamiltonian dynamics. We also show that there may be interesting non-quantum theories satisfying only the first two of our postulates, which would allow for higher-order interference in experiments while still respecting the contextuality analogue of the local orthogonality principle.Comment: 21 + 6 pages, 1 figure. v4: published version (includes several minor corrections

Entropy, majorization and thermodynamics in general probabilistic theories

In this note we lay some groundwork for the resource theory of thermodynamics in general probabilistic theories (GPTs). We consider theories satisfying a purely convex abstraction of the spectral decomposition of density matrices: that every state has a decomposition, with unique probabilities, into perfectly distinguishable pure states. The spectral entropy, and analogues using other Schur-concave functions, can be defined as the entropy of these probabilities. We describe additional conditions under which the outcome probabilities of a fine-grained measurement are majorized by those for a spectral measurement, and therefore the "spectral entropy" is the measurement entropy (and therefore concave). These conditions are (1) projectivity, which abstracts aspects of the Lueders-von Neumann projection postulate in quantum theory, in particular that every face of the state space is the positive part of the image of a certain kind of projection operator called a filter; and (2) symmetry of transition probabilities. The conjunction of these, as shown earlier by Araki, is equivalent to a strong geometric property of the unnormalized state cone known as perfection: that there is an inner product according to which every face of the cone, including the cone itself, is self-dual. Using some assumptions about the thermodynamic cost of certain processes that are partially motivated by our postulates, especially projectivity, we extend von Neumann's argument that the thermodynamic entropy of a quantum system is its spectral entropy to generalized probabilistic systems satisfying spectrality.Comment: In Proceedings QPL 2015, arXiv:1511.0118

Discoveries in the Biology of Oxidized Chlorine

Chlorine participates in a biogeochemical cycle that rivals that of elements like nitrogen and sulfur in chemical diversity. Nature produces a variety of organic and inorganic chlorine-containing molecules that participate in different biological, chemical, and geological processes. The most defining feature of chlorine is its high electronegativity, perhaps best illustrated by the very high reduction potential of molecules in which chlorine is at a higher oxidation state, such as hypochlorous acid (HOCl), chlorite (ClO2-), chlorate (ClO3-), and perchlorate (ClO4-). Compared to research on the biogeochemical cycles for other elements, the mechanisms by which oxidized chlorine molecules are produced and consumed are relatively understudied. One reason is that descriptions of the biogeochemical chlorine cycle have been incomplete, omitting important processes involving inorganic chlorine molecules. Another reason is that the microbiology of oxidized chlorine has been studied almost exclusively through a reductionist approach that can preclude the discovery of ecological interactions. Yet another reason is that the genes known to be involved in the metabolism of oxidized chlorine have yet to be used to find new organisms and processes that metabolize oxidized chlorine. Here, a more holistic approach, enabled by improvements in genome sequencing, is used to better describe the biology of oxidized chlorine across several research projects. The first chapter of this dissertation provides the first review of the entire biogeochemical chlorine cycle, emphasizing connections between the various biological, chemical, and geological processes that interconvert chlorine between different chemical forms. The second chapter of this dissertation is a published research article describing the reduction of perchlorate in microbial communities for the first time by using bioinformatics techniques to obtain genomes from metagenomes. Instead of being dominated by the specific bacteria known to respire perchlorate, perchlorate-reducing communities contain diverse organisms that interact via the chemical intermediates of dissimilatory perchlorate reduction.The third chapter of this dissertation is a published research article investigating the mechanism of one such interaction between perchlorate-reducing bacteria and chlorate-reducing bacteria. A combination of genomics, strain isolation, genetics, metabolite measurements, and theoretical modeling are used to learn that these two metabolisms, which have been studied separately for several decades, have a conserved interaction due to the accumulation of chlorate by perchlorate-reducing bacteria.The fourth chapter of this dissertation is a brief report characterizing a possible perchlorate reductase or chlorate reductase first identified in perchlorate-reducing communities. An organism with this reductase, which is always found in genomes adjacent to the chlorite-degrading enzyme chlorite dismutase (Cld), is capable of chlorate reduction but not perchlorate reduction, indicating the enzyme is a chlorate reductase.The fifth chapter of this dissertation extends the above comparative genomics analysis to identify any gene or organism linked to the enzyme Cld. Because Cld is biomarker for chlorite and other chlorine oxyanions, this approach was able to expand the environments, organisms, and processes known to participate in oxidized chlorine biology beyond the organisms and genes described above. Specifically, more was learned about the reduction of perchlorate and chlorate in the environment; the potential oxidation of chloride beyond hypochlorous acid by chemical or biological activity; and the connection between chlorite and reactive chlorine stress response.Together, this research has answered important questions about the reduction of chlorine while opening new questions about the oxidation of chlorine and the role of oxidized chlorine species in the environment

Better bound on the exponent of the radius of the multipartite separable ball

We show that for an m-qubit quantum system, there is a ball of radius asymptotically approaching kappa 2^{-gamma m} in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices, for an exponent gamma = (1/2)((ln 3/ln 2) - 1), roughly .29248125. This is much smaller in magnitude than the best previously known exponent, from our earlier work, of 1/2. For normalized m-qubit states, we get a separable ball of radius sqrt(3^(m+1)/(3^m+3)) * 2^{-(1 + \gamma)m}, i.e. sqrt{3^{m+1}/(3^m+3)}\times 6^{-m/2} (note that \kappa = \sqrt{3}), compared to the previous 2 * 2^{-3m/2}. This implies that with parameters realistic for current experiments, NMR with standard pseudopure-state preparation techniques can access only unentangled states if 36 qubits or fewer are used (compared to 23 qubits via our earlier results). We also obtain an improved exponent for m-partite systems of fixed local dimension d_0, although approaching our earlier exponent as d_0 approaches infinity.Comment: 30 pp doublespaced, latex/revtex, v2 added discussion of Szarek's upper bound, and reference to work of Vidal, v3 fixed some errors (no effect on results), v4 involves major changes leading to an improved constant, same exponent, and adds references to and discussion of Szarek's work showing that exponent is essentially optimal for qubit case, and Hildebrand's alternative derivation for qubit case. To appear in PR

Generalization of entanglement to convex operational theories: Entanglement relative to a subspace of observables

We define what it means for a state in a convex cone of states on a space of observables to be generalized-entangled relative to a subspace of the observables, in a general ordered linear spaces framework for operational theories. This extends the notion of ordinary entanglement in quantum information theory to a much more general framework. Some important special cases are described, in which the distinguished observables are subspaces of the observables of a quantum system, leading to results like the identification of generalized unentangled states with Lie-group-theoretic coherent states when the special observables form an irreducibly represented Lie algebra. Some open problems, including that of generalizing the semigroup of local operations with classical communication to the convex cones setting, are discussed.Comment: 19 pages, to appear in proceedings of Quantum Structures VII, Int. J. Theor. Phy

Compressibility of Mixed-State Signals

We present a formula that determines the optimal number of qubits per message that allows asymptotically faithful compression of the quantum information carried by an ensemble of mixed states. The set of mixed states determines a decomposition of the Hilbert space into the redundant part and the irreducible part. After removing the redundancy, the optimal compression rate is shown to be given by the von Neumann entropy of the reduced ensemble.Comment: 7 pages, no figur

Efficient solvability of Hamiltonians and limits on the power of some quantum computational models

We consider quantum computational models defined via a Lie-algebraic theory. In these models, specified initial states are acted on by Lie-algebraic quantum gates and the expectation values of Lie algebra elements are measured at the end. We show that these models can be efficiently simulated on a classical computer in time polynomial in the dimension of the algebra, regardless of the dimension of the Hilbert space where the algebra acts. Similar results hold for the computation of the expectation value of operators implemented by a gate-sequence. We introduce a Lie-algebraic notion of generalized mean-field Hamiltonians and show that they are efficiently ("exactly") solvable by means of a Jacobi-like diagonalization method. Our results generalize earlier ones on fermionic linear optics computation and provide insight into the source of the power of the conventional model of quantum computation.Comment: 6 pages; no figure

Experimentally realizable quantum comparison of coherent states and its applications

When comparing quantum states to each other, it is possible to obtain an unambiguous answer, indicating that the states are definitely different, already after a single measurement. In this paper we investigate comparison of coherent states, which is the simplest example of quantum state comparison for continuous variables. The method we present has a high success probability, and is experimentally feasible to realize as the only required components are beam splitters and photon detectors. An easily realizable method for quantum state comparison could be important for real applications. As examples of such applications we present a "lock and key" scheme and a simple scheme for quantum public key distribution.Comment: 14 pages, 5 figures, version one submitted to PRA. Version two is the final accepted versio

Lossless quantum data compression and variable-length coding

In order to compress quantum messages without loss of information it is necessary to allow the length of the encoded messages to vary. We develop a general framework for variable-length quantum messages in close analogy to the classical case and show that lossless compression is only possible if the message to be compressed is known to the sender. The lossless compression of an ensemble of messages is bounded from below by its von-Neumann entropy. We show that it is possible to reduce the number of qbits passing through a quantum channel even below the von-Neumann entropy by adding a classical side-channel. We give an explicit communication protocol that realizes lossless and instantaneous quantum data compression and apply it to a simple example. This protocol can be used for both online quantum communication and storage of quantum data.Comment: 16 pages, 5 figure

Three-dimensionality of space and the quantum bit: an information-theoretic approach

It is sometimes pointed out as a curiosity that the state space of quantum two-level systems, i.e. the qubit, and actual physical space are both three-dimensional and Euclidean. In this paper, we suggest an information-theoretic analysis of this relationship, by proving a particular mathematical result: suppose that physics takes place in d spatial dimensions, and that some events happen probabilistically (not assuming quantum theory in any way). Furthermore, suppose there are systems that carry "minimal amounts of direction information", interacting via some continuous reversible time evolution. We prove that this uniquely determines spatial dimension d=3 and quantum theory on two qubits (including entanglement and unitary time evolution), and that it allows observers to infer local spatial geometry from probability measurements.Comment: 13 + 22 pages, 9 figures. v4: some clarifications, in particular in Section V / Appendix C (added Example 39