1,721,013 research outputs found
Universal Probability Distribution for the Wave Function of a Quantum System Entangled with Its Environment
No full textA quantum system (with Hilbert space H1) entangled with its environment (with Hilbert space H2) is usually not attributed a wave function but only a reduced density matrix ρ1. Nevertheless, there is a precise way of attributing to it a random wave function ψ1, called its conditional wave function, whose probability distribution μ1 depends on the entangled wave function ψ∈H1⊗H2 in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of H2 but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about μ1, e.g., that if the environment is sufficiently large then for every orthonormal basis of H2, most entangled states ψ with given reduced density matrix ρ1 are such that μ1 is close to one of the so-called GAP (Gaussian adjusted projected) measures, GAP(ρ1). We also show that, for most entangled states ψ from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval [E,E+δE]) and most orthonormal bases of H2, μ1 is close to GAP(tr2ρmc) with ρmc the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then μ1 is close to GAP(ρβ) with ρβ the canonical density matrix on H1 at inverse temperature β=β(E). This provides the mathematical justification of our claim in [J. Statist. Phys. 125:1193 (2006), http://arxiv.org/abs/quant-ph/0309021] that GAP measures describe the thermal equilibrium distribution of the wave function.Peer reviewe
Is the hypothesis about a low entropy initial state of the Universe necessary for explaining the arrow of time?
No full textAccording to statistical mechanics, microstates of an isolated physical system (say, a gas in a box) at time t0 in a given macrostate of less-than-maximal entropy typically evolve in such a way that the entropy at time t increases with |t-t0| in both time directions. In order to account for the observed entropy increase in only one time direction, the thermodynamic arrow of time, one usually appeals to the hypothesis that the initial state of the Universe was one of very low entropy. In certain recent models of cosmology, however, no hypothesis about the initial state of the Universe is invoked. We discuss how the emergence of a thermodynamic arrow of time in such models can nevertheless be compatible with the above-mentioned consequence of statistical mechanics, appearances to the contrary notwithstanding
Arrival Times Versus Detection Times
No full textHow to compute the probability distribution of a detection time, i.e., of the time which a detector registers as the arrival time of a quantum particle, is a long-debated problem. In this regard, Bohmian mechanics provides in a straightforward way the distribution of the time at which the particle actually does arrive at a given surface in 3-space in the absence of detectors. However, as we discuss here, since the presence of detectors can change the evolution of the wave function and thus the particle trajectories, it cannot be taken for granted that the arrival time of the Bohmian trajectories in the absence of detectors agrees with the one in the presence of detectors, and even less with the detection time. In particular, we explain why certain distributions that Das and Dürr (Sci. Rep. 9: 2242, 2019) presented as the distribution of the detection time in a case with spin, based on assuming that all three times mentioned coincide, are actually not what Bohmian mechanics predicts
On the spin dependence of detection times and the nonmeasurability of arrival times
No full textAccording to a well-known principle of quantum physics, the statistics of the outcomes of any quantum experiment are governed by a Positive-Operator-Valued Measure (POVM). In particular, for experiments designed to measure a specific physical quantity, like the time of a particle’s first arrival at a surface, this principle establishes that if the probability distribution of that quantity does not arise from a POVM, no such experiment exists. Such is the case with the arrival time distributions proposed by Das and Dürr, due to the nature of their spin dependence
Who’s Afraid of the Measurement Problem?
No full textScientific realists usually claim that quantum mechanics can be made compatible with scientific realism by solving the measurement problem, even if there is disagreement about which solution is best. In this paper I argue this is due to having different views about what it means to make quantum theory compatible with scientific realism: ‘relaxed’ realists think it is enough to solve the adequacy problem, ‘modest’ realists believe that there is also a precision problem, while ‘robust’ realists insist that quantum theory still needs to be suitably completed. These attitudes are connected with the type of explanation one favors: while relaxed realists favor principle theories, robust realists prefer constructive theories, and modest realists provide non-constructive dynamical hybrids as long as they preserve locality and separability
Any orthonormal basis in high dimension is uniformly distributed over the sphere
No full textLet Xd be a real or complex Hilbert space of finite but large dimension d, let S(Xd ) denote the unit sphere of Xd, and let u denote the normalized uniform measure on S(Xd ). For a finite subset B of S(Xd ), we may test whether it is approximately uniformly distributed over the sphere by choosing a partition A1, . . . , Am of S(Xd ) and checking whether the fraction of points in B that lie in Ak is close to u(Ak) for each k = 1, . . . , m. We show that if B is any orthonormal basis of Xd and m is not too large, then, if we randomize the test by applying a random rotation to the sets A1, . . . , Am, B will pass the random test with probability close to 1. This statement is related to, but not entailed by, the law of large numbers. An application of this fact in quantum statistical mechanics is briefly described
Universal Probability Distribution for the Wave Function of a Quantum System Entangled with its Environment
No full textA quantum system (with Hilbert space (Formula presented.)) entangled with its environment (with Hilbert space (Formula presented.)) is usually not attributed to a wave function but only to a reduced density matrix (Formula presented.). Nevertheless, there is a precise way of attributing to it a random wave function (Formula presented.) , called its conditional wave function, whose probability distribution (Formula presented.) depends on the entangled wave function (Formula presented.) in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of (Formula presented.) but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about (Formula presented.) , e.g., that if the environment is sufficiently large then for every orthonormal basis of (Formula presented.) , most entangled states (Formula presented.) with given reduced density matrix (Formula presented.) are such that (Formula presented.) is close to one of the so-called GAP (Gaussian adjusted projected) measures, (Formula presented.). We also show that, for most entangled states (Formula presented.) from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval (Formula presented.)) and most orthonormal bases of (Formula presented.) , (Formula presented.) is close to (Formula presented.) with (Formula presented.) the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then (Formula presented.) is close to (Formula presented.) with (Formula presented.) the canonical density matrix on (Formula presented.) at inverse temperature (Formula presented.). This provides the mathematical justification of our claim in Goldstein et al. (J Stat Phys 125: 1193–1221, 2006) that GAP measures describe the thermal equilibrium distribution of the wave function
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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