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William K Wootters - One of the best experts on this subject based on the ideXlab platform.
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optimal information transfer and Real Vector Space quantum theory
arXiv: Quantum Physics, 2016Co-Authors: William K WoottersAbstract:Consider a photon that has just emerged from a linear polarizing filter. If the photon is then subjected to an orthogonal polarization measurement—e.g., horizontal vs vertical—the photon’s preparation cannot be fully expressed in the outcome: a binary outcome cannot reveal the value of a continuous variable. However, a stream of identically prepared photons can do much better. To quantify this effect, one can compute the mutual information between the angle of polarization and the observed frequencies of occurrence of “horizontal” and “vertical.” Remarkably, one finds that the quantum-mechanical rule for computing probabilities—Born’s rule—maximizes this mutual information relative to other conceivable probability rules. However, the maximization is achieved only because linear polarization can be modeled with a Real state Space; the argument fails when one considers the full set of complex states. This result generalizes to higher dimensional Hilbert Spaces: in every case, one finds that information is transferred optimally from preparation to measurement in the Real-Vector-Space theory but not in the complex theory. Attempts to modify the statement of the problem so as to see a similar optimization in the standard complex theory are not successful (with one limited exception). So it seems that this optimization should be regarded as a special feature of Real-Vector-Space quantum theory.
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communicating through probabilities does quantum theory optimize the transfer of information
Entropy, 2013Co-Authors: William K WoottersAbstract:A quantum measurement can be regarded as a communication channel, in which the parameters of the state are expressed only in the probabilities of the outcomes of the measurement. We begin this paper by considering, in a non-quantum-mechanical setting, the problem of communicating through probabilities. For example, a sender, Alice, wants to convey to a receiver, Bob, the value of a continuous variable, θ, but her only means of conveying this value is by sending Bob a coin in which the value of θ is encoded in the probability of heads. We ask what the optimal encoding is when Bob will be allowed to flip the coin only a finite number of times. As the number of tosses goes to infinity, we find that the optimal encoding is the same as what nature would do if we lived in a world governed by Real-Vector-Space quantum theory. We then ask whether the problem might be modified, so that the optimal communication strategy would be consistent with standard, complex-Vector-Space quantum theory.
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Real Vector Space quantum theory with a universal quantum bit
Physical Review A, 2013Co-Authors: Antoniya A Aleksandrova, Victoria Borish, William K WoottersAbstract:We explore a model of the world based on Real-Vector-Space quantum theory. In our model the familiar complex phase appearing in quantum states is replaced by a single binary object that we call the ubit, which is not localized and which can interact with any object in the world. Ordinary complex-Vector-Space quantum theory can be recovered from this model if we simply impose a certain restriction on the sets of allowed measurements and transformations (Stueckelberg's rule), but in this paper we try to obtain the standard theory, or a close approximation to it, without invoking such a restriction. We look particularly at the effective theory that applies to a subsystem when the ubit is interacting with a much larger environment. In a certain limit it turns out that the ubit-environment interaction has the effect of enforcing Stueckelberg's rule automatically, and we obtain a one-parameter family of effective theories--modifications of standard quantum theory--that all satisfy this rule. The one parameter is the ratio s/omega, where s quantifies the strength of the ubit's interaction with the rest of the world and omega is the ubit's rotation rate. We find that when this parameter is small but not zero, the effective theory is similar to standard quantum theory but is characterized by spontaneous decoherence of isolated systems.
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entanglement sharing in Real Vector Space quantum theory
arXiv: Quantum Physics, 2010Co-Authors: William K WoottersAbstract:The limitation on the sharing of entanglement is a basic feature of quantum theory. For example, if two qubits are completely entangled with each other, neither of them can be at all entangled with any other object. In this paper we show, at least for a certain standard definition of entanglement, that this feature is lost when one replaces the usual complex Vector Space of quantum states with a Real Vector Space. Moreover, the difference between the two theories is extreme: in the Real-Vector-Space theory, there exist states of arbitrarily many binary objects, "rebits," in which every rebit in the system is maximally entangled with each of the other rebits.
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limited holism and Real Vector Space quantum theory
arXiv: Quantum Physics, 2010Co-Authors: Lucien Hardy, William K WoottersAbstract:Quantum theory has the property of "local tomography": the state of any composite system can be reconstructed from the statistics of measurements on the individual components. In this respect the holism of quantum theory is limited. We consider in this paper a class of theories more holistic than quantum theory in that they are constrained only by "bilocal tomography": the state of any composite system is determined by the statistics of measurements on pairs of components. Under a few auxiliary assumptions, we derive certain general features of such theories. In particular, we show how the number of state parameters can depend on the number of perfectly distinguishable states. We also show that Real-Vector-Space quantum theory, while not locally tomographic, is bilocally tomographic.
Erik Walsberg - One of the best experts on this subject based on the ideXlab platform.
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coarse dimension and definable sets in expansions of the ordered Real Vector Space
Illinois Journal of Mathematics, 2020Co-Authors: Erik WalsbergAbstract:Let E⊆R. Suppose there is an s>0 such that |{k∈Z,−m≤k≤m−1:[k,k+1]∩E≠∅}|≥ms for all sufficiently large m∈N. Then there is an n∈N and a linear T:Rn→R such that T(En) is dense. As a corollary, we show that if E is in addition nowhere dense, then (R,<,+,0,(x↦λx)λ∈R,E) defines every bounded Borel subset of every Rn.
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coarse dimension and definable sets in expansions of the ordered Real Vector Space
arXiv: Logic, 2019Co-Authors: Erik WalsbergAbstract:Suppose $E \subseteq \mathbb{R}$ is nowhere dense. If $(\mathbb{R}, 0$ we have $$ | \{ k \in \mathbb{Z}, -m \leq k \leq m - 1 : [k,k+1] \cap E \neq \emptyset \} | < m^s $$ for sufficiently large $m \in \mathbb{N}$. Then there is an $n \in \mathbb{N}$ and a linear $T : \mathbb{R}^n \to \mathbb{R}$ such that $T(E^n)$ is dense. It follows that if $E$ is in addition nowhere dense then $(\mathbb{R},<,+,0,(x \mapsto \lambda x)_{\lambda \in \mathbb{R}}, E)$ defines every bounded Borel subset of every $\mathbb{R}^n$.
Melvyn B Nathanson - One of the best experts on this subject based on the ideXlab platform.
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every finite subset of an abelian group is an asymptotic approximate group
Journal of Number Theory, 2018Co-Authors: Melvyn B NathansonAbstract:Abstract If A is a nonempty subset of an additive abelian group G, then the h-fold sumset is h A = { x 1 + ⋯ + x h : x i ∈ A i for i = 1 , 2 , … , h } . We do not assume that A contains the identity, nor that A is symmetric, nor that A is finite. The set A is an ( r , l ) -approximate group in G if there exists a subset X of G such that | X | ≤ l and r A ⊆ X A . The set A is an asymptotic ( r , l ) -approximate group if the sumset hA is an ( r , l ) -approximate group for all sufficiently large h. It is proved that every polytope in a Real Vector Space is an asymptotic ( r , l ) -approximate group, that every finite set of lattice points is an asymptotic ( r , l ) -approximate group, and that every finite subset of every abelian group is an asymptotic ( r , l ) -approximate group.
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every finite subset of an abelian group is an asymptotic approximate group
arXiv: Number Theory, 2015Co-Authors: Melvyn B NathansonAbstract:If $A$ is a nonempty subset of an additive group $G$, then the $h$-fold sumset is \[ hA = \{x_1 + \cdots + x_h : x_i \in A_i \text{ for } i=1,2,\ldots, h\}. \] The set $A$ is an $(r,\ell)$-approximate group in $G$ if $A$ is a nonempty subset of a group $G$ and there exists a subset $X$ of $G$ such that $|X| \leq \ell$ and $rA \subseteq XA$. We do not assume that $A$ contains the identity, nor that $A$ is symmetric, nor that $A$ is finite. The set $A$ is an asymptotic $(r,\ell)$-approximate group if the sumset $hA$ is an $(r,\ell)$-approximate group for all sufficiently large $h$. It is proved that every polytope in a Real Vector Space is an asymptotic $(r,\ell)$-approximate group, that every finite set of lattice points is an asymptotic $(r,\ell)$-approximate group, and that every finite subset of an abelian group is an asymptotic $(r,\ell)$-approximate group.
Mineyama Ryosuke - One of the best experts on this subject based on the ideXlab platform.
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Cannon-Thurston maps for Coxeter groups including affine special subgroups
2014Co-Authors: Mineyama RyosukeAbstract:For a Coxeter group $W$ we have an associating bi-linear form $B$ on a Real Vector Space. We assume that $B$ has the signature $(n-1,1)$. In this case we have the Cannon-Thurston map for $W$, that is, a $W$-equivariant continuous surjection from the Gromov boundary of $W$ to the limit set of $W$. We focus on the case where Coxeter groups contain affine special subgroups.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1312.317
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Cannon-Thurston maps for Coxeter groups with signature $(n-1,1)$
2014Co-Authors: Mineyama RyosukeAbstract:For a Coxeter group $W$ we have an associating bi-linear form $B$ on suitable Real Vector Space. We assume that $B$ has the signature $(n-1,1)$ and all the bi-linear form associating rank $n' (\ge 3)$ Coxeter subgroups generated by subsets of $S$ has the signature $(n',0)$ or $(n'-1,1)$. Under these assumptions, we see that there exists the Cannon-Thurston map for $W$, that is, the $W$-equivariant continuous surjection from the Gromov boundary of $W$ to the limit set of $W$. To see this we construct an isometric action of $W$ on an ellipsoid with the Hilbert metric. As a consequence, we see that the limit set of $W$ coincides with the set of accumulation points of roots of $W$.Comment: 24 page
V M Gichev - One of the best experts on this subject based on the ideXlab platform.
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polar representations of compact groups and convex hulls of their orbits
Differential Geometry and Its Applications, 2010Co-Authors: V M GichevAbstract:Abstract The paper contains a characterization of compact groups G ⊆ GL ( v ) , where v is a finite-dimensional Real Vector Space, which have the following property SP : the family of convex hulls of G-orbits is a semigroup with respect to the Minkowski addition. If G is finite, then SP holds if and only if G is a Coxeter group; if G is connected then SP is equivalent to the property to be polar. In general, G satisfies SP if and only if it is polar and its Weyl group is a Coxeter group.
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polar representations of compact groups and convex hulls of their orbits
arXiv: Metric Geometry, 2009Co-Authors: V M GichevAbstract:The paper contains a characterization of compact groups $G\subseteq\GL(V)$, where $V$ is a finite dimensional Real Vector Space, which have the following property \SP{}: the family of convex hulls of $G$-orbits is a semigroup with respect to the Minkowski addition. If $G$ is finite, then \SP{} holds if and only if $G$ is a Coxeter group; if $G$ is connected then \SP{} is true if and only if $G$ is polar. In general, $G$ satisfies \SP{} if and only if it is polar and its Weyl group is a Coxeter group.
