arXiv subjects

William K. Wootters

Publications and source records attributed to William K. Wootters.

At least 19 recordsLinked to original sources

arXiv2024

Interpreting symplectic linear transformations in a two-qubit phase space

For the continuous Wigner function and for certain discrete Wigner functions, permuting the values of the Wigner function in accordance with a symplectic linear transformation is equivalent to performing a certain unitary transformation on the state. That is, performing this unitary transformation is simply a matter of moving Wigner-function values around in phase space. This result holds in particular for the simplest discrete Wigner function defined on a $d \times d$ phase space when the Hilbert-space dimension $d$ is odd. It does not hold for a $d \times d$ phase space if the dimension is even. Here we show, though, that a generalized version of this correspondence does apply in the case of a two-qubit phase space. In this case, a symplectic linear permutation of the points of the phase space, together with a certain reinterpretation of the Wigner function, is equivalent to a unitary transformation.

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arXiv2022

A quantum prediction as a collection of epistemically restricted classical predictions

Spekkens has introduced an epistemically restricted classical theory of discrete systems, based on discrete phase space. The theory manifests a number of quantum-like properties but cannot fully imitate quantum theory because it is noncontextual. In this paper we show how, for a certain class of quantum systems, the quantum description of an experiment can be decomposed into classical descriptions that are epistemically restricted, though in a different sense than in Spekkens' work. For each aspect of the experiment -- the preparation, the transformations, and the measurement -- the epistemic restriction limits the form of the probability distribution an imagined classical observer may use. There are also global constraints that the whole collection of classical descriptions must satisfy. Each classical description generates its own prediction regarding the outcome of the experiment. One recovers the quantum prediction via a simple but highly nonclassical rule: the "nonrandom part" of the predicted quantum probabilities is obtained by summing the nonrandom parts of the classically predicted probabilities. By "nonrandom part" we mean the deviation from complete randomness, that is, from what one would expect upon measuring the fully mixed state.

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arXiv2022

A classical formulation of quantum theory?

We explore a particular way of reformulating quantum theory in classical terms, starting with phase space rather than Hilbert space, and with actual probability distributions rather than quasiprobabilities. The classical picture we start with is epistemically restricted, in the spirit of a model introduced by Spekkens. We obtain quantum theory only by combining a collection of restricted classical pictures. Our main challenge in this paper is to find a simple way of characterizing the allowed sets of classical pictures. We present one promising approach to this problem and show how it works out for the case of a single qubit.

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arXiv2020

Transition probabilities and transition rates in discrete phase space

The evolution of the discrete Wigner function is formally similar to a probabilistic process, but the transition probabilities, like the discrete Wigner function itself, can be negative. We investigate these transition probabilities, as well as the transition rates for a continuous process, aiming particularly to give simple criteria for deciding when a set of such quantities corresponds to a legitimate quantum process. We also show how the transition rates for any Hamiltonian evolution can be worked out by expanding the Hamiltonian as a linear combination of displacement operators in the discrete phase space.

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arXiv2018

The Boltzmann distribution and the quantum-classical correspondence

In this paper we explore the following question: can the probabilities constituting the quantum Boltzmann distribution, $P^B_n \propto e^{-E_n/kT}$, be derived from a requirement that the quantum configuration-space distribution for a system in thermal equilibrium be very similar to the corresponding classical distribution? It is certainly to be expected that the quantum distribution in configuration space will approach the classical distribution as the temperature approaches infinity, and a well-known equation derived from the Boltzmann distribution shows that this is generically the case. Here we ask whether one can reason in the opposite direction, that is, from quantum-classical agreement to the Boltzmann probabilities. For two of the simple examples we consider---a particle in a one-dimensional box and a simple harmonic oscillator---this approach leads to probability distributions that provably approach the Boltzmann probabilities at high temperature, in the sense that the Kullback-Leibler divergence between the distributions approaches zero.

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arXiv2015

Entanglement Sharing in Real-Vector-Space Quantum Theory

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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arXiv2015

The rebit three-tangle and its relation to two-qubit entanglement

The three-tangle is a measure of three-way entanglement in a system of three qubits. For a pure state, it can be understood as the residual entanglement not accounted for by pairwise entanglements between individual qubits. Here we define and evaluate the analogous quantity for three rebits (that is, binary systems in the real-amplitude variant of quantum theory). We find that the resulting formula is the same as in the complex case, except that an overall absolute value sign is missing. As a result, the rebit three-tangle can be negative, expressing the possibility of non-monogamous entanglement in real-amplitude quantum theory (for entanglement based on the convex-roof construction). We then relate the entanglement among three rebits to the entanglement of two qubits, by re-expressing the two-qubit state as a three-rebit state in the ubit model.

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arXiv2015

States that "look the same" with respect to every basis in a mutually unbiased set

A complete set of mutually unbiased bases in a Hilbert space of dimension $d$ defines a set of $d+1$ orthogonal measurements. Relative to such a set, we define a "MUB-balanced state" to be a pure state for which the list of probabilities of the $d$ outcomes of one of these measurements is independent of the choice of measurement, up to permutations. In this paper we explicitly construct a MUB-balanced state for each prime power dimension $d$ for which $d = 3$ (mod 4). These states have already been constructed by Appleby in unpublished notes, but our presentation here is different in that both the expression for the states themselves and the proof of MUB-balancedness are given in terms of the discrete Wigner function, rather than the density matrix or state vector. The discrete Wigner functions of these states are "rotationally symmetric" in a sense roughly analogous to the rotational symmetry of the energy eigenstates of a harmonic oscillator in the continuous two-dimensional phase space. Upon converting the Wigner function to a density matrix, we find that the states are expressible as real state vectors in the standard basis. We observe numerically that when $d$ is large (and not a power of 3), a histogram of the components of such a state vector appears to form a semicircular distribution.

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arXiv2014

Unextendible Mutually Unbiased Bases from Pauli Classes

We provide a construction of sets of (d/2+1) mutually unbiased bases (MUBs) in dimensions d=4,8 using maximal commuting classes of Pauli operators. We show that these incomplete sets cannot be extended further using the operators of the Pauli group. However, specific examples of sets of MUBs obtained using our construction are shown to be strongly unextendible; that is, there does not exist another vector that is unbiased with respect to the elements in the set. We conjecture the existence of such unextendible sets in higher dimensions (d=2^{n}, n>3) as well. Furthermore, we note an interesting connection between these unextendible sets and state-independent proofs of the Kochen-Specker Theorem for two-qubit systems. Our construction also leads to a proof of the tightness of a H_{2} entropic uncertainty relation for any set of three MUBs constructed from Pauli classes in d=4.

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arXiv2013

Real-Vector-Space Quantum Theory with a Universal Quantum Bit

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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arXiv2013

Optimal Information Transfer and Real-Vector-Space Quantum Theory

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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arXiv2011

Geometric local invariants and pure three-qubit states

We explore a geometric approach to generating local SU(2) and $SL(2,\mathbb{C})$ invariants for a collection of qubits inspired by lattice gauge theory. Each local invariant or 'gauge' invariant is associated to a distinct closed path (or plaquette) joining some or all of the qubits. In lattice gauge theory, the lattice points are the discrete space-time points, the transformations between the points of the lattice are defined by parallel transporters and the gauge invariant observable associated to a particular closed path is given by the Wilson loop. In our approach the points of the lattice are qubits, the link-transformations between the qubits are defined by the correlations between them and the gauge invariant observable, the local invariants associated to a particular closed path are also given by a Wilson loop-like construction. The link transformations share many of the properties of parallel transporters although they are not undone when one retraces one's steps through the lattice. This feature is used to generate many of the invariants. We consider a pure three qubit state as a test case and find we can generate a complete set of algebraically independent local invariants in this way, however the framework given here is applicable to mixed states composed of any number of $d$ level quantum systems. We give an operational interpretation of these invariants in terms of observables.

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arXiv2010

Entanglement cost of two-qubit orthogonal measurements

The "entanglement cost" of a bipartite measurement is the amount of shared entanglement two participants need to use up in order to carry out the given measurement by means of local operations and classical communication. Here we numerically investigate the entanglement cost of generic orthogonal measurements on two qubits. Our results strongly suggest that for almost all measurements of this kind, the entanglement cost is strictly greater than the average entanglement of the eigenstates associated with the measurements, implying that the nonseparability of a two-qubit orthogonal measurement is generically distinct from the nonseparability of its eigenstates.

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arXiv2010

Limited Holism and Real-Vector-Space Quantum Theory

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.

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arXiv2010

Entanglement and Composite Bosons

We build upon work by C. K. Law [Phys. Rev. A 71, 034306 (2005)] to show in general that the entanglement between two fermions largely determines the extent to which the pair behaves like an elementary boson. Specifically, we derive upper and lower bounds on a quantity that governs the bosonic character of a pair of fermions when N such pairs approximately share the same wavefunction. Our bounds depend on the purity of the single-particle density matrix, an indicator of entanglement, and demonstrate that if the entanglement is sufficiently strong, the quantity in question approaches its ideal bosonic value.

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arXiv2009

Entanglement Cost of Nonlocal Measurements

For certain joint measurements on a pair of spatially separated particles, we ask how much entanglement is needed to carry out the measurement exactly. For a class of orthogonal measurements on two qubits with partially entangled eigenstates, we present upper and lower bounds on the entanglement cost. The upper bound is based on a recent result by D. Berry [Phys. Rev. A 75, 032349 (2007)]. The lower bound, based on the entanglement production capacity of the measurement, implies that for almost all measurements in the class we consider, the entanglement required to perform the measurement is strictly greater than the average entanglement of its eigenstates. On the other hand, we show that for any complete measurement in d x d dimensions that is invariant under all local Pauli operations, the cost of the measurement is exactly equal to the average entanglement of the states associated with the outcomes.

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arXiv2007

Discrete phase space and minimum-uncertainty states

The quantum state of a system of qubits can be represented by a Wigner function on a discrete phase space, each axis of the phase space taking values in a finite field. Within this framework, we show that one can make sense of the notion of a "rotationally invariant state" of any collection of qubits, and that any such state is, in a well defined sense, a state of minimum uncertainty.

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arXiv2005

Distinguishing unentangled states with an unentangled measurement

In a 1991 paper, Asher Peres and the author theoretically analyzed a set of unentangled bipartite quantum states that could apparently be distinguished better by a global measurement than by any sequence of local measurements on the individual subsystems. The present paper returns to the same example, and shows that the best result so far achieved can alternatively be attained by a measurement that, while still global, is "unentangled" in the sense that the operator associated with each measurement outcome is a tensor product.

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