Quantum Dynamics as an analog of Conditional Probability

Quantum theory can be regarded as a non-commutative generalization of classical probability. From this point of view, one expects quantum dynamics to be analogous to classical conditional probabilities. In this paper, a variant of the well-known isomorphism between completely positive maps and bipartite density operators is derived, which makes this connection much more explicit. The new isomorphism is given an operational interpretation in terms of statistical correlations between ensemble preparation procedures and outcomes of measurements. Finally, the isomorphism is applied to elucidate the connection between no-cloning/no-broadcasting theorems and the monogamy of entanglement, and a simplified proof of the no-broadcasting theorem is obtained as a byproduct.Comment: 16 pages, 3 figures. v2 Presentation greatly improved, references updated and typos fixe

"It from bit" and the quantum probability rule

I argue that, on the subjective Bayesian interpretation of probability, "it from bit" requires a generalization of probability theory. This does not get us all the way to the quantum probability rule because an extra constraint, known as noncontextuality, is required. I outline the prospects for a derivation of noncontextuality within this approach and argue that it requires a realist approach to physics, or "bit from it". I then explain why this does not conflict with "it from bit". This version of the essay includes an addendum responding to the open discussion that occurred on the FQXi website. It is otherwise identical to the version submitted to the contest.Comment: First prize winner of 2013 fqxi.org essay contest, "It from bit, or bit from it?". See http://fqxi.org/community/forum/topic/1938 and links therein. v1: LaTeX 10 pages v2: 14 pages. Updated for publication in Springer Frontiers Collection volum

Conditional Density Operators and the Subjectivity of Quantum Operations

Assuming that quantum states, including pure states, represent subjective degrees of belief rather than objective properties of systems, the question of what other elements of the quantum formalism must also be taken as subjective is addressed. In particular, we ask this of the dynamical aspects of the formalism, such as Hamiltonians and unitary operators. Whilst some operations, such as the update maps corresponding to a complete projective measurement, must be subjective, the situation is not so clear in other cases. Here, it is argued that all trace preserving completely positive maps, including unitary operators, should be regarded as subjective, in the same sense as a classical conditional probability distribution. The argument is based on a reworking of the Choi-Jamiolkowski isomorphism in terms of "conditional" density operators and trace preserving completely positive maps, which mimics the relationship between conditional probabilities and stochastic maps in classical probability.Comment: 10 Pages, Work presented at "Foundations of Probability and Physics-4", Vaxjo University, June 4-9 200

Maximally epistemic interpretations of the quantum state and contextuality

We examine the relationship between quantum contextuality (in both the standard Kochen-Specker sense and in the generalised sense proposed by Spekkens) and models of quantum theory in which the quantum state is maximally epistemic. We find that preparation noncontextual models must be maximally epistemic, and these in turn must be Kochen-Specker noncontextual. This implies that the Kochen-Specker theorem is sufficient to establish both the impossibility of maximally epistemic models and the impossibility of preparation noncontextual models. The implication from preparation noncontextual to maximally epistemic then also yields a proof of Bell's theorem from an EPR-like argument.Comment: v1: 4 pages, revTeX4.1, some overlap with arXiv:1207.7192. v2: Changes in response to referees including revised proof of theorem 1, more rigorous discussion of measure theoretic assumptions and extra introductory materia

Uncertainty from the Aharonov-Vaidman Identity

In this article, I show how the Aharonov-Vaidman identity $A \left \vert \psi\right \rangle = \left \langle A \right \rangle \left \vert \psi\right \rangle + \Delta A \left \vert \psi^{\perp}_A \right \rangle$ can be used to prove relations between the standard deviations of observables in quantum mechanics. In particular, I review how it leads to a more direct and less abstract proof of the Robertson uncertainty relation $\Delta A \Delta B \geq \frac{1}{2} \left \vert \left \langle [A,B] \right \rangle \right \vert$ than the textbook proof. I discuss the relationship between these two proofs and show how the Cauchy-Schwarz inequality can be derived from the Aharonov-Vaidman identity. I give Aharonov-Vaidman based proofs of the Maccone-Pati uncertainty relations and I show how the Aharonov-Vaidman identity can be used to handle propagation of uncertainty in quantum mechanics. Finally, I show how the Aharonov-Vaidman identity can be extended to mixed states and discuss how to generalize the results to the mixed case.Comment: 31 pages, 1 figure, pdfLaTe

Is the quantum state real? An extended review of ψ\psi-ontology theorems

Towards the end of 2011, Pusey, Barrett and Rudolph derived a theorem that aimed to show that the quantum state must be ontic (a state of reality) in a broad class of realist approaches to quantum theory. This result attracted a lot of attention and controversy. The aim of this review article is to review the background to the Pusey--Barrett--Rudolph Theorem, to provide a clear presentation of the theorem itself, and to review related work that has appeared since the publication of the Pusey--Barrett--Rudolph paper. In particular, this review: Explains what it means for the quantum state to be ontic or epistemic (a state of knowledge); Reviews arguments for and against an ontic interpretation of the quantum state as they existed prior to the Pusey--Barrett--Rudolph Theorem; Explains why proving the reality of the quantum state is a very strong constraint on realist theories in that it would imply many of the known no-go theorems, such as Bell's Theorem and the need for an exponentially large ontic state space; Provides a comprehensive presentation of the Pusey--Barrett--Rudolph Theorem itself, along with subsequent improvements and criticisms of its assumptions; Reviews two other arguments for the reality of the quantum state: the first due to Hardy and the second due to Colbeck and Renner, and explains why their assumptions are less compelling than those of the Pusey--Barrett--Rudolph Theorem; Reviews subsequent work aimed at ruling out stronger notions of what it means for the quantum state to be epistemic and points out open questions in this area. The overall aim is not only to provide the background needed for the novice in this area to understand the current status, but also to discuss often overlooked subtleties that should be of interest to the experts.Comment: 88 pages, 15 figures, and a lot of sleepless nights. v2 is the journal version. Reformatted in journal format, references updated, many typo corrections and other minor updates. TeX source had to be modified slightly to compile using the arXiv autocompiler, so I recommend downloading the journal version from http://quanta.ws/ojs/index.php/quanta/article/view/2

ψ\psi-epistemic models are exponentially bad at explaining the distinguishability of quantum states

The status of the quantum state is perhaps the most controversial issue in the foundations of quantum theory. Is it an epistemic state (state of knowledge) or an ontic state (state of reality)? In realist models of quantum theory, the epistemic view asserts that nonorthogonal quantum states correspond to overlapping probability measures over the true ontic states. This naturally accounts for a large number of otherwise puzzling quantum phenomena. For example, the indistinguishability of nonorthogonal states is explained by the fact that the ontic state sometimes lies in the overlap region, in which case there is nothing in reality that could distinguish the two states. For this to work, the amount of overlap of the probability measures should be comparable to the indistinguishability of the quantum states. In this letter, I exhibit a family of states for which the ratio of these two quantities must be $\leq 2de^{-cd}$ in Hilbert spaces of dimension $d$ that are divisible by $4$. This implies that, for large Hilbert space dimension, the epistemic explanation of indistinguishability becomes implausible at an exponential rate as the Hilbert space dimension increases.Comment: 4 pages and 1 line, revTeX4-1. v2 Added two references, submitted to Phys. Rev. Lett. v3 Edited to match published versio

The non-Abelian state-dependent gauge field in optics

The covariant formulation of the quantum dynamics in CP(1) should lead to the observable geometrodynamical effects for the local dynamical variable of the light polarization states.Comment: 8 pages, 3 figures, LaTe

Pre- and Post-selection paradoxes and contextuality in quantum mechanics

Many seemingly paradoxical effects are known in the predictions for outcomes of intermediate measurements made on pre- and post-selected quantum systems. Despite appearances, these effects do not demonstrate the impossibility of a noncontextual hidden variable theory, since an explanation in terms of measurement-disturbance is possible. Nonetheless, we show that for every paradoxical effect wherein all the pre- and post- selected probabilities are 0 or 1 and the pre- and post-selected states are nonorthogonal, there is an associated proof of contextuality. This proof is obtained by considering all the measurements involved in the paradoxical effect -- the pre-selection, the post-selection, and the alternative possible intermediate measurements -- as alternative possible measurements at a single time.Comment: 5 pages, 1 figure. Submitted to Phys. Rev. Lett. v2.0 revised in the light of referee comments, results unchange