Singular value decomposition in bra-ket notation

In linear algebra, and therefore quantum information, the singular value decomposition (SVD) is elementary, ubiquitous, and beautiful. However, I only recently realized that its expression in bra-ket notation is very elegant. The SVD is equivalent to the statement that any operator \hat{M} can be expressed as

(1)   \begin{align*} \hat{M} = \sum_i \vert A_i \rangle \lambda_i \langle B_i \vert \end{align*}

where \vert A_i \rangle and \vert B_i \rangle are orthonormal sets of vectors, possibly in Hilbert spaces with different dimensionality, and the \lambda_i \ge 0 are the singular values.

That’s it.… [continue reading]

Comments on Bousso’s communication bound

Bousso has a recent paper bounding the maximum information that can be sent by a signal from first principles in QFT:

I derive a universal upper bound on the capacity of any communication channel between two distant systems. The Holevo quantity, and hence the mutual information, is at most of order E\Delta t/\hbar, where E the average energy of the signal, and \Delta t is the amount of time for which detectors operate. The bound does not depend on the size or mass of the emitting and receiving systems, nor on the nature of the signal. No restrictions on preparing and processing the signal are imposed. As an example, I consider the encoding of information in the transverse or angular position of a signal emitted and received by systems of arbitrarily large cross-section. In the limit of a large message space, quantum effects become important even if individual signals are classical, and the bound is upheld.

Here’s his first figure:

This all stems from vacuum entanglement, an oft-neglected aspect of QFT that Bousso doesn’t emphasize in the paper as the key ingredient.I thank Scott Aaronson for first pointing this out.a   The gradient term in the Hamiltonian for QFTs means that the value of the field at two nearby locations is always entangled. In particular, the value of \phi(x) and \phi(x+\Delta x) are sometimes considered independent degrees of freedom but, for a state with bounded energy, they can’t actually take arbitrarily different values as \Delta x becomes small, or else the gradient contribution to the Hamiltonian violates the energy bound. Technically this entanglement exists over arbitrary distances, but it is exponentially suppressed on scales larger than the Compton wavelength of the field.… [continue reading]

How to think about Quantum Mechanics—Part 1: Measurements are about bases

[This post was originally “Part 0”, but it’s been moved. Other parts in this series: 1,2,3,4,5,6,7.]

In an ideal world, the formalism that you use to describe a physical system is in a one-to-one correspondence with the physically distinct configurations of the system. But sometimes it can be useful to introduce additional descriptions, in which case it is very important to understand the unphysical over-counting (e.g., gauge freedom). A scalar potential V(x) is a very convenient way of representing the vector force field, F(x) = \partial V(x), but any constant shift in the potential, V(x) \to V(x) + V_0, yields forces and dynamics that are indistinguishable, and hence the value of the potential on an absolute scale is unphysical.

One often hears that a quantum experiment measures an observable, but this is wrong, or very misleading, because it vastly over-counts the physically distinct sorts of measurements that are possible. It is much more precise to say that a given apparatus, with a given setting, simultaneously measures all observables with the same eigenvectors. More compactly, an apparatus measures an orthogonal basis – not an observable.We can also allow for the measured observable to be degenerate, in which case the apparatus simultaneously measures all observables with the same degenerate eigenspaces. To be abstract, you could say it measures a commuting subalgebra, with the nondegenerate case corresponding to the subalgebra having maximum dimensionality (i.e., the same number of dimensions as the Hilbert space). Commuting subalgebras with maximum dimension are in one-to-one correspondence with orthonormal bases, modulo multiplying the vectors by pure phases.a   You can probably start to see this by just noting that there’s no actual, physical difference between measuring X and X^3; the apparatus that would perform the two measurements are identical.… [continue reading]

Bleg: Classical theory of measurement and amplification

I’m in search of an authoritative reference giving a foundational/information-theoretic approach to classical measurement. What abstract physical properties are necessary and sufficient?

Motivation: The Copenhagen interpretation treats the measurement process as a fundamental primitive, and this persists in most uses of quantum mechanics outside of foundations. Of course, the modern view is that the measurement process is just another physical evolution, where the state of a macroscopic apparatus is conditioned on the state of a microscopic quantum system in some basis determined by their mutual interaction Hamiltonian. The apparent nonunitary aspects of the evolution inferred by the observer arises because the measured system is coupled to the observer himself; the global evolution of the system-apparatus-observer system is formally modeled as unitary (although the philosophical meaningfulness/ontology/reality of the components of the wavefunction corresponding to different measurement outcomes is disputed).

Eventually, we’d like to be able to identify all laboratory measurements as just an anthropocentric subset of wavefunction branching events. I am very interested in finding a mathematically precise criteria for branching.Note that the branches themselves may be only precisely defined in some large-N or thermodynamic limit.a   Ideally, I would like to find a property that everyone agrees must apply, at the least, to laboratory measurement processes, and (with as little change as possible) use this to find all branches — not just ones that result from laboratory measurements.Right now I find the structure of spatially-redundant information in the many-body wavefunction to be a very promising approach.b  

It seems sensible to begin with what is necessary for a classical measurement since these ought to be analyzable without all the philosophical baggage that plagues discussion of quantum measurement.… [continue reading]

Comments on an essay by Wigner

[PSA: Happy 4th of July. Juno arrives at Jupiter tonight!]

This is short and worth reading:

The sharp distinction between Initial Conditions and Laws of Nature was initiated by Isaac Newton and I consider this to be one of his most important, if not the most important, accomplishment. Before Newton there was no sharp separation between the two concepts. Kepler, to whom we owe the three precise laws of planetary motion, tried to explain also the size of the planetary orbits, and their periods. After Newton's time the sharp separation of initial conditions and laws of nature was taken for granted and rarely even mentioned. Of course, the first ones are quite arbitrary and their properties are hardly parts of physics while the recognition of the latter ones are the prime purpose of our science. Whether the sharp separation of the two will stay with us permanently is, of course, as uncertain as is all future development but this question will be further discussed later. Perhaps it should be mentioned here that the permanency of the validity of our deterministic laws of nature became questionable as a result of the realization, due initially to D. Zeh, that the states of macroscopic bodies are always under the influence of their environment; in our world they can not be kept separated from it.

This essay has no formal abstract; the above is the second paragraph, which I find to be profound. Here is the PDF. The essay shares the same name and much of the material with Wigner’s 1963 Nobel lecture [PDF].The Nobel lecture has a nice bit contrasting invariance principles with covariance principles, and dynamical invariance principles with geometrical invariance principles.[continue reading]

My talk on ideal quantum Brownian motion

I have blogged before about the conceptual importance of ideal, symplectic covariant quantum Brownian motion (QBM). In short: QBM is to open quantum systems as the harmonic oscillator is to closed quantum systems. Like the harmonic oscillator, (a) QBM is universal because it’s the leading-order behavior of a taylor series expansion; (b) QBM evolution has a very intuitive interpretation in terms of wavepackets evolving under classical flow; and (c) QBM is exactly solvable.

If that sounds like a diatribe up your alley, then you are in luck. I recently ranted about it here at PI. It’s just a summary of the literature; there are no new results. As always, I recommend downloading the raw video file so you can run it at arbitrary speed.

Abstract: In the study of closed quantum system, the simple harmonic oscillator is ubiquitous because all smooth potentials look quadratic locally, and exhaustively understanding it is very valuable because it is exactly solvable. Although not widely appreciated, Markovian quantum Brownian motion (QBM) plays almost exactly the same role in the study of open quantum systems. QBM is ubiquitous because it arises from only the Markov assumption and linear Lindblad operators, and it likewise has an elegant and transparent exact solution. QBM is often introduced with specific non-Markovian models like Caldeira-Leggett, but this makes it very difficult to see which phenomena are universal and which are idiosyncratic to the model. Like frictionless classical mechanics or nonrenormalizable field theories, the exact Markov property is aphysical, but handling this subtlety is a small price to pay for the extreme generality.
[continue reading]

Redundant consistency

I’m happy to announce the recent publication of a paper by Mike, Wojciech, and myself.

The Objective Past of a Quantum Universe: Redundant Records of Consistent Histories
C. Jess Riedel, Wojciech H. Zurek, and Michael Zwolak
Motivated by the advances of quantum Darwinism and recognizing the role played by redundancy in identifying the small subset of quantum states with resilience characteristic of objective classical reality, we explore the implications of redundant records for consistent histories. The consistent histories formalism is a tool for describing sequences of events taking place in an evolving closed quantum system. A set of histories is consistent when one can reason about them using Boolean logic, i.e., when probabilities of sequences of events that define histories are additive. However, the vast majority of the sets of histories that are merely consistent are flagrantly nonclassical in other respects. This embarras de richesses (known as the set selection problem) suggests that one must go beyond consistency to identify how the classical past arises in our quantum universe. The key intuition we follow is that the records of events that define the familiar objective past are inscribed in many distinct systems, e.g., subsystems of the environment, and are accessible locally in space and time to observers. We identify histories that are not just consistent but redundantly consistent using the partial-trace condition introduced by Finkelstein as a bridge between histories and decoherence. The existence of redundant records is a sufficient condition for redundant consistency. It selects, from the multitude of the alternative sets of consistent histories, a small subset endowed with redundant records characteristic of the objective classical past. The information about an objective history of the past is then simultaneously within reach of many, who can independently reconstruct it and arrive at compatible conclusions in the present.
[continue reading]

KS entropy generated by entanglement-breaking quantum Brownian motion

A new paper of mine (PRA 93, 012107 (2016), arXiv:1507.04083) just came out. The main theorem of the paper is not deep, but I think it’s a clarifying result within a formalism that is deep: ideal quantum Brownian motion (QBM) in symplectic generality. In this blog post, I’ll refresh you on ideal QBM, quote my abstract, explain the main result, and then — going beyond the paper — show how it’s related to the Kolmogorov-Sinai entropy and the speed at which macroscopic wavefunctions branch.

Ideal QBM

If you Google around for “quantum Brownian motion”, you’ll come across a bunch of definitions that have quirky features, and aren’t obviously related to each other. This is a shame. As I explained in an earlier blog post, ideal QBM is the generalization of the harmonic oscillator to open quantum systems. If you think harmonic oscillator are important, and you think decoherence is important, then you should understand ideal QBM.

Harmonic oscillators are ubiquitous in the world because all smooth potentials look quadratic locally. Exhaustively understanding harmonic oscillators is very valuable because they are exactly solvable in addition to being ubiquitous. In an almost identical way, all quantum Markovian degrees of freedom look locally like ideal QBM, and their completely positive (CP) dynamics can be solved exactly.

To get true generality, both harmonic oscillators and ideal QBM should be expressed in manifestly symplectic covariant form. Just like for Lorentz covariance, a dynamical equation that exhibits manifest symplectic covariance takes the same form under linear symplectic transformations on phase space. At a microscopic level, all physics is symplectic covariant (and Lorentz covariant), so this better hold.… [continue reading]