Branches and matrix-product states

I’m happy to use this bully pulpit to advertise that the following paper has been deemed “probably not terrible”, i.e., published.

Here’s the figureThe editor tried to convince me that this figure appeared on the cover for purely aesthetic reasons and this does not mean my letter is the best thing in the issue…but I know better!a   and caption:

It is my highly unusual opinion that identifying a definition for the branches in the wavefunction is the most conceptually important problem is physics. The reasoning is straightforward: (1) quantum mechanics is the most profound thing we know about the universe, (2) the measurement process is at the heart of the weirdness, and (3) the critical roadblock to analysis is a definition of what we’re talking about. (Each step is of course highly disputed, and I won’t defend the reasoning here.) In my biased opinion, the paper represents the closest yet anyone has gotten to giving a mathematically precise definition.

On the last page of the paper, I speculate on the possibility that branch finding may have practical (!) applications for speeding up numerical simulations of quantum many-body systems using matrix-product states (MPS), or tensor networks in general. The rough idea is this: Generic quantum systems are exponentially hard to simulate, but classical systems (even stochastic ones) are not. A definition of branches would identify which degrees of freedom of a quantum system could be accurately simulated classically, and when. Although classical computational transitions are understood in many certain special cases, our macroscopic observations of the real world strongly suggest that all systems we study admit classical descriptions on large enough scales.Note that whether certain degrees of freedom admit a classical effective description is a computational question.[continue reading]

Comments on Cotler, Penington, & Ranard

One way to think about the relevance of decoherence theory to measurement in quantum mechanics is that it reduces the preferred basis problem to the preferred subsystem problem; merely specifying the system of interest (by delineating it from its environment or measuring apparatus) is enough, in important special cases, to derive the measurement basis. But this immediately prompts the question: what are the preferred systems? I spent some time in grad school with my advisor trying to see if I could identify a preferred system just by looking at a large many-body Hamiltonian, but never got anything worth writing up.

I’m pleased to report that Cotler, Penington, and Ranard have tackled a closely related problem, and made a lot more progress:

The paper has a nice, logical layout and is clearly written. It also has an illuminating discussion of the purpose of nets of observables (which appear often in the algebraic QFT literature) as a way to define “physical” states and “local” observables when you have no access to a tensor decomposition into local regions.

For me personally, a key implication is that if I’m right in suggesting that we can uniquely identify the branches (and subsystems) just from the notion of locality, then this paper means we can probably reconstruct the branches just from the spectrum of the Hamiltonian.

Below are a couple other comments.

Uniqueness of locality, not spectrum fundamentality

The proper conclusion to draw from this paper is that if a quantum system can be interpreted in terms of spatially local interactions, this interpretation is probably unique. It is tempting, but I think mistaken, to also conclude that the spectrum of the Hamiltonian is more fundamental than notions of locality.… [continue reading]

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:

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. For massless fields (infinite Compton wavelength), the entanglement is long range, but the amount you can actually measure is suppressed exponentially on a scale given by the length of your measuring apparatus.

In this case Bob’s measuring apparatus has effective size c \Delta t, which of courseYou can tell this is a HEP theorist playing with some recently-learned quantum information because he sets c=1 but leaves \hbar explicit. 😀b   Bousso just calls \Delta t. (It may actually be of size L = c \Delta t or, like a radio antenna, it may effectively be this size by integrating the measurement over a time long enough for a wave of that length to pass by.) Such a device is necessarily noisy when trying to measure modes whose wavelength is longer than this scale. So Alice can only communicate to Bob with high fidelity through excitations of energy at least \hbar/\Delta t.… [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: .]

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:

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.a  

Some comments:

  • It is very satisfying to see Wigner — the titan — highlight the deep importance of the seminal work by the grandfather of my field, Dieter Zeh. Likewise for his comments on Bell:

    As to the J.S. Bell inequalities, I consider them truly important, inasmuch as they prove that in the case considered by him, one cannot define a non-negative probability function which describes the state of his system in the classical sense, i.e., gives nonnegative probabilities for all possible events….

    This is a very interesting and very important observation and it is truly surprising that it has not been made before. Perhaps some of those truly interested in the epistemology of quantum mechanics took it for granted but they did not demonstrate it.

  • I like the hierarchy of regularity that Wigner draws: data ➢ laws ➢ symmetries. Symmetries are strong restrictions on, but do not determine, laws in the same way that laws are strong restrictions on, but do not determine, data.
  • It is interesting that Wigner tried to embed relativistic restrictions into the description of initial states:

    Let me mention, finally, one effect which the theory of relativity should have introduced into the description of the initial conditions and perhaps also into the description of all states.

[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.

If you want more reading, see the blog post linked above and the citations in the introduction of my related PRA.

The fact that this isn’t taught in every graduate quantum mechanics class is more evidence that graduate education is awful and stagnant. Professors choose topics based on historical accident and inertia rather than trying to give their students a thorough, deep, and clarifying understanding of quantum mechanics.… [continue reading]