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Diagonal operators in the coherent state basis

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I asked a question back in November on Physics.StackExchange, but that didn’t attract any interest from anyone. I started thinking about it again recently and figured out a good solution. The question and answer are explained below.I posted the answer on Physics.SE too since they encourage the answering of one’s own question. How lonely is that?!?a  

Q: Is there a good notion of a “diagonal” operator with respect the overcomplete basis of coherent states?
A: Yes. The operators that are “coherent-state diagonal” are those that have a smooth Glauber–Sudarshan P transform.

The primary motivation for this question is to get a clean mathematical condition for diagonality (presumably with a notion of “approximately diagonal”) for the density matrix of a system of a continuous degree of freedom being decohered. More generally, one might like to know the intuitive sense in which X, P, and X+P are all approximately diagonal in the basis of wavepackets, but RXR^\dagger is not, where R is the unitary operator which maps

(1)   \begin{align*} \vert x \rangle \to (\vert x \rangle + \mathrm{sign}(x) \vert - x \rangle) / \sqrt{2}. \end{align*}

(This operator creates a Schrodinger’s cat state by reflecting about x=0.)

For two different coherent states \vert \alpha \rangle and \vert \beta \rangle, we want to require an approximately diagonal operator A to satisfy \langle \alpha \vert A \vert \beta \rangle \approx 0, but we only want to do this if \langle \alpha \vert \beta \rangle \approx 0. For \langle \alpha \vert \beta \rangle \approx 1, we sensibly expect \langle \alpha \vert A \vert \beta \rangle to be within the eigenspectrum of A.

One might consider the negativity of the Wigner-Weyl transformCase has a pleasingly gentle introduction.b   of the density matrix (i.e. the Wigner phase-space quasi-probability distribution aka the Wigner function) as a sign of quantum coherence, since it is known that coherent superpositions (which are clearly not diagonal in the coherent state basis) have negative oscillations that mark the superposition, and also that these oscillations are destroyed by decoherence.… [continue reading]

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Comments on Tegmark’s ‘Consciousness as a State of Matter’

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[Edit: Scott Aaronson has posted on his blog with extensive criticism of Integrated Information Theory, which motivated Tegmark’s paper.]

Max Tegmark’s recent paper entitled “Consciousness as a State of Matter” has been making the rounds. See especially Sabine Hossenfelder’s critique on her blog that agrees in several places with what I say below.

Tegmark’s paper didn’t convince me that there’s anything new here with regards to the big questions of consciousness. (In fairness, I haven’t read the work of neuroscientist Giulio Tononi that motivated Tegmark’s claims). However, I was interested in what he has to say about the proper way to define subsystems in a quantum universe (i.e. to “carve reality at its joints”) and how this relates to the quantum-classical transition. There is a sense in which the modern understanding of decoherence simplifies the vague questions “How does (the appearance of) a classical world emerge in a quantum universe? ” to the slightly-less-vague question “what are the preferred subsystems of the universe, and how do they change with time?”. Tegmark describes essentially this as the “quantum factorization problem” on page 3. (My preferred formulation is as the “set-selection problem” by Dowker and Kent. Note that this is a separate problem from the origin of probability in quantum mechanicsThe problem of probability as described by Weinberg: “The difficulty is not that quantum mechanics is probabilistic—that is something we apparently just have to live with. The real difficulty is that it is also deterministic, or more precisely, that it combines a probabilistic interpretation with deterministic dynamics.” HT Steve Hsu.a  .)

Therefore, my comments are going to focus on the “object-level” calculations of the paper, and I won’t have much to say about the implications for consciousness except at the very end.… [continue reading]

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Argument against EA warmth risk

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When I’m trying to persuade someone that people ought to concentrate on effectiveness when choosing which charities to fund, I sometime hear the worry that this sort of emphasis on cold calculation risks destroying the crucial human warmth and emotion that should surround charitable giving. It’s tempting to dismiss this sort of worry out of hand, but it’s much more constructive to address it head on.I also think it gestures at a real aspect of “EA cultural”, although the direction of causality is unclear. It could just be that EA ideas are particularly attractive to us cold unfeeling robots.a   This situations happened to me today, and I struggled for a short and accessible response. I came up with the following argument later, so I’m posting it here.

It’s often noticed that many of the best surgeons treat their patients like a broken machine to be fixed, and lack any sort of bedside manner. Surgeons are also well known for their gallows humor, which has been thought to be a coping mechanism to deal with death and with the unnatural act of cutting open a living human body. Should we be worried that surgery dehumanizes the surgeon? Well, yes, this is a somewhat valid concern, which is even being addressed (with mixed results).

But in context this is only a very mild concern. The overwhelmingly most important thing is that the surgery is actually performed, and that it is done well. If someone said “I don’t think we should have doctors perform surgery because of the potential for it to take the human warmth out of medicine”, you’d rightly call them crazy! No one wants to die from a treatable appendicitis, no matter how comforting the warm and heartfelt doctors are.… [continue reading]

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State of EA organizations

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Below is a small document I prepared to summarize the current slate of organizations that are strongly related to effective altruismThere is a bit of arbitrariness about what to include.  There are some organizations that are strongly aligned with EA principles even if they do not endorse that name or the full philosophy.I am not including cause-level organization for mainstream causes, e.g. developing world health like Innovations for Poverty Action. I am including all existential risk organizations, since these are so unusual and potentially important for EA.a  . (A Google doc is available here, which can be exported to many other formats.) If I made a mistake please comment below or email me. Please feel free to take this document and build on it, especially if you would like to expand the highlights section.

By Category

Charity Evaluation

Existential risk

Meta

Former names

“Effective Fundraising” → Charity Science (Greatest Good Foundation)
“Singularity Institute” → Machine Intelligence Research Institute
“Effective Animal Activism” → Animal Charity Evaluators

Organizational relationships

FHI and CEA share office space at Oxford.  CEA is essentially an umbrella organization.  It contains GWWC and 80k, and formerly contained TLYCS and ACE.  Now the latter two organizations operate independently.

MIRI and CFAR currently share office space in Berkeley and collaborate from time to time.… [continue reading]

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New review of decoherence by Schlosshauer

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Max Schlosshauer has a new review of decoherence and how it relates to understanding the quantum-classical transition. The abstract is:

I give a pedagogical overview of decoherence and its role in providing a dynamical account of the quantum-to-classical transition. The formalism and concepts of decoherence theory are reviewed, followed by a survey of master equations and decoherence models. I also discuss methods for mitigating decoherence in quantum information processing and describe selected experimental investigations of decoherence processes.

I found it very concise and clear for its impressive breadth, and it has extensive cites to the literature. (As you may suspect, he cites me and my collaborators generously!) I think this will become one of the go-to introductions to decoherence, and I highly recommend it to beginners.

Other introductory material is Schlosshauer’s textbook and RMP (quant-ph/0312059), Zurek’s RMP (quant-ph/0105127) and Physics Today article, and the textbook by Joos et al.… [continue reading]

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Entanglement never at first order

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When two initially uncorrelated quantum systems interact through a weak coupling, no entanglement is generated at first order in the coupling constant. This is a useful and very easy to prove fact that I haven’t seen pointed out anywhere, although I assume someone has. I’d love a citation reference if you have one.

Suppose two systems \mathcal{A} and \mathcal{B} evolve under U = \exp(- i H t) where the Hamiltonian coupling them is of the form

(1)   \begin{align*} H=H_A + H_B + \epsilon H_I, \end{align*}

with H_A = H_A \otimes I_B and H_B = I_A \otimes H_B as usual. We’ll show that when the systems start out uncorrelated, \vert \psi^0 \rangle = \vert \psi_A^0 \rangle \otimes \vert \psi_B^0 \rangle, they remain unentangled (and therefore, since the global state is pure, uncorrelated) to first order in \epsilon. First, note that local unitaries cannot change the entanglement, so without loss of generality we can consider the modified unitary

(2)   \begin{align*} U' = e^{+i H_A t} e^{+i H_B t} e^{-i H t} \end{align*}

which peels off the unimportant local evolution of \mathcal{A} and \mathcal{B}. Then the Baker–Campbell–Hausdorff formula gives

(3)   \begin{align*} U' = e^{+i H_A t} e^{+i H_B t} e^{-i (H_A + H_B) t} e^{-i \epsilon H_I t} e^{Z_2} e^{Z_3} \cdots \end{align*}

where the first few Z‘s are given by

(4)   \begin{align*} Z_2 &= \frac{(-i t)^2}{2} [H_A+H_B,\epsilon H_I] \\ Z_3 &= \frac{(-i t)^3}{12} \Big( [H_A+H_B,[H_A+H_B,\epsilon H_I]]- [\epsilon H_I,[H_A+H_B,\epsilon H_I]] \Big) \\ Z_4 &= \cdots. \end{align*}

The key feature here is that every commutators in each of the Z‘s contains at least one copy of \epsilon H_I, i.e. all the Z‘s are at least first order in \epsilon. That allows us to write

(5)   \begin{align*} U' = e^{-i \epsilon H'_I t} \big(1 + O(\epsilon^2) \big) \end{align*}

for some new H'_I that is independent of \epsilon. Then we note just that a general Hamiltonian cannot produce entanglement to first order:

(6)   \begin{align*} \rho_A &= \mathrm{Tr}_B \left[ U' \vert \psi^0 \rangle \langle \psi^0 \vert {U'}^\dagger \right] \\ &= \vert \psi'_A \rangle \langle \psi'_A \vert + O(\epsilon^2) \end{align*}

where

(7)   \begin{align*} \vert \psi'_A \rangle &= \left( I - i \epsilon t \langle \psi^0_B \vert H_I' \vert \psi^0_B \rangle \right) \vert \psi^0_A \rangle . \end{align*}

This is potentially a very important (negative) result when considering decoherence detection of very weakly coupled particles. If the coupling is so small that terms beyond first order are negligible (e.g. relic neutrinos), then there is no hope of being sensitive to any decoherence.

Of course, non-entangling (unitary) effect may be important. Another way to say this result is: Two weakly coupled systems act only unitarily on each other to first order in the coupling constant.… [continue reading]

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Follow up on contextuality and non-locality

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This is a follow up on my earlier post on contextuality and non-locality. As far as I can tell, Spekken’s paper is the gold standard for how to think about contextuality in the messy real world. In particular, since the idea of “equivalent” measurements is key, we might never be able to establish that we are making “the same” measurement from one experiment to the next; there could always be small microscopic differences for which we are unable to account. However, Spekken’s idea of forming equivalence classes from measurement protocols that always produce the same results is very natural. It isolates, as much as possible, the inherent ugliness of a contextual model that gives different ontological descriptions for measurements that somehow always seem to give identical results.

I also learned an extremely important thing in my background reading. Apparently John Bell discovered contextuality a few years before Kochen and Specker (KS).This is according to Mermin’s RMP on contextuality and locality. I haven’t gone back and read Bell’s papers to make sure he really did describe something equivalent to the KS theorem.a   More importantly, Bell’s theorem on locality grew out of this discovery; the theorem is just a special case of contextuality where “the context” is a space-like separated measurement.

So now I think I can get behind Spekken’s idea that contextuality is more important than non-locality, especially non-locality per se. It seems very plausible to me that the general idea of contextuality is driving at the key thing that’s weird about quantum mechanics (QM) and that — if QM is one day more clearly explained by a successor theory — we will find that the non-local special case of contextuality isn’t particularly different from local versions.… [continue reading]

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Wigner function = Fourier transform + Coordinate rotation

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[Follow-up post: In what sense is the Wigner function a quasiprobability distribution?]

I’ve never liked how people introduce the Wigner function (aka the Wigner quasi-probability distribution). Usually, they just write down a definition like

(1)   \begin{align*} W(x,p) = \frac{1}{\pi \hbar} \int \mathrm{d}y \rho(x+y, x-y) e^{-2 i p y/\hbar} \end{align*}

and say that it’s the “closest phase-space representation” of a quantum state. One immediately wonders: What’s with the weird factor of 2, and what the heck is y? Usually, the only justification given for the probability interpretation is that integrating over one of the variables recovers the probability distribution for the other (if it were measured):

(2)   \begin{align*} \int \! \mathrm{d}p \, W(x,p) = \vert \rho(x,x) \vert^2 , \\ \int \! \mathrm{d}x \, W(x,p) = \vert \hat{\rho}(p,p) \vert^2 , \end{align*}

where \hat{\rho}(p,p') is just the density matrix in the momentum basis. But of course, that doesn’t really tell us why we should think of W(x,p), as having anything to do with the (rough) value of x conditional on a (rough) value of p.

Well now I have a much better idea of what the Wigner function actually is and how to interpret it. We start by writing it down in sane variables (and suppress \hbar):

(3)   \begin{align*} W(\bar{x},\bar{p}) = \frac{1}{2 \pi} \int \! \mathrm{d}\Delta x \,\rho \left(\bar{x}+\frac{\Delta x}{2}, \bar{x}-\frac{\Delta x}{2} \right) e^{-i \bar{p} \Delta x}. \end{align*}

So the first step in the interpretation is to consider the function

(4)   \begin{align*} M(\bar{x},\Delta x) \equiv \rho \left(\bar{x}+\frac{\Delta x}{2}, \bar{x}-\frac{\Delta x}{2} \right) , \end{align*}

which appears in the integrand. This is just the (position-space) density matrix in rotated coordinates \bar{x} \equiv (x+x')/2 and \Delta x = x-x'. There is a strong sense in which the off-diagonal terms of the density matrix represent the quantum coherence of the state between different positions, so \Delta x indexes how far this coherence extends; large values of \Delta x indicate large spatial coherence. On the other hand, \bar{x} indexes how far down the diagonal of the density matrix we move; it’s the average position of the two points between which the off-diagonal terms of the density matrix measures coherence. (See the figure below.)… [continue reading]