In general, relative arrangements of states is decided by higher-order overlaps (also known as Bargmann invariants) i.e traces of products of density matrices \(tr(\rho_1 \rho_2 … \rho_k)\). Specifically, it is possible to use these invariants to construct “physically meaningful” covariance matrix of vectors \(\langle \psi_i| \psi_j \rangle\) which encodes the relational information between quantum states. Exactly which Bargmann invariants one needs to consider depends on the orthogonality relations between states in question -in general, the triad phase (ie. phase of \(tr(\rho_1 \rho_2\rho_3) \) ) does not suffice and higher-order invariants are needed.

The same invariants \(tr(\rho_1 \rho_2 … \rho_k)\) are sufficient to characterize classes of unitary equivalence of tuples of mixed quantum states. This is in fact closely connected to the problem of characterizing orbits in the problem simultaneous conjugation of tuples of matrices. Avi Wigderson discusses this problem in his expository book “Mathematics and Computation” -p. 151 of https://www.math.ias.edu/files/Book-online-Aug0619.pdf#page=1)

Anyway, it’s interesting that this kind of question appeared in the fermionic context!

]]>They couldn’t find this identity in the literature, so on a recommendation they emailed Terry Tao about it. Though initially skeptical because it seemed the kind of thing he would have heard about if true (plus his gut feel that at least some knowledge of the matrix values is necessary), Tao was persuaded by trying examples and responded a couple of hours later with 3 proofs.

I think Tao’s blog has a post with details.

]]>Dammit! DPPs were the first thing Ben had me work on when I was at Penn, and I’ve been kind of keeping them in my back pocket in case they ever proved useful for quantum computing. (This was exceedingly unlikely, but I’m always hopefully that directions that never panned out would someday prove useful.)

Now Scott Aaronson’s gone ahead and told the whole quantum computing community about them. My back pocket is immeasurably reduced.

]]>(Incidentally, ‘t Hooft’s prediction doesn’t follow in any way from his own premises—once you have a superdeterministic conspiracy theory, it could “account for” scalable QC working in exactly the same way it could “account for” Bell/CHSH violations. But that’s a separate point.)

]]>Our theory comes with a firm prediction:

Yes, by making good use of quantum features, it will be possible in principle, to build a computer vastly superior to conventional computers, but no, these will not be able to function better than a classical computer would do, if its memory sites would be scaled down to one per Planckian volume element (or perhaps, in view of the holographic principle, one memory site per Planckian surface element), and if its processing speed would increase accordingly, typically one operation per Planckian time unit of 1e-43 seconds.

Such scaled classical computers can of course not be built, so that this quantum computer will still be allowed to perform computational miracles, but factoring a number with millions of digits into its prime factors will not be possible—unless fundamentally improved classical algorithms turn out to exist. If engineers ever succeed in making such quantum computers, it seems to me that the CAT is falsified; no classical theory can explain perfect quantum computers.

The book is also on the arxiv (https://arxiv.org/abs/1405.1548v3) though I don’t know if the preprint version is exactly the same. The Springer and Arxiv links to the book are both from the author’s web site.

Any idea how large a Google-style quantum supremacy experiment it would take to fulfill the criterion in the doubly quoted paragraph?

]]>I wonder what do you think about Doron Zeilberger’s opinion #181 concerning this year’s Abel-prize:

https://sites.math.rutgers.edu/~zeilberg/Opinion181.html

?

The writer is an interesting person, a self-avowed Marxist whose day job is to write for Investors’ Chronicle (as capitalist a magazine as you will find in the UK). He makes the point that Alasdair MacIntyre distinguished between goods of excellence such as mastery of a craft and goods of effectiveness such as wealth and power.

Ideally, one would want everyone to seek goods of excellence, that is to do what they do best and be rewarded for it, but in a capitalist society the rewards of the goods of effectiveness can be such as to mislead people into following them (I know this is true because it happened to me before I realised I was following the wrong path). I see your ‘blankfaces’ as people who have chosen the wrong path and have either not realised this or, worse still, have realised it and found themselves trapped. The author makes the good point that sloughing off ambition is a counsel of perfection: easy enough to say if you are debt-free and your mortgage paid off; far harder if you are still struggling to repay student debts.

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