Happy New Year to You!

MI

]]>For this reason, it seems to me that fewer researchers are regarding quantum computing research narrowly, as being about building quantum computers, because it has become clear that this achievement is a long way off. More researchers are regarding quantum computing broadly, as being about physical systems conceived as quantum computational processes, because this point of view yields immediate practical benefits *and* suggests wonderful new mathematical and physical questions.

]]>Perhaps my favorite piece is one entitled “What is Quantum Theory?”, which deals with one of my obsessions. Wilczek claims that perhaps we still don’t properly understand the significance of quantum theory, especially what it has to do with symmetries. He notes that Hermann Weyl, soon after the discovery of quantum mechanics, realized that the Heisenberg commutation relations are the relations of a Lie algebra (called the Heisenberg Lie algebra), and that this exponentiates to a symmetry group (the Heisenberg group to mathematicians, Weyl group to physicists). Wilczek goes on to speculate that:

The next level in understanding may come when an overarching symmetry is found, melding the conventional symmetries and Weyl’s symmetry of quantum kinematics (made more specific, and possibly modified) into an organic whole.

I’ve never seen the information-theory blogosphere to be so dead and/or downbeat as it is right now (except for that active and always-optimistic *Nuit Blanche*, that is). This gloom surely isn’t appropriate to the Holiday season, and so I’ll post a brief appreciation of Wilczek’s essay, that emphasizes two of its aspects that (to me) seem directly applicable to opportunities for younger researchers in quantum information science.

p. 4: “Symmetry has proven a fruitful guide to the fundamentals of physical reality, both when it is observed and when it is not!”

Quantum information science has (AFAICT) only one fundamental symmetry, but it is an extraordinarily powerful one. That symmetry is Nielsen and Chuang’s Theorem 8.2: *Unitary freedom in the operator-sum representation.* This is a regrettably awkward name for what is arguably the most fundamental mathematical theorem in the book, so I am going to abbreviate it as UNFOSURE—suggestions for a better name would be very welcome!

We first notice that UNFOSURE is a gauge invariance, in the sense that when we unravel a quantum trajectory, we are free to choose our UNFOSURE representation independently (or even adaptively) at every point in the state-space. This point of view has three benefits: (1) it immediately links UNFOSURE to Terry Tao’s wonderful on-line discussion of gauge theory, (2) it links UNFOSURE to field theory and general relativity, which as Wilczek reminds us, historically made little progress until gauge invariance was appreciated as central to field-theoretic dynamics, and (3) UNFOSURE guarantees that pure states of open quantum systems are unobservable, so that open quantum systems become paradigmatic examples of Wilczek’s principle “Symmetry is a fruitful guide both when it is observed and when it is not!”

This sets the stage for a second appreciation of Wilczek’s essay:

Under the influence of information technology, attention has turned from the issue, famously pioneered by Gödel and Turing, of determining the limits of what is computationally possible, to the more down-to earth problem determining the limits of what is computationally practical.

If we take *practical* in it’s most literal and down-to-earth sense, namely, asking what we can compute in-practice, right now, with tools presently available, then we see that UNFOSURE is among the most valuable mathematical tools for practical calculations and simulations in quantum information science.

This comes about for the same that gauge invariance is among the most valuable mathematical tools for practical calculations in field theory, that reason being, if we are clever and diligent, we can hope to find an UNFOSURE “gauge” that makes our calculations and simulations vastly simpler and more efficient.

At present, gauge theory is much more fully developed than UNFOSURE theory … so much so, that UNFOSURE theory does not (at present) even have a name. The point of this post being, the as-yet uncharted territory of UNFOSURE invariance is fertile habitat for young researchers in quantum information science. It could even happen, perhaps, that both gauge and UNFOSURE invariance might someday be subsumed in a fundamentally informatic point of view.

Of course, a rich essay like Wilczek’s has many alternative readings … that above is just one possibility, which is Holiday-spirited to the best of my capabilities!

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