Hmmm ... the question asked intimately blends quantum mechanics and general relativity, in the sense that time (as we experience it in everyday life) is associated to our ability to causally order events.

For example, we readily communicate information forward in time, but not backwards in time, and in particular, we cannot send information faster than speed-of-light. How does this work?

Even more challenging: how can we reduce these physical puzzles to well-posed mathematical problems?

The discussion in Nielsen and Chuang's textbook *Quantum Computation and Quantum Information* of "The Principles of Deferred and Implicit Measurement" bears directly upon these mysteries ... and in turn, the Nielsen and Chuang discussion derives largely from work by Kraus, Lindblad, and Choi ... and in turn, Kraus, Lindblad, and Choi based their work largely on theorems derived in a dry 1955 article by W. Forrest Stinespring titled "Positive Functions on {$C^\ast$}-Algebras."

So we are lead to ask, how does one explicitly link Stinespring's dry algebraic theorems to the juicy physical mysteries of causality, relativity, and quantum mechanics?

Well, I had occasion earlier this week to post about this link on Scott Aaronson's *Shtetl Optimized* weblog, and I append that discussion.

The short answer is "A seminal 1955 experiment by Hanbury Brown and Twiss established the connexion" ... and the details are very interesting.

Entire books have been written upon this subject, and so I hope MathOverflow readers don't mind a fairly length answer ... which nonetheless covers only a tiny fraction of this fascinating topic ...

(from a post on *Shtetl Optimized*)

One wonderful aspect of (of many IMHO) of Scott [Aaronson] and Alex [Arkhipov's] new class of linear optics experiments is the motivation these experiments provide for students to go beyond Feynman's celebrated *Lectures on Physics* in understanding the physics of photon counting.

[note added: in particular, photon counting as a causal communication channel.]

The quantum physics of photon detection is a subtle topic that even Richard Feynman got wrong on occasion. The story of Feynman's mistake is vividly told in the *Physics Today's* obituary for Robert Hanbury Brown (volume 55(7), 2002), which tells of Feynman standing up during a talk Hanbury Brown, proclaiming (wrongly) "It can't work!", and walking out of the lecture.

The quantum physics associated to this Feynman story is summarized in series of six short letters, totaling 12 pages in all, that appeared in *Nature* during 1955-6. These letters describe what is today called the "Hanbury Brown and Twiss Effect"—the first-ever observation of higher-order photon counting correlations.

The story of the Hanbury Brown and Twiss Effect, as recounted on the pages of *Nature*, in effect has six thrilling episodes:

**Episode 1:** Hanbury Brown and Twiss announce (in effect) "In the laboratory, we observe nontrivial correlations in photons generated by glowing gases." (*Correlation between photons in two coherent beams of light*, Nature 177(4497), 1956).

**Episode 2:** Brannen and Ferguson announce (in effect) "The claims of Hanbury Brown and Twiss, if true, would require major revision of some fundamental concepts of quantum mechanics; moreover when we did a more careful experiment, we saw nothing." (*The question of correlation between photons in coherent light rays*, Nature 178(4531), 1956).

**Episode 3:** Not yet having seen Brannen and Ferguson's criticism, Hanbury Brown and Twiss further announce (in effect) "We observe nontrivial correlations even in photons from the star Sirius, and our theory allows us to determine its diameter" (*Test of new type of stellar interferometer on Sirius*, Nature 178(4541), 1956).

**Episode 4:** Hanbury Brown and Twiss reply "The experiment of Brannen and Ferguson was grossly lacking in sensitivity; had they analyzed their experiment properly, they would have *expected* to see no effect" (*The question of correlation between photons in coherent light rays*, Nature 178(4548), 1956).

**Episode 5:** In an accompanying letter, Ed Purcell announces (in effect) "Hanbury Brown and Twiss are right, moreover their theoretical predictions and their experiments data are in accord with quantum mechanics as properly understood." (Nature 178(4548), 1956).

**Episode 6:** Hanbury Brown and Twiss announce (in effect) "When the experimental methods of Brannen and Ferguson are implemented with higher sensitivity, and analyzed with due respect for quantum theory as explained by Purcell, the results wholly confirm our earlier findings." (*Correlation between photons, in coherent beams of light, detected by a coincidence counting technique*, Nature 180(4581), 1956).

When we read the 12-page story of Hanbury Brown and Twiss side-by-side with the discussion of photon counting in *The Feynman Lectures on Physics*, we are struck by three aspects of the Hanbury Brown and Twiss experiments that are *not* emphasized in the *Feynman Lectures*.

First, the Hanbury Brown and Twiss articles exhibit a charming physicality that is largely absent from the Feynman Lectures. For example, Hanbury Brown and Twiss describe the use of an "integrating motor" to measure the total current associated to photon detection during an experimental run. Modern physics students will wonder "What the heck is an integrating motor?", yet in the physics literature of the 1950s this concept was viewed as being so intuitively obvious as to require no explanation: the total number of revolutions of an electric motor (as counted by purely mechanical means!) *obviously* can be made proportional to the integral of the current flowing through it ... that's how electric meters work, right?"

As Ed Purcell's letter to *Nature* rightly observes, the observation of subtle quantum correlations with purely mechanical counters "adds lustre to the notable achievement of Hanbury Brown and Twiss."

Second, the experimental protocol of Hanbury Brown and Twiss includes elements that are highly sophisticated from the viewpoint of modern quantum information theory. In particular, while aligning their apparatus, they reverse the flow of photons by placing their eyes at the position of the source, and while physically looking at two photodetectors through a half-silvered mirror, they adjust the mirrors such that the images of the photodectors are coherently superimposed. We nowadays appreciate that from the viewpoint of QED, this time-reversed coherence is necessary to ensure that quantum fluctuations in the photon detector currents are deterministically associated to quantum fluctuations in the photon source currents.

Third, it follows that in the observations of Sirius recorded by Hanbury Brown and Twiss, their experimental record of correlated photocurrents here on earth is deterministically associated to currents that span the surface of the remote star Sirus -- eight light-years away! This counterintuitive implication was why many theoretical physicists (including Feynman) at first considered the results of Hanbury Brown and Twiss to be (literally) incredible.

Nowadays we appreciate that this seeming paradox is naturally reconciled via the quantum informatic mechanism that Nielsen and Chuang call the "Principles of Deferred and Implicit Measurement" -- principles that are formally associated to work by Kraus and Lindblad in the 1970s; principles that were not readily appreciated by Feynman and his colleagues in the 1950s.

[note added: although Stinespring published his theorems in 1955, it took decades for physicists to appreciate their implications.]

Moreover, the experiments of Hanbury Brown and Twiss were vastly wasteful of photonic resources. The star Sirius emits about $10^{46}$ photons/second, of which Hanbury Brown and Twiss detected about $10^{9}$ two-photon entangled states/second ... the relative production efficiency thus was a dismal $10^{-37}$. Even today, more than 50 years later, the production of six-photon entangled states still is dismally inefficient: in recent experiments $10^{18}$ photons/second of pump power yield about one six-photon state per thousand seconds, for a relative production efficiency of order $10^{-21}$.

We see that one of the fundamental challenges (among many!) that Scott and Alex's experiment poses for 21st century physicists, is to devise methods for generating entangled photon states that are *exponentially* more efficient than existing methods. To achieve this, modern physicists will have to do exactly what Hanbury Brown and Twiss did ... "look" at the photon detectors from the time-reversed viewpoint of the photon source ... and then (by careful design) arrange for the photon source currents to have near-unity correlation with the photon detector currents.

This is an immense practical challenge in cavity quantum electrodynamics, that we are certain to learn a great deal in trying to solve. At present we are similarly far from having scalable quantum-coherent n-photon sources, as we are far from having scalable quantum-coherent n-gate quantum computers.

These considerations are why, from an engineeering point-of-view, it is prudent to regard n-photon linear optics experiments, not as being obviously easier than building n-gate quantum circuits, but rather as being comparably challenging from a technical point-of-view. And this is why it will not be surprising (to me) if the Aaronson/Arkhipov distribution-sampling algorithms prove in the long run to be similarly seminal mathematically and theoretically—and similarly challenging experimentally—to Peter Shor's number-factoring algorithms.

**Summary:** A satisfactory understanding of mathematical/physical time is intimately bound-up with our understanding of experiments like that of Hanbury Brown and Twiss ... and even after many decades of work, we still have a long way to go, to achieve this understanding.

In particular, despite more than a century of work, we still lack a mathematical roadmap that naturally accommodates the quantum dynamics of field theory, the informatic causality of Stinespring/Kraus/Choi/Lindblad, and the dynamical state-space geometry of Riemann and Einstein ... see the concluding section of Ashtekar and Schilling's arxiv manuscript *Geometrical formulation of quantum mechanics*, and also Troy Schilling's thesis *Geometry of Quantum Mechanics* (Penn State, 1996) for further discussion.

Added comment: Troy Schilling's 1996 thesis *Geometry of Quantum Mechanics* is well-conceived, and I have often wondered about Schilling's subsequent career. If anyone has information, please post a comment.