“Born in Leningrad in 1950, from childhood till (approximately) 1973 I was quite sure that nature is a large classical mechanism.” Boris Tsirelson, thinker, theorist, Russian Jew, refusenik, émigré, now Israeli, and (if at first hesitant) quantum physicist, is the person best suited to introduce him to this blog. Three findings in mathematics were named after him: Tsirelson’s bound, an upper limit to quantum mechanical correlations, which many readers of this blog will recognize, Tsirelson space, which is an example of a Banach space, and the Tsirelson drift, a counterexample in the theory of stochastic differential equations.

In his blog contribution, Tsirelson carefully and with a self-ironic undertone traces the origin of “his bound”, as he is fond of calling it, the emergence of which is intertwined with the history of quantum physics, Bell’s inequalities, the social and political situation in Europe in cold and post-cold war era, his personal life and the inner workings of his mind. Boris Tsirelson is currently Professor of Mathematics at Tel Aviv University.

By Boris Tsirelson

Born in Leningrad in 1950, from childhood till (approximately) 1973 I was quite sure that nature is a large classical mechanism. Phrases like “the wave function is a wave of probability” disturbed me, so I asked: is it a physical field, or a probability distribution? A physical field interferes, but cannot propagate in the high-dimensional configuration space. A probability distribution propagates in the configuration space, but cannot interfere, since possibilities cannot interact with one another.

I was not deeply attached to such details of classical physics as particles and waves. A monstrous fractal (or a fractalous monster?) instead of a particle would astonish but not shock me, as far as signals propagate within the monster not faster than light. However, I was deeply attached to the idea of space-time as a partially ordered set that separates elements of reality and constrains signaling.

The Universe as a cellular automaton (like Conway’s “game of life”), if space-time is discrete, or the limit of some sequence of cellular automatons, if space-time is continuous. Is this paradigm falsifiable? That is, could it ever conflict with empirical evidence (rather than theories)? I was pretty sure that it is not falsifiable (at least in the absence of strong gravitation effects like black holes), and therefore it is pointless to doubt this paradigm. Accordingly, quantum theory must be only the partial truth.

Then I became aware of Bell’s theorem. It was quite a shock. Irrespective of all the loopholes, the very idea that “my” paradigm is falsifiable (and probably will be falsified) was a shock. At that moment I told myself that this is *The Error* of my (scientific) life; and I still think so.

Naturally, I was excited. Over several years I had reached a reasonably good understanding of the quantum description (including the approach via the universal wave function). I gave some talks on Bell’s theorem at mathematical seminars. The matter was far from fashionable, and my talks went unnoticed.

In one talk, which I gave at Anatoly Vershik’s seminar in Leningrad, he asked me: “Well, by classical theory, this probability cannot exceed 3/4. What about the quantum bound? Probably, 1? Or maybe not?”

I replied: “What’s the difference? Rejection of all classical theories by a quantum process is *The Point*. Do you expect a similar rejection of all quantum theories by some process in the near future?”

Then Vershik said: “I urge you to investigate it anyway.”

Well, I did; just a cute mathematical problem, of little importance, but still a challenge, and not a hard one. (The “Tsirelson space” in the theory of Banach spaces was much harder a challenge.) Seeking a way to add to (what is now called) the Bell operator some squares and get a constant, I realized that the symmetry of the Bell operator should be the key to success. And it really was.

So, I got “my bound” (probability 0.853…, between 0.75 and 1). It was unnoticed. I did some more math, toward generalizations, and in 1980 published an article in “Letters in Mathematical Physics”, which was illegal for Soviet citizens without academic affiliation; being a “refusenik” I was lucky to work as a programmer in industry. As young Soviet mathematician, I believed naively that a paper published in English in a Western physical journal should be read by some physicists. No reaction for years. I tried to convince some mathematicians that this topic may be thought of as math not physics; no success either. I told to myself: well, I’ll wait for 10 years; if nothing happens till 1990, I’ll leave this topic. Meanwhile I published a paper in a Soviet journal (1985 in Russian; 1987 English translation) and, in collaboration with Leonid Khalfin, two papers in proceedings of an international symposium (1985, 1987). By the way, “time of classical factorization” from section 5 of our paper of 1987 reappeared in 2002 as “decoherence time” in a paper of Lajos Diosi and Claus Kiefer, and again in 2007 as ESD (“entanglement sudden death”). Also, example 4.9 from our paper of 1985, published in the same year also by Peter Rastall, reappeared in 1994 as the “Popescu-Rohrlich box”.

But what about the first paper (of 1980)? Being written in math style (as all my papers), it was first cited in 1987 by physically oriented mathematicians Lawrence Landau and Stephen Summers, and mathematically oriented physicist Reinhard Werner. I had to smile when looking at Summers and Werner’s works; their 1985 paper presents their new inequality, but their 1987 paper acknowledges that the inequality is due to me. The same three authors mentioned me also in 1988, and then again later; but nobody else joined us.

In 1991 I was, at last, permitted to emigrate from Soviet Union to Israel. To this end, “perestroika” was necessary, but not sufficient; a special effort by Vitali Milman was instrumental. I got a position at Tel Aviv University. For several years I was quite overloaded. Two of my papers appeared in 1992: one with Vershik, another with Khalfin. The latter paper treats the “time of classical factorization” (basically, the “sudden death of entanglement”) in sections 4 and 7. In the same year I wrote an invited survey for Hadronic Journal Supplement 1993, summarizing all that I had achieved, and returned to mathematics, not being aware that my first article was cited 8 times in 1992 (Braunstein, Mann, Revsen, Nakamura; Gisin, Peres; Landau; Vinduska; Popescu, Rohrlich).

The deficit of feedback and acknowledgment, in combination with me being a “refusenik” for a long time, had led to very sketchy proofs, or no published proofs at all, for many claims made in my papers. Some calculations behind the first article (of 1980) contain mistakes corrected in a paper of 2012 by Elie Wolfe and Susanna Yelin (32 years later! Did anyone else read theorem 2 in my paper of 1980?). Much worse, my survey of 1993 claims a result that was in fact an open problem for decades! It should be called Tsirelson’s scandalous failure, but instead it is now called “Tsirelson’s problem”, my scruples aside.

Quantum computation, revealed by Peter Shor in 1994-96, was a revolution in quantum probability. I asked myself, what prevented me from doing the same. The answer was not comforting. I had all the needed prerequisites; but my mathematical strength is not sufficient. I observe that my achievements in mathematics are greater than my strength. What does it mean? Just a good luck? Or maybe an ability to choose a problem that is too hard for mathematicians weaker than me and unnoticed or underestimated by those stronger than me? I do not know. About half of my topics are suggested by Anatoly Vershik (quantum Bell-type inequalities; black noise). The other half is my own choice (Banach space; Gaussian isoperimetry; triple points).

In 1997 the resilience threshold revealed by Dorit Aharonov and Michael Ben-Or impressed me. It is somewhat similar to my work of 1978 on probabilistic automaton, but much deeper, and definitely exceeds my mathematical strength. The new, most exciting subfield of quantum probability appeared to be too hard for me.

Another subfield of quantum probability, suggested by Vershik, suited me: noncommutative dynamics, in particular, the noncommutative counterpart of black noise, so-called Arveson systems of type II or III. Participating in relevant conferences, occasionally I discussed Bell inequalities with other participants. In the last decade of the 20th century I was usually able to convince a skeptic. Frustratingly, in the first decade of the 21th century I faced a small serried cohort of “Bell-deniers”. My gut feeling was telling me that most (but not all) of them realize in the depth of their hearts that they are wrong, but cannot afford to acknowledge defeat and exchange a high status in the opposition for a low status in the coalition.

New progress in quantum Bell-type inequalities (namely, their large-scale theory) is made in the current decade (2010-…) by mathematicians (Marius Junge, Carlos Palazuelos, David Perez-Garcia, Oded Regev, William Slofstra, Michael Wolf) and computer scientists (Harry Buhrman, Thomas Vidick, Ronald de Wolf). At last, mathematicians agree that this is math. Ironically, since then I am an expert in this topic no more.

In addition, an anecdote from my youth. In 1965, participating in the Soviet Academic Achievement Olympiad for scholars, 8th graders, city-round in Leningrad, I received a first rank diploma in physics and a second rank diploma in mathematics. I felt then that physics was less challenging, and concentrated on mathematics, the more so as physicists were regrettably passive in attracting young people. (Now I understand that they were the normal ones; mathematicians, on the other hand, were surprisingly active, see www.tau.ac.il/~tsirel/Research/myspace/remins.html.)

In the following year I received a first rank diploma in mathematics and nothing in physics; here is why: In the Physics Olympiad, a judge listening to my solutions unexpectedly asked: “By the way, what is the Lebesgue integral?” I started to answer. He said: “Enough; I see that our physics problems are too easy for a young mathematician such as yourself; we rather are looking for people with a strong physics intuition.” (Was it the real reason or just an excuse?)

I did not really regret the missed award; instead I turned arrogant toward physicists (but not physics), as you can feel from my papers. Do not blame me too much; I am a victim of the unfortunate incident described above. :-)

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