# Is the Heisenberg picture propagating operators “backwards in time”?

A recent arXiv post ignited an interesting discussion with students and colleagues, demonstrating once more how the Heisenberg picture in quantum mechanics can easily be misunderstood to the point of becoming almost paradoxical. Here I intend to briefly summarize what I think may be the crux of the problem (or problems). The argument below follows a discussion on the topic that I had with Masanao Ozawa few years ago; however, any error or misunderstanding in it is to be entirely attributed to me.

## One-step evolutions

Suppose that we are following the evolution of a quantum system from an initial time $t=t_0$ to a later time $t=t_1\ge t_0$ , and that the unitary operator evolving the state of the system is $U(t_1,t_0)$, so that $\rho(t_1)=U(t_1,t_0) \rho(t_0) U(t_1,t_0)^\dagger$.

The latter is called the Schrödinger picture of the evolution. In this picture, states evolve in time, while observables (like the Hamiltonian) do not.

The Heisenberg picture is meant to do the opposite: it keeps states “freezed”, while observables evolve. It can be also understood as a “pullback” operation: very much like when one looks at a rotation from the viewpoint of vectors (Schrödinger picture) or the viewpoint of the coordinate system (Heisenberg picture).

For the two pictures to give consistent predictions, that is, $Tr[\rho(t_1)\ H(t_0)]=Tr[\rho(t_0)\ H(t_1)]$, it is prescribed that, if an observable at time $t_0$ is denoted as $H(t_0)$, the same observable at the later time will be $H(t_1)=U(t_1,t_0)^\dagger H(t_0) U(t_1,t_0)$. From this relation, we see that the state evolves according to $U(t_1,t_0)$, while the observable evolves according to $U(t_1,t_0)^\dagger$ . It is quite tempting at this point to interpret this by saying that “states evolve forward in time, while observables evolve backwards in time”. If only two times are considered, that seems just a curious though innocuous way of phrasing it. Indeed I have heard a lot of researchers explaining the Heisenberg picture this way. I myself would have nodded my head hearing this some years ago. However, I now see why this interpretation can be in fact very confusing, potentially leading to wrong calculations, when more than two times are considered.

## Two-step evolutions: the wrong approach

Imagine now to fix three instants in times, $t_0\le t_1\le t_2$ and two unitary operators:  one, $U(t_1,t_0)$, describing the evolution of states from $t_0$ to $t_1$ as before; and another one, $U(t_2,t_1)$, propagating states from $t_1$ to $t_2$ . The problem is: how should one model the evolution of an observable $H$ from $t_1$ to $t_2$ ? A naive guess based on the “backwards-in-time evolution” intuition would suggest a scheme like the following: But what should be the evolution operator describing the box denoted by question marks? As the arXiv post mentioned at the beginning of this post argues, one could be tempted to say that the right evolution operator is $U(t_2,t_1)^\dagger$, probably by symmetry with the Schrödinger’s branch evolving forward in time. This naive guess leads to the equation $H(t_2)= U(t_2,t_1)^\dagger H(t_1) U(t_2,t_1) = U(t_2,t_1)^\dagger U(t_1,t_0)^\dagger H(t_0) U(t_1,t_0) U(t_2,t_1)$ .

Problem is, this is of course wrong! The correct thing to do is to understand that the total evolution of the state from $t_0$ to $t_2$ is given by the unitary operator $U(t_2,t_0)=U(t_2,t_1)U(t_1,t_0)$. Consequently, one has that $H(t_2)= U(t_1,t_0)^\dagger U(t_2,t_1)^\dagger H(t_0) U(t_2,t_1) U(t_1,t_0)$.

This is the correct description of $H(t_2)$ in the Heisenberg picture.

## Another, more subtle, source of confusion

We have seen how the naive “backwards in time” interpretation is wrong. However, at this point, another structure emerges that still suggests some kind of “time-reversal”. I am speaking now of the fact that, in the correct equation, that is, $H(t_2)= U(t_1,t_0)^\dagger U(t_2,t_1)^\dagger H(t_0) U(t_2,t_1) U(t_1,t_0)$, the order of the propagators is reversed with respect to the one that is used for states, that is $\rho(t_2)= U(t_2,t_1) U(t_1,t_0) \rho(t_0)U(t_1,t_0)^\dagger U(t_2,t_1)^\dagger$.

Given that the equation itself is correct, in what follows I am simply criticizing its interpretation. I would like to argue, in particular, that, even though the evolution operators act in reverse order on the observable, the Heisenberg picture should not (or, at least, need not) be interpreted or explained as “backwards in time” evolution.

The point is that $U(t_2,t_1)$, on its own, has no meaning in the Heisenberg picture. In the Heisenberg picture, all operators must be evolved consistently. In particular, the operator $U(t_2,t_1)$, which is defined formally at $t=t_0$ , when applied at time $t_1$ , must also be consistently evolved before being applied on anything. (Better said, the Hamiltonian generating the unitary evolves in time and, with it, the unitary operator it generates.)

Hence, in the Heisenberg picture, the propagator of observables from $t_1$ to $t_2$ is not $U(t_2,t_1)^\dagger$ but its evolved version, that is, $\tilde U(t_2,t_1)^\dagger=U(t_1,t_0)^\dagger U(t_2,t_1)^\dagger U(t_1,t_0)$.

If we substitute this into the initial formula, then we indeed obtain that $H(t_2)=\tilde U(t_2,t_1)^\dagger H(t_1) \tilde U(t_2,t_1)=U(t_1,t_0)^\dagger U(t_2,t_1)^\dagger H(t_0) U(t_2,t_1) U(t_1,t_0)$ , as it was computed at point 2 above. However, once written as above, it gives us a very clear understanding of what is going on in the Heisenberg picture.

Summarizing, the Heisenberg picture is indeed a pullback transformation, but a pullback that happens forward in time. After all, both Heisenberg and Schrödinger pictures provide equivalent representations of exactly the same process, which of course happens forward in time.

# Birkhoff-von Neumann Prize

I was delighted to learn that I was awarded with the “Birkhoff-von Neumann Prize” by the International Quantum Structures Association. I feel very honored and humbled — at once! — to join a list including such superb colleagues. Thank you very much!

Francesco Buscemi is Associate Professor at the Department of Mathematical Informatics of Nagoya University, Japan. His results solved some long-standing open problems in the foundations of quantum physics, using ideas from mathematical statistics and information theory. He established, in a series of single-authored papers, the theory of quantum statistical morphisms and quantum statistical comparison, generalizing to the noncommutative setting some fundamental results in mathematical statistics dating back to works of David Blackwell and Lucien Le Cam. In particular, Prof. Buscemi successfully applied his theory to construct the framework of “semiquantum nonlocal games,” which extend Bell tests and are now widely used in theory and experiments to certify, in a measurement device-independent way, the presence of non-classical correlations in space and time.

In such an occasion, it is impossible not to remember Professor Paul Busch, gentleman scientist, President of IQSA until his sudden death, of which I learned almost simultaneously with my award.

# The “No-Hypersignaling Principle”

An important consequence of special relativity, in particular, of the constant and finite speed of light, is that space-like separated regions in spacetime cannot communicate. This fact is often referred to as the “no-signaling principle” in physics. However, even when signaling is in fact possible, there still are obvious constraints on how signaling can occur: for example, by sending one physical bit, no more than one bit of information can be communicated; by sending two physical bits, no more than two bits of information can be communicated; and so on. Such extra constraints, that by analogy we call “no-hypersignaling,” are not dictated by special relativity, but by the physical theory describing the system being transmitted. If the physical bit is described by classical theory, then the no-hypersignaling principle is true by definition. It is not so in quantum theory, where the validity of the no-hypersignaling principle becomes a non-trivial mathematical theorem relying on a recent result by Péter E. Frenkel and Mihály Weiner (whose proof, using the “supply-demand theorem” for bipartite graphs, is very interesting in itself).

As one may suspect, the no-hypersignaling principle does not hold in general: it is possible to construct artificial worlds in which the no-hypersignaling principle is violated. Such worlds are close relatives of the “box world,” a toy-model theory used to describe conceptual devices called Popescu-Rohrlich boxes. Exploring such alternative box worlds, one further discovers that the no-hypersignaling principle is logically independent of both the conventional no-signaling principle and the information causality principle, however related these two may seem to be with no-hypersignaling.

This means that the no-hypersignaling principle needs to be either assumed from the start, or derived from presently unknown physical principles analogous to the finite and constant speed of light behind Einstein’s no-signaling principle.

The paper was published on Physical Review Letters, but is also available free of charge on the arXiv.

# Popper against the ideas of dignity, wholeness, real truth, and essentiality in science Thus I freely admit that in arriving at my proposals I have been guided, in the last analysis, by value judgments and predilections. But I hope that my proposals may be acceptable to those who value not only logical rigour but also freedom from dogmatism; who seek practical applicability, but are even more attracted by the adventure of science, and by discoveries which again and again confront us with new and unexpected questions, challenging us to try out new and hitherto undreamt-of answers.

Karl Popper, The Logic of Scientific Discovery. 2nd Edition (Routledge, 1999), p.38.

# A chat with Jesse Casman about students and quantum foundations

I have been recently interviewed by Jesse Casman about why younger students should get a course about quantum foundations and computing, and how IBM’s Quantum Experience and their QISKit can help. Here is the full text. # Trip to Bihar, India

I was invited to talk at the 3rd International Conference on Quantum Foundations in Patna, the capital city of the Indian state of Bihar. Great hospitality and many brilliant students eager to discuss and interact with the international community. A visit to the remains of the ancient university of Nalanda completed the program.