If you’re a fan of the quantum switch, like me, the arXiv brought two great news this month: first, an Argentinian-Brazilian-Chilean collaboration finally managed to implement a superposition of more than two causal orders, as I had been bothering people to do since I first worked on the quantum switch. After the experiment was done they invited me, Alastair Abbott, and Cyril Branciard to help with the theory, which we gladly accepted. With their fancy new laboratory in Concepción, they managed to put 4 quantum gates in a superposition of 4 different causal orders, using the path degree of freedom (in optical fibres) of a photon as the control system, and the polarisation degree of freedom as the target system. Using the original algorithm I had developed for the quantum switch, that would have been pointless, as that algorithm required a target system of dimension at least $n!$ (and maybe even $n!^{n-1}$) to make use of a superposition of $n$ gates. To get around that they came up with a new algorithm, that seems just as hard as the old one, but only needs a target system of dimension 2 to work.

They also did an elegant refutation of a criticism that was levelled to previous versions of the experiment: people complained that depending on the state of the control system, the photons were hitting the waveplates implementing the unitaries at different points, which acted differently. In this way the experiment wasn’t actually probing the same unitaries in different orders, but rather different unitaries, which isn’t really interesting. Well here the experimentalists have shown that if you let it all four paths shine at a waveplate implement a unitary simultaneously, you get a beautiful interference pattern, showing that the paths are indistinguishable at the waveplate, and thus the same unitary is implemented, independently of the control system.

The other piece of great news appeared on the arXiv today: Barrett, Lorenz, and Oreshkov managed to show that no bipartite causally nonseparable process is purifiable. This means that if you have a process matrix, encoding the causal relationship between two parties, Alice and Bob, and this process matrix is not simply a mixture of saying that Alice acts before Bob or Bob acts before Alice, but rather encodes an indefinite causal order, then it is not possible to purify this matrix, that is, it is not possible to recover this process from a larger process, with more parties, that is compatible with the principle of conservation of information.

Now I hold the principle of conservation of information very close to my heart, and proposed a purification postulate, that if a process is not purifiable then it is unphysical. If you believe that, then the result of Barrett et al. implies that the only way to put two parties in an indefinite causal order is really just the quantum switch1. I had asked this question 3 years ago, managed only to show that a few processes are not purifiable, and speculated that none of them are. Now they came and did the hard part, solving the question completely for two parties.

UPDATE 13.03: And today the arXiv brought a paper by Yokojima et al. that proved the same result independently. They formulated it in a different way, that I find more clear: if you have a pure process, which encodes the causal relationships between Alice, Bob, a global past, and a global future, then this process is necessarily a variation of the quantum switch.

This paper also has a well-hidden gem: they showed that if you try to superpose two different causal orders, without using a control system to switch between them, then this will never be a valid process matrix. This plays wells in the theme “the quantum switch is the only thing that exists”, and allows us to be careless about talking about superpositions of causal orders. Since we can’t have superpositions without a control, then there’s no need to distinguish those with a control from those without.

Apparently this had been shown by Fabio Costa some years ago, but he kept the result a secret and got scooped.

This is cool. As I understand it is also related to the controlling of blackboxed unitaries https://arxiv.org/abs/1309.7976

Though I’m puzzled about one thing. If we have an unknown unitary then multiplying it by a phase doesn’t change its action. But controlled version of a unitary does depend on such phase. In other words, U and -U are indistinguishable blackboxes, but 1⊕U is not the same thing as 1⊕-U. How this is interpreted in the physical realization?

Thanks! It is somewhat seem related. The most straightforward way to simulate the quantum switch with a causally ordered circuit uses controlled unitaries, but it turns out that you can have an equally effective simulation using just controlled-swaps and non-controlled unitaries, so on this circuit level there’s actually no relationship. But if you look at the interferometric implementations, you see that the interferometer that allows one to control a blackbox unitary is very similar to the interferometer that allows you to implement a quantum switch (and in fact it is how I came up with the implementation in the first place).

As for your question, indeed if you have $U$ just as a non-controlled unitary in your circuit, then its phase will not have any physical effect, so we say that for any phase $\varphi$ the matrix $1\oplus e^{i\varphi}U$ is a valid controlled version of $U$. Now when you have a physical realisation of the controlled unitary the phase will have a physical effect, and will be fixed by the details of the experiment. For example in the interferometric realisation it will be determined by the relative phase between the arms of the interferometer. As far as I’m aware this phase you never actually matter, though, for example in Shor’s algorithm you do phase estimation of the modular exponentiation unitary, but only phase differences will be relevant for the result.

So, can we think that in reality every quantum gate of a circuit always corresponds to a concrete unitary (probably unknown) and not to a “unitary up to some phase”? Most quantum computation books deal with a standard circuit model and this idea seems elusive (coz global phases are irrelevant).

Also this raises a question about the notion of a quantum state, which doesn’t depend on a global phase. If we can track “phases” of every quantum gate in a circuit then we should know the exact “phase” of the output. So, maybe quantum states physically are indeed vectors and not rays, we just can’t feel the difference in a common scenarios.

In other words, global phase factors are not unphysical, they are just hidden and inaccessible.

Sorry if this a little bit off-topic.

Well, a quantum circuit abstracts away a

lotof the physics going on, it isn’t really the appropriate description to look at if you are asking these questions.But if you consider the physical unitary transformation generated by evolving the (effective) Hamiltonian for a given amount of time, this unitary will have a well-defined phase, so it might be tempting to take this as the “true” phase of the unitary. There are two problems with it: the first is that we know that a constant shift in the energy of the Hamiltonian has no effect on the dynamics, precisely because it just changes the global phase of the unitary. And since the Hamiltonian is just effective anyway, there’s no reason to take seriously which constant shift is applied to it. The second problem is that if we do make this unitary controlled, in order to be able to measure the phase, we see that the phase we’ll see depends on how precisely we do the controlling. In the interferometric case, it will not depend only on the waveplates we are using, but also on the length of the arms of the interferometer.

But lets say you find a canonical energy shift to apply to the Hamiltonian, and mod out the phase you get from the specific implementation of the controlled unitary, so that you do have a definition of the global phase of the unitary. Is this enough to find out the global phase of a quantum state? Actually not! Because even if you know all the phases that were applied to it in the circuit, still there’s nothing telling you what was the state’s initial global phase.

Also, remember that charge conservation comes from Noether’s theorem precisely by saying that the global phase of a quantum state makes no difference for its dynamics, so it’s not as if we would ever be able to find it out. I think the best way to describe this ghostly phase is really to say that it is unphysical. You can also describe quantum states via density matrices and unitaries by their Choi operators, where this phase never shows up.

Thanks. I think I should read more books on physics rather than on quantum computations to understand this deeper :)

Anyway, those great examples, the quantum switch and controlled unitaries, convincingly show that the standard circuit model should be extended.

I’m glad you find these examples great, but I no longer think that the possibility of controlling blackbox unitaries means that the circuit model should be extended. Instead, what it shows is that physical unitaries always have the property that allows them to be controlled (they will act trivially on some subspace), and if this is taken into account it can easily be shown with the standard circuit model that they can indeed be controlled.

Maybe a different way to put it is that physical unitaries are never completely blackbox.

Do you know why in the “Hadamard promise problem” the authors consider only real Hadamard matrices? It looks like complex Hadamard matrices of the Butson-type (https://en.wikipedia.org/wiki/Butson-type_Hadamard_matrix) also satisfy the math, but the target system dimension must be $q$ in such case. Also your original construction is exactly of this kind with $q=n!$ and a Fourier matrix of size $q \times q$.

Well, precisely because we wanted to get away with a target system of dimension 2. Of course, we could just use the complex version, as in the original approach, and only use $n$ orders out of the $n!$ possible ones, but that would still require a target system of dimension $n$. In the experiment $n=4$, and that would make for a much harder experiment (using OAM instead of polarisation), for no benefit.