Tomography turns out to be harder than expected

Recently, I had a very interesting discussion with Joel Klassen, one of the PhD students here at the Institute for Quantum Computing (IQC). He's been working on a problem that is closely related to that of quantum marginals.

Before explaining what Joel and his colleagues did, let me remind you about the problem of quantum marginals. Suppose you have several quantum systems (let's call them A, B, C, etc...) and you wish to know the global quantum state of all of them: ABCD..., but you are only given information about some of the marginals: for instance you know the state of AB, and that of BD, and that of AEF, etc... Can you still infer the global state? What would suffice for you to do so? These are the main questions in the quantum marginal problem.

Joel and his colleagues worked on the next question in line, which is given that you do find a state that is compatible with the marginals, is it uniquely determined by those marginals?

In quantum state tomography, it is prohibitively expensive to measure a complete set of observables to fully characterize a large quantum state. If you only needed to measure the marginals though, it would be much easier. But if you are trying to determine what quantum state you possess based solely on your marginals, and your global state isn't uniquely determined (i.e., if there are other states that have the same marginals), then the state you infer may not be the correct one! So it is important to be able to test for this.

Let's break down the uniqueness of the reconstructed state into two categories: we could say that the state is Uniquely Determined among Pure states (UDP) or Uniquely Determined among All states (UDA). It is clear that if a state is unique among all states, it is also unique among just the pure ones (because they are a subset of all states), but does the converse hold? This is a very important question, because checking if a state is unique in a space of pure states is immensely easier than checking if it is unique in a space of mixed states! So if the conjecture "UDP implies UDA" is true, we only need to check uniqueness among pure states. Three-dimensional caricatures of the possible shapes of state space, and the space of reduced density matrices as projections. Pure states are given by the extreme points. (a) A sphere, for which all boundary points are extreme points. Only the points on the boundary of the projected circle have a unique preimage in the state space, and so are UDA. All the interior points have multiple extreme points in their preimage, so they are not UDP. Thus UDP implies UDA. (b) A polytope, for which the five vertices are extreme points. The four corner points have a unique preimage in the state space, and so are UDA. However, one interior point located at the centre has multiple preimages where only one is an extreme point, so it is UDP but not UDA.

At this point you might think that this cannot be a reasonable conjecture. After all, there are way fewer pure states than mixed states, so we should expect the set of mixed states which are uniquely determined to be larger than the set of pure states which are uniquely determined. And yet, prior to this work, no counterexample to the "UDP implies UDA" conjecture had been found. What's more, there are some limited cases which support the conjecture, for example, if you reconstruct the state of three qubits by looking at pairs of them, then UDP does indeed imply UDA.

I would like to take a brief detour to make it clear what it means to look for counterexamples to a conjecture. Suppose you *think* that A implies B. This would mean that the set of elements that satisfy A is contained in the set of elements that satisfy B: if an element satisfies A then it always satisfies B because the A-set sits inside the B-set.

But if your conjecture is not true, there must be elements of the A-set that are not contained in the B-set, those are the counterexamples: elements that satisfy A without satisfying B. You just need to find a single one to show that the conjecture was false. If a conjecture is really bad it could be that the whole A-set is outside the B-set, in which case any object would serve as counterexample. But some other conjectures are very hard to disprove because they are true for most objects and the portion of the A-set that sits outside the B-set is small.

In our case, the UDA set sits entirely inside the UDP set (because if a state is unique among all states, then of course it is also unique among just the pure ones), and the conjecture states that also the converse is true, meaning that the two sets really are the same. So a counterexample would be a state that is UDP but not UDA. If you find even a single state like this, then the whole "UDP implies UDA" conjecture falls apart.

This is precisely what Joel's group did: they found a counterexample, and the way they found it is quite clever too. They first needed a large class of states which are UDP (so it's easier to look for counterexamples, as there should be more of them in a larger set). Finding this class analytically is difficult in general.

To simplify things they looked at the 2-qubit marginals of the symmetric (under swapping) pure states of 4 qubits, |ψ〉ψ. One can impose symmetry by using very specific basis states, such as |0000〉0000, |1111〉1111, |0011〉+|0101〉+|1001〉+|1100〉0011010110011100, etc.

Symmetry is a very nice property, it makes everything so much easier because the marginals are all the same! What's more, all pure states compatible with the marginals of a symmetric state must also be symmetric, so they only needed a class of states uniquely determined among pure symmetric states.

Once they had found a class of UDP states, they needed to see if any of those states were not UDA (i.e., counterexamples to the conjecture). Now, imagine a 2-qubit marginal of |ψ〉ψ that is separable: if it is separable its state must be in the form ∑ipiρ̂ i⊗ρ̂ iipiρiρi for a suitable choice of pipi and ρ̂ iρi(see? ρ̂ iρi must occur twice because of symmetry). But then there clearly is another state besides |ψ〉ψ that gives the same marginal: ∑ipiρ̂ i⊗ρ̂ i⊗ρ̂ i⊗ρ̂ iipiρiρiρiρi! So all they needed to do to obtain a counterexample to the conjecture was to check if their large class of UDP states intersected the set of states with separable 2-qubit marginals! Lo and behold there is a whole region of states which are UDP and not UDA, so the conjecture is indeed false.

A skeptical reader might begin to wonder if these cases are nothing but mathematical curiosities. In anticipation of such criticisms, Joel and his theoretical colleagues called on the aid of experimentalists Tao Xin and Dawei Lu to hunt these UDP but not UDA states down in the wild. Their experimental group are able to instantiate a system of four qubits with very high degrees of control using a solution of crotonic acid, each molecule of which contains four qubits (that's four carbon-13 atoms to our physicist and chemist readers). Using NMR spectroscopy, Dawei and Tao were able to confirm that they could replicate such states, and that these states were robust against noise.

Now, despite the conjecture being false in general, not all is lost. In fact, there can be particular situations where the conjecture is true, such as the 3-qubit case. It is worth investigating how to characterize these special cases, because viable quantum technologies will make use of large collections of qubits, and strategies for verifying their correct function will probably employ these cases.

From a more philosophical point of view, this piece of research is telling us something new about the shape of the space of quantum states and about the inherent limitations that one must face when presented with the task of learning about the quantum states of large systems. After all, the way we interact with the universe is by local interactions, so by necessity we probe nature through small marginals, and then make inferences about the global structure. However it seems that the true global structure could be hidden among a crowd of imposters.

A preprint of this work is available on the arxiv, by T. Xin, D. Lu, J. Klassen, N. Yu, Z. Ji, J. Chen, X. Ma, G. Long, B, Zeng and R. Laflamme).

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