Nonlocality without entanglement

After much work we finally uploaded our paper titled “A framework for bounding nonlocality of state discrimination” on the arXiv (this is joint work with Andrew Childs, Debbie Leung, and Laura Mančinska).

Let me informally explain what quantum nonlocality without entanglement is about and briefly summarize the main results of our paper.

Can you tell an apple from an orange?

Let me describe a slightly strange game. Imagine that some crazy biotechnology corporation like Monsanto comes up with a way to grow a new genetically modified fruit called abo. An abo fruit consists of two halves: each of the halves is either an apple (A), banana (B), or orange (O).

In total there are 9 different possible abo fruits: apple-apple, apple-banana, apple-orange, banana-apple, banana-banana, etc. Moreover, let’s assume that each combination is equally likely.

Now, two friends Amandine and Bobby that live in two different places decide to play the following game. They ask somebody to buy an abo fruit, cut it in two halves and mail one half to each of them. Their task is to determine which of the 9 possible combinations it is. (We assume that abo fruits don’t go bad, since they have been properly sprayed with Monsanto’s chemicals.)

Since Amandine and Bobby live in different places, we allow them to talk over Skype to make their decision. In fact, they can even use the video mode to show each other the half of the fruit they have. Technically, this is called LOCC (local operations and classical communication).

If you think about this for a bit, you should realize that this is clearly a very silly game! Each of the parties can easily identify their half of the fruit as either an apple, orange, or banana, and inform the other party. Thus, no fun in the classical case.

Quantum fruits?

Imagine that Monsanto’s technology advances to the point when they can create superpositions of different plant DNA and produce quantum abo fruits, where each half of the fruit is an arbitrary combination of an apple, banana, and orange. For example,

\displaystyle    \frac{|\text{apple}\rangle + |\text{banana}\rangle}{\sqrt{2}}    \otimes    |\text{orange}\rangle

is a valid quantum abo fruit (Amandine has something between an apple and banana, but Bobby has an orange). If you buy it at supermarket, it might look something like this:

To be fair, we should allow only 9 different kinds of quantum abo fruits to be produced. Moreover, we require that they can be discriminated globally, i.e., when Amandine and Bobby are together and perform a joint observation of the whole fruit (in other words, the corresponding quantum states form an orthonormal product basis).

Now the question is: can Amandine and Bobby always tell the 9 quantum abo fruits apart over Skype? Surprisingly, the answer to this question is “No”.

Sausage states…

Let us consider a specific construction (also known as domino states). If we replace fruits by numbers, we can write these 9 states as

|\psi_1\rangle = |1\rangle \otimes |1\rangle
|\psi_{2,3}\rangle = |0\rangle \otimes |0 \pm 1\rangle
|\psi_{4,5}\rangle = |2\rangle \otimes |1 \pm 2\rangle
|\psi_{6,7}\rangle = |1 \pm 2\rangle \otimes |0\rangle
|\psi_{8,9}\rangle = |0 \pm 1\rangle \otimes |2\rangle

where |i \pm j\rangle := (|i\rangle \pm |j\rangle) / \sqrt{2}. Clearly, these all are product states, and one can easily check that they are mutually orthogonal.

Alternatively, we can depict them as “sausages” on a grill, where Amandine has the first half (rows), but Bobby has the second half of the state (columns):

These states were introduced by Charles Bennett, David DiVincenzo, Christopher Fuchs, Tal Mor, Eric Rains, Peter Shor, John Smolin, and William Wootters in 1998 in their paper “Quantum nonlocality without entanglement”.

They show that these states cannot be locally discriminated even if Amandine and Bobby are allowed to talk on Skype as long as they want. No matter what they do, they will always have at least some small amount (at least 0.00000531 bits) of uncertainty left about what their joint state is. (Technically, we say that these states cannot be discriminated by LOCC asymptotically.) Thus, despite there being no entanglement, these states exhibit some nonlocal properties. That’s why this phenomenon is called quantum nonlocality without entanglement.

Intuitively, the problem is that Amandine cannot just identify her half of the state as being either 0, 1, or 2, since this would destroy the superposition in case the state was, say |\psi_8\rangle or |\psi_9\rangle, and a similar argument holds for Bobby. In other words, the reduced states on Amandine’s side are not orthonormal, so there is no preferred basis in which to perform the first measurement. However, it is very hard to make this argument rigorous and obtain a quantitative estimate of the degree of failure.

Our contributions and the proof idea

We provide a framework for bounding the amount of nonlocality in a given set of bipartite quantum states in terms of a lower bound on the probability of error in any LOCC discrimination protocol. The main idea is to establish a trade-off between disturbance and information gain, i.e., to show that the more information we learn about a quantum system, the more we disturb it.

To explain this more formally, let us consider the problem of discriminating arbitray n bipartite states |\psi_1\rangle, \dotsc, |\psi_n\rangle. Assume that Amandine and Bobby execute some protocol for discriminating these states, and we stop them at some point during the protocol. Let p_1, \dotsc, p_n be the posterior probability distribution and |\phi_1\rangle, \dotsc, |\phi_n\rangle be the corresponding post-measurement states.

Definition. We say that states |\psi_i\rangle satisfy a trade-off between disturbance and information gain with constant \eta if

\eta \, \varepsilon \leq \delta

holds in every branch when the protocol has been stopped, where

\varepsilon = \max_k p_k - \frac{1}{n}

is the information gain and

\delta = \max_{i \neq j} |\langle\phi_i|\phi_j\rangle|

is the disturbance. The largest constant \eta that satisfies this inequality we call the nonlocality constant of the given states.

Intuitively, \varepsilon measures how far the posterior probability distribution is from uniform, but \delta measures how nonorthogonal the post-measurement states have become.

Now our main result can be stated as follows:

Theorem. Any LOCC protocol for discriminating states drawn uniformly from |\psi_1\rangle, \dotsc, |\psi_n\rangle errs with probability

p_{\text{error}} \geq \displaystyle    \frac{2}{27} \, \frac{\eta^2}{n^5}

where \eta is the nonlocality constant of these states.

Our proof of this theorem is based on Helstrom bound. Our second main contribution is a systematic method for bounding the nonlocality constant \eta for a large class of product bases. In particular, we obtain specific bounds for domino states, rotated domino states, and a more general family of domino-type states.

For more details see our paper.