Simulating equatorial measurements on GHZ states with finite expected communication cost
Abstract
The communication cost of simulating probability distributions obtained by measuring quantum states is a natural way to quantify quantum nonlocality. While much is known in the case of bipartite entanglement, little has been done in the multipartite setting. In this paper, we focus on the GHZ state. Specifically, equatorial measurements lead to correlations similar to the ones obtained with Bell states. We give a protocol to simulate these measurements on the partite GHZ state using Ê bits of communication on average.
1 Introduction
The issue of nonlocality in quantum physics was raised in 1935 by Einstein, Podolsky and Rosen [5]. Thirty years later, John Bell proved that quantum physics yields correlations that cannot be reproduced by classical local hidden variable theories [2]. This momentous discovery led to the more general question of quantifying quantum nonlocality. Not only is this question relevant for the foundations of quantum physics, but it is directly related to our understanding of the computational power of quantum resources.
A natural quantitative approach to nonlocality is to study the amount of resources required to reproduce probabilities obtained by measuring quantum states. In this paper, we consider the simulation of these distributions using classical communication. This approach was introduced independently by several authors [8, 3, 10]. It led to a series of results, culminating with the protocol of Toner and Bacon to simulate von Neumann measurements on Bell states with a single bit of communication [11]. Later, Regev and Toner extended this result by giving a simulation of binary von Neumann measurements on arbitrary bipartite states using two classical bits [9].
We focus here on multipartite entanglement, and more specifically on GHZ states [7]. Unlike the bipartite case, which has been the topic of intensive investigation, the simulation of multipartite entanglement is still teeming with major open problems.
The easiest situation arises in the case of equatorial measurements on a GHZ state because all the marginal probability distributions obtained by tracing out one or more of the parties are uniform. Hence, it suffices in this case to simulate the partite correlation, henceforth called the full correlation. (Once this has been achieved, all the marginals can easily be made uniform [6].) Making the best of this observation, Bancal, Branciard and Gisin have given a protocol to simulate equatorial measurements on the tripartite and fourpartite GHZ states at an expected cost of 10 and 20 bits of communication [1]. However, the amount of communication entailed by their protocol is unbounded in the worst case. More recently, Branciard and Gisin impoved this in the tripartite case with a protocol using 3 bits of communication in the worst case [4].
In this paper, we give a protocol to simulate equatorial measurements on the partite GHZ state. For any measurements, our protocol has an expected cost of bits of communication, where the expectation is taken over the inner randomness of the protocol.
The paper is organized as follows. In the Section 2, we give the structure of the distribution arising from equatorial measurements on GHZ states, and then show how to simulate it. The main technical tool that we use is a protocol to sample uniform vectors on connected subsets of the circle. We call this task Uniform Vector Sampling. The protocol to sample those vectors is given in Section 3.
The structure of our simulation is inspired by the protocol proposed by Toner and Bacon to simulate von Neumann measurements on Bell states. It can be divided in two parts. In the first part, the players communicate to sample shared random vectors on the sphere. The distribution of these vectors depend on the player’s input. In the second part, they apply a postprocessing to compute their outputs. While the sampling part in the protocol of Toner and Bacon is simple, it is more involved in our case. The goal of Uniform Vector Sampling is to sample vectors with the correct distribution.
2 Simulating equatorial measurements
We consider the family of GHZ states , and the distribution generated by the following process: players each receive a qubit of . Each player applies a bipartite measurement to its share. Let denote the output of the player. The problem is to simulate the probability distribution over the player’s output using hidden variables and communication.
The measurement operators corresponding to equatorial measurements are on the equator of the Bloch sphere and therefore, can be parametrized by a single polar angle. Denote the angle corresponding to the player’s measurement. It is known that the distribution arising from such measurements is fully characterized by the full correlation (see e.g. [1]).
Proposition 1.
The distributions of the outputs is characterized by the following relations:

The full correlation is given by .

The marginal distributions are given by for all .
We denote the sphere in dimension 3 and the sphere in dimension 2. A vector on is parametrized by a single polar angle, or equivalently a real number modulo . An interval on is a connected subset of or equivalently of .
Our simulation is based on a procedure to sample uniform vectors on intervals of , when the description of this subset is shared among several players. For , we introduce the following task, called Uniform Vector Sampling and denoted . The players each receive the angles , respectively. Each player computes a message depending on his input and on a public random variable and sends it to a referee. At the end, the referee has to output a uniform angle on the interval . We measure the communication cost of a protocol for by considering the total length of all messages sent from the players to the referee.
Theorem 1.
For any , there exists a protocol for with expected communication cost at most .
We now show how to simulate equatorial measurement on GHZ states, given a protocol for . Toner and Bacon proposed a simulation of binary measurements on Bell states, using a single bit of communication. In the bipartite, the correlation between the player’s output is a scalar product of two vectors on . We sketch their protocol. For a vector , denote the half sphere centered on , . We denote the sign function with range by . Toner and Bacon prove the following theorem.
Theorem 2 ([11]).
Let be vectors in , and be uniformly distributed on . Then
To complete the simulation, it suffices to notice that shared uniform vectors on can be sampled efficiently by players using shared randomness and communication, even if only one player has a full description of . The idea is to first sample a uniform random vector on the sphere, and then the player that knows tells the other if he has to flip the random vector in order to get a vector in . This requires to send a single bit of communication.
Our simulation is based on the following observation. Consider and . These are unit vectors on , embeded in to apply Theorem 2. For these vectors, we have . Therefore, if and are two vectors sampled uniformly on , Theorem 2 gives .
We now describe the simulation in more details. The players are denoted . Before receiving their inputs, they prepare a shared variable , used for . In addition, they prepare some shared uniform random bits for . In our simulation, we only need to apply Uniform Vector Sampling with .

For , the players run the protocol for , sending their messages to .

Using the messages he received, sets uniform on the interval .

Player samples and uniformly on and for , sets

For , the player outputs .

The player outputs , where we defined .
After step 3, player has the complete description of two angles and uniformly distributed on . The purpose of step 4 is to transform the angles in uniform random vectors on , where is the vector with coordinates . Since is on the equator, it is sufficient to assign a random latitude to the vectors whose longitudes are and . Finally, after steps 5 and 6, we have
Sampling the angles and , can be done with expected bits of communication. Therefore, the whole protocol can be done with expected bits of communication.
3 Uniform Vector Sampling
The goal of this section is to prove Theorem 1. Observe that in the simulation, we only need the case . Nevertheless, our inductive proof requires to prove the stronger statement given in Section 2.
The base case:
For , there is a single input . Fix . Let be chose uniformly at random on . The player sends
to the referee, who computes . The resulting angle is uniformly distributed on . Notice that since , the length of the message is at most .
The induction step
Let . The following Lemma is the main technical tool that we use for the induction. It explains how to generate uniformly distributed variables from specific nonuniform ones. We first give and prove Lemma 1, and then use it to prove the induction.
Lemma 1.
Let denote the uniform distribution on and denote the uniform distribution on . Let be the distribution on defined by the following procedure:

Pick an integer with probability , and uniform in .

If , sample .

Otherwise, sample .

Set .
Then is the uniform distribution on .
Proof.
Denote and . We define the density functions associated to the distributions and ,
By definition, the density of for a fixed is
where denotes the convolution product of two functions. By direct calculation, we have
and
Let denote the density of the distribution . We now calculate . Notice that , and for , and have disjoint supports. Assume that (the other case is similar). In that case, for any . Let and notice that for any . We have
which concludes the proof. ∎
The induction hypothesis is that for any , it is possible for players to each send a message to another party such that he outputs an angle uniformly distributed on .
Before receiving their inputs, the players prepare the following random elements:

an integer chosen with probability ,

uniform in ,

the random elements required to run ,

the random elements required to run .
The protocol proceeds as follows. The players first players send to the referee the messages corresponding to . The referee uses them to prepare uniform on . The th player sends to the referee the message corresponding to . The referee uses it to prepare uniform on . Finally, the referee outputs .
We now analyze the protocol and prove that is uniform on the interval . To apply Lemma 1, we need to rescale the random variables and . We split the term in two parts and think of each as a shift of and in a direction that depends on the bit . Each angle is shifted in the same direction.
Let and be uniform random variables on and and be uniform random variables on . Let denote the random variable , that is, the random variable generated by for a fixed value of . The shifted random variable is uniform

either on if ,

or on if .
Using, and , we can rewrite
Similarly, let denote the random variable . Using and , it can be written
For the sum, we get the expression
where is the sum of and if and the sum of and if . According to Lemma 1, when taking the expectation over and , is uniform on ]. In consequence, is uniform on the interval .
It remains to bound the expected length of messages. Denote the expected sum of the messages length. We already know that for any . Fix . Analyzing our protocol, we get the induction:
The induction hypothesis is that . We plus this expression and get
which concludes the proof.
4 Conclusion
We gave a protocol to simulate equatorial measurements on the partite GHZ state, using bits on average. Our protocol is in two parts. Firstly, we reduce the problem to sampling vectors on regions of the . Secondly, we give a procedure to sample the vectors, called Uniform Vector Sampling. This scheme is inspired by the protocol of Toner and Bacon to simulate von Neumann measurements on Bell States.
Our work leads to an obvious question. Is it possible to transform our protocol into a protocol that is bounded in the worst case? To solve this question, it enough to give a protocol for that use bounded communication in the worst case. Uniform Vector Sampling could also be considered as a task of independent interest or be applied in other contexts.
Our work, like others on the same topic, considers only equatorial measurements. The simulation of more general measurements is an intriguingly hard question. The main difference is that they lead to nonuniform marginals. In the bipartite case, an analogous problem arises when considering nonmaximally entangled states. It may seem that modifying local marginals is easy once the correlation is simulated. Unfortunately, local transforms usually also modify the full correlation.
Aknowledgement
We thank Nicolas Gisin and Cyril Branciard for interesting discussions.
References
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