Device-independent dimension test in a multiparty Bell experiment

A device-independent dimension test for a Bell experiment aims to estimate the underlying Hilbert space dimension that is required to produce given measurement statistical data without any other assumptions concerning the quantum apparatus. Previous work mostly deals with the two-party version of this problem. In this paper, we propose a very general and robust approach to test the dimension of any subsystem in a multiparty Bell experiment. Our dimension test stems from the study of a new multiparty scenario which we call prepare-and-distribute. This is like the prepare-and-measure scenario, but the quantum state is sent to multiple, non-communicating parties. Through specific examples, we show that our test results can be tight. Furthermore, we compare the performance of our test to results based on known bipartite tests, and witness remarkable advantage, which indicates that our test is of a true multiparty nature. We conclude by pointing out that with some partial information about the quantum states involved in the experiment, it is possible to learn other interesting properties beyond dimension.

Introduction.-Suppose we have an unknown quantum system and we want to assess its quantum properties. One way to tackle this problem is by using only classical information obtained by interacting with the target system classically and thus no (possibly unrealistic) assumptions need to be made concerning the quantum states and/or measurements involved. For this purpose, often what people do is choose different means/settings to measure the system, then collect the corresponding statistical data, which is of course classical. It is wellknown, on the other hand, that if one wants to describe a quantum system completely using only classical information, the amount of information needed will increase exponentially with the size of the quantum system, which is usually much more than what is collected through measurements. Therefore, it would seem that we cannot infer any useful information about the quantum state using a limited amount of statistical data alone.
Interestingly, these tasks are indeed possible in some cases, and the information inferred is said to be deviceindependent. Clearly, they are attractive not only mathematically, but also from an application standpoint. For example, when a businessman wants to sell a quantum product, it would help if he can convince potential clients that the product is behaving as advertised. Instead of taking the machine apart piece by piece and trying to convince the buyer that there is nothing funny going on, e.g., something maliciously entangled with his company laboratory, he can choose to interact with it via measurements to obtain a small number of outcome statistics, and invoke device-independent results from the literature.
Bell experiments are typical settings to demonstrate phenomena of device-independence. In such a setting, a number of spatially separated parties share a quantum state and each party chooses one local measurement from a finite selection to measure his/her subsystem. The statistical data for all possible choices of measurements is recorded as a correlation. For bipartite cases of Bell experiments, it has been shown that the dimension of each party can be estimated in a device-independent manner [1,2] (see also [3]). Concretely, using the fact that some entangled quantum states can produce correlations violating certain Bell inequalities, the concept of dimension witness was proposed to estimate the underlying dimension, where the key idea is to build a relation between dimension and the extent that Bell inequalities are violated [1]. To make applications easier, a new easy-tocompute technique for this problem has also been provided [2], which is independent of any Bell inequalities. Other examples of device-independence on Bell experiments include assessing the amount of entanglement in some bipartite cases [4], and even pinning down the underlying quantum states completely, a task known as selftesting [5][6][7][8][9].
Though more than one approach has been discovered to deal with device-independent dimension estimation of bipartite Bell experiments, multipartite versions have not been found to the best of our knowledge. This problem is not only important and realistic, but also interesting in its own right as the generalization from bipartite to multipartite cases enriches the mathematics needed considerably as it is much more complicated. In this paper, we develop a general technique for this problem which results in an easy-to-compute lower bound for the underlying dimension of any subsystem in a general multiparty Bell experiment. To this end, we define a multiparty quantum scenario called prepare-and-distribute, and then propose an efficient way to estimate the distances between quantum states in this scenario based on measurement statistical data only. This allows us to identify device-independently a desired lower bound for the target dimension in the multipartite Bell setting.
Through important examples, we show that our result can be tight. At the same time, since we are interested in the dimensions of individual parties, in principle we can also use methods for bipartite cases (e.g. in Ref. [2]) to tackle our problem. By a concrete example, we illustrate that our new result in this paper is much better than generalizations from known bipartite results. This demonstrates that it is of a true multiparty nature. We also point out that with more information on the target quantum state, it is possible to learn other quantum properties beyond dimension in some circumstances.
Multiparty Bell Scenario.-In a multiparty Bell scenario, we have k + 1 physically separated parties, sharing a quantum state ρ acting on a (k + 1)-partite Hilbert space k+1 i=1 C di , where d i is the dimension of the i-th subsystem. Each party has a local measurement apparatus, which allows for various measurement settings which can be applied to their subsystems.
As not to be bound to 26 parties, we shall call one of them Alice, and the rest of the parties Bob-1, Bob-2, up to Bob-k. Alice will have measurement settings given by a finite set X and Bob-j will have measurement settings from a finite set Y j . Thus, when they measure the shared quantum state ρ with their chosen settings, the probability that Alice gets outcome a (from a finite set A) and Bob-j gets outcome b j (from a finite set B j ) is given by The set of all joint conditional probabilities p(ab 1 · · · b k |xy 1 · · · y k ) is called a (k + 1)-correlation (or just correlation when k is clear from context). The prepare-and-distribute scenario.-We now define a new (k+1)-party quantum scenario that is useful for the purposes of this paper. Suppose a single party, say Paula, prepares a k-partite quantum state ρ x , for some x ∈ X, and distributes it to k different, physically separated parties, which we call Roger-1, . . . , Roger-k. Then Roger-j measures his corresponding subsystem with available local POVM indexed by y j and gets the outcome b j . The measurement settings and outcomes share the same notation as in the previous discussion about multiparty Bell experiments for reasons that will be clear shortly. Like a (k + 1)-party Bell correlation, a prepare-and-distribute correlation can be defined as below with similar notations, A prepare-and-distribute experiment involving three parties can be seen in FIG. 2. Later we will discuss the close relationship between multiparty Bell scenarios and prepare-and-distribute scenarios. Bounding distances between quantum states in a prepare-and-distribute scenario.-In this section, we consider the following problem: Suppose we are given a prepare-and-distribute setting and the corresponding correlation data p(b 1 · · · b k |xy 1 · · · y k ), can we give a nontrivial estimation for the distance between two arbitrary preparations ρ x and ρ x ′ ? The answer is affirmative.
In this paper, we choose the concept of fidelity to measure the distances between quantum states [10]. For two quantum states σ 1 and σ 2 acting on the same Hilbert space, their fidelity is defined as A useful property of fidelity is that for any two quantum states, if one measures them using the same measure-ment, then the fidelity between the two outcome distributions (as classical-quantum states) is no less than that between the two original quantum states [10]. Therefore, by compositing all the local POVMs on Rogers as a whole, we can immediately get an upper bound for F(ρ x , ρ x ′ ) as below: If we want to optimize, we can indeed take the minimum over all measurement settings y 1 , . . . , y k and the bound still holds.
As a crucial part of our discussion later, we introduce a new method to estimate F(ρ x , ρ x ′ ), which has a much better performance than the simple bound above. To this end, we need the expansive property of fidelity [10], which means that for any quantum states ρ, σ and any completely positive and trace-preserving (CPTP) map Φ. The CPTP map of relevance here is the map where |b 1 b 1 | is the quantum state of Roger-1, and ρ x,y1,b1 is the joint state of the rest of (unmeasured) Rogers. Note that this map effectively measures Roger-1's part of the state, obtains outcome b 1 which he stores classically, and then the rest of Rogers are left with the state ρ x,y1,b1 . Now, starting with ρ x and ρ x ′ , we have that ). Note that this bound is valid for any y 1 ∈ Y 1 , and thus we can take the minimum over y 1 , similar to the discussion after (3).
We can now continue this argument for each subsequent measurement one at a time. In the second step, we consider Roger-2's local measurement y 2 ∈ Y 2 on ρ x,y1,b1 and ρ x ′ ,y1,b1 , which results in where we similarly define ρ x,y1,b1,y2,b2 as the quantum state of the other k − 2 Rogers after Roger-1 and Roger-2 perform POVMs y 1 and y 2 , and get outcomes b 1 and b 2 respectively. Note that in (6) we included the minimization over y 2 explicitly. Continuing further in this manner, we eventually end up with the entire state being measured, and are left with the relation that F(ρ x,y1,b1,...,y k−1 ,b k−1 , ρ x ′ ,y1,b1,...,y k−1 ,b k−1 ) ≤ min y k b k p(b k |b 1 · · · b k−1 xy 1 · · · y k ) p(b k |b 1 · · · b k−1 x ′ y 1 · · · y k ). (7) Then by the chain rule in probability theory, we obtain the following lemma. For simplicity, we define the vectors b = b 1 · · · b k and y = y 1 · · · y k . Lemma 1. In a prepare-and-distribute experiment generating the correlation p( b|x y), it holds that where, for a function f ( y, b), we define Here AMS is short for alternating minimization and summation. Note that this bound is valid for any ordering of the Rogers, so in (9) we also have the freedom to optimize over such orderings.
Clearly, the bound given by the above lemma is stronger than (3). Later we will see that the gap can be very large.
Dimension estimations in the multiparty Bell scenario.-We now turn to the main problem of the current paper: In a multiparty Bell scenario, can we test the Hilbert space dimension of a specific party in a device-independent manner? We designate Alice as the party whose Hilbert space dimension we are testing and, after fixing Alice, we may assume that the shared quantum state ρ is pure (one of the Bobs can hold the purification and measure it trivially to obtain the same correlation data). Now let us explain the relation between multiparty Bell scenarios and prepare-and-distribute scenarios that we mentioned earlier. Suppose Alice measures her subsystem with any specific measurement x. Then different outcomes a will force the other subsystems to collapse onto different quantum states ρ x,a , which means she essentially "prepares and distributes" ρ x,a on the other subsystems with probability p(a|x). Meanwhile, due to nosignalling, for any x, we have that where A is Alice's Hilbert space. In this way, for any x, x ′ ∈ X we have that . (11) Note that Then by Lemma 1, we have that Tr( (13) So far, what we have done is upper bound the purity of the joint state of the Bobs. We now argue how this implies a dimension bound for Alice. Since ρ is pure, we have that where B is the combined Hilbert space of all the Bobs. Since Tr B (ρ) is a quantum state on A, we have that where dim(A) is Alice's Hilbert space dimension. By combining (13), (14), and (15), and using the chain rule of probability theory (p(a b|x y) = p(a|x)p( b|ax y)), we have the main result of this paper, below.
Theorem. In a multiparty Bell experiment generating the correlation p(a b|x y), the Hilbert space dimension of Alice is at least (16) Note that the dimension of any other subsystem can be tested similarly by defining that party to be Alice. First, let us examine the (k + 1)-party PR-box [11,12] where the correlation probability p(ab 1 . . . b k |xy 1 . . . y k ) can be expressed as Then the bound (16) shows that Alice's dimension must be infinite, which can be seen as follows. We choose x = 0, x ′ = 1, then when a = a ′ , let y that optimizes (16) be 1 · · · 1, otherwise let it be 0 · · · 0. This reproves that like in the bipartite case, multiparty PR-Box cannot be produced by any finite-dimensional Hilbert spaces. Next we consider a finite-dimensional example. The GHZ correlation is generated by the k-qubit quantum state and each party has binary measurement settings and outcomes given by the following measurements: Pauli-X : andî is the imaginary unit. Then we have that where h denotes the Hamming weight of a binary vector. It can be verified that if we choose x = 0, x ′ = 1 and y 1 = · · · = y k−1 = 0, the lower bound for Alice's dimension is 2 for any k, which is obviously tight.
Remarkable advantage over bipartite results.-Though we are focusing on multiparty Bell scenarios in this paper, one could in principle apply bipartite results by interpreting the correlation as a bipartite one by combining the Bobs into a single party. Since device-independent dimension tests already exist for bipartite cases (for example [2]), this provides a simple solution for our problem. In this situation, a natural question is whether the new result we provide in the current paper can beat this "bipartite approach". In fact, the following example shows that this is the case, and moreover, the advantage can be great.
Consider a three-party Bell experiment in which each party has two binary POVMs, and the correlation is given as First thing we note is that this correlation is nonsignalling, so it is a reasonable candidate to test our bound. We focus on the dimension of Alice's subsystem. By straightforward calculation, one can verify that the lower bound provided by Ref. [2] is 4, while the lower bound given by (16) in this paper is infinite. This means that this correlation cannot be produced by any finitedimensional quantum system. Clearly, this example indicates that the result in the current paper enjoys remarkable advantage over the bipartite results in Ref. [2], for example, and thus we believe is of a true multipartite nature.
It should be pointed out that the correlation (20) is also an example illustrating the fact that considering a different ordering of the Bobs in our bound results in a different performance. In fact, if we switch the roles of Bob-1 and Bob-2, the dimension bound will decay to finite.
Purity and entanglement test.-Going back to the proof of our theorem, we can see from (15) that the purity of Tr B (ρ) is the quantity that we actually test. Recall that the purity of a quantum state σ is defined as Tr(σ 2 ). It turns out that the purity contains much more information than just a bound on the dimension. For example, for a bipartite pure state, the purity of reduced density matrices can be used to lower bound the amount of entanglement. Unfortunately, in multipartite cases the situation is much more complicated. On one hand, the concepts of entanglement measures have not yet been fully understood for multipartite quantum states, and on the other hand, mathematical difficulties also arise in these cases [13]. However, in our setting if somehow more information on the structure of the shared quantum state is already known, it is possible to draw nontrivial conclusions on entanglement of multiparty quantum states. As an example, suppose in addition to the correlation data, we are told that the shared quantum state can be transferred to a state of the form m i=1 a i |i ⊗n by local unitary operations. Then like in bipartite cases [14], we can give a nontrivial estimation for the amount of entanglement based on only the purity estimation of the reduced density matrices. It should be pointed out that because of the need of extra (quantum) information, this would no longer be fully device-independent, but still could be interesting nonetheless, as sometimes these assumptions may be reasonable. Rigorous device-independent techniques to test multipartite entanglement, for example [15,16], rely on multipartite Bell inequalities that often involve complicated geometrical characterizations of multipartite quantum correlation sets. With these extra assumptions that we discussed, our approach avoids such multipartite Bell inequalities which will be very convenient for certain applications.
Conclusions.-In this work, we defined the prepareand-distribute scenario, and developed an efficient tech-nique for estimating distances between quantum preparations based only on measurement correlation data. This allowed us to derive a device-independent lower bound for the Hilbert space dimension of any given party in a multiparty Bell scenario and gave examples showing that the result can be tight. Furthermore, by comparing the performance of our bound with methods based on bipartite dimension bounds, we showed that our bound is much stronger, revealing its multipartite nature. Moreover, our bound involves only simple functions of the correlation data, thus being easy to calculate (all the examples in this paper can be computed by hand), and allowing it to enjoy a robustness against experimental uncertainty during the process of gathering the correlation data. We believe these features allow this work to have great potential for future applications. In particular, we hope it will prove itself useful in future quantum experiments involving multipartite device-independent tasks.