A convergent inflation hierarchy for quantum causal structures

TL;DR

A first version of the quantum inflation hierarchy that is provably convergent is constructed, which takes an additional parameter, r, which can be interpreted as an upper bound on the Schmidt rank of the observables involved, and provides a family of increasingly strict and ultimately complete compatibility tests for correlations that are compatible with a given causal structure under this Schmidt rank constraint.

Abstract

A causal structure is a description of the functional dependencies between random variables. A distribution is compatible with a given causal structure if it can be realized by a process respecting these dependencies. Deciding whether a distribution is compatible with a structure is a practically and fundamentally relevant, yet very difficult problem. Only recently has a general class of algorithms been proposed: These so-called inflation techniques associate to any causal structure a hierarchy of increasingly strict compatibility tests, where each test can be formulated as a computationally efficient convex optimization problem. Remarkably, it has been shown that in the classical case, this hierarchy is complete in the sense that each non-compatible distribution will be detected at some level of the hierarchy. An inflation hierarchy has also been formulated for causal structures that allow for the observed classical random variables to arise from measurements on quantum states - however, no proof of completeness of this quantum inflation hierarchy has been supplied. In this paper, we construct a first version of the quantum inflation hierarchy that is provably convergent. From a technical point of view, convergence proofs are built on de Finetti Theorems, which show that certain symmetries (which can be imposed in convex optimization problems) imply independence of random variables (which is not directly a convex constraint). A main technical ingredient to our proof is a Quantum de Finetti Theorem that holds for general tensor products of $C^*$-algebras, generalizing previous work that was restricted to minimal tensor products.

Figures (8)

• Figure 1: The triangle scenario -- a conceptually simple, but mathematically non-trivial causal structure that serves as the guiding example in this paper. Round vertices denote latent variables that are not directly accessible, while observed variables are written in squares. Arrows represent causal relations. In the classical case depicted in Fig. (a), the graph denotes the hypothesis that the observed variables $A, B, C$ arise in a process where (1) the latent variables $X, Y, Z$ are chosen from a product distribution, and (2) each observed variable $A, B, C$ is computed as a function of its graph-theoretical parents and additional independent randomness. Given a joint distribution of the observed variables and a candidate causal structure, we aim to decide whether the distribution is compatible with a process as outlined above. For example, take $A,B,C$ to be binary random variables. It is easy to see that a joint distribution where the outcomes are random but perfectly correlated is not compatible with the triangle scenario. In the quantum case, shown in Fig. (b), each of the round nodes represents a bi-partite quantum state. One subsystem is distributed along each outgoing arrow. At each square vertex, a bi-partite measurement is performed on the two incoming quantum systems, and the result is assigned to an observed classical random variable.
• Figure 2: The Bell scenario, where the latent variable is a quantum state. The set of correlations that can arise from this quantum version of the Bell scenario is larger than its classical counterpart. This set of quantum correlations can be categorized by the NPA hierarchy, a converging hierarchy of semidefinite programs.
• Figure 3: (a) The $n=2$ inflation of the classical triangle scenario. In the inflation procedure, the latent variables are copied and the observable variables are indexed according to these copies. For example, the variable $A_{12}$ is the result of $Z_1$ and $X_2$. (b) The $n=2$ inflation of the quantum triangle scenario. Again the latent variables are copied, but instead of obtaining copies of the observable variables, Alice, Bob and Charlie now have a choice of measurement operators. The measurements are performed over pairs of copies using the same measurement operators $\{E_a\}_a, \{F_b\}_b, \{G_c\}_c$, labeled by the copies they act on. For example, $E_a^{12}$ is acting on the parts of the quantum states $\rho_{CA}^1$ and $\rho_{AB}^2$ that are sent to Alice.
• Figure 4: We associate to each quantum system an observable algebra. This means that the POVM elements of Alice generate a subalgebra of a larger algebra $\mathcal{A}_- \otimes \mathcal{A}_+$, and similar for Bob and Charlie. It is with respect to the splitting $\mathcal{C}_+ \mathcal{A}_- \mid \mathcal{A}_+ \mathcal{B}_- \mid \mathcal{B}_+ \mathcal{C}_-$ that the global state is supposed to factorize.
• Figure 5: (a) The instrumental scenario as an example of a latent exogenous causal structure. The variable $A$ is both a parent and a child and thus the causal structure is not a network scenario. By splitting $A$ into $A$ and $A_1^\#$, as in (b), and post-selecting $A_1^\#$ on the outcome of $A$, the instrumental scenario can be modeled by a network scenario, which happens to be the Bell scenario. This process is an example of maximal interruption.

Theorems & Definitions (24)

• Theorem 1
• Definition 2
• Lemma 3
• proof
• Lemma 4
• proof : Proof (of Lemma \ref{['lem:all the same']})
• Lemma 5
• proof
• Lemma 6
• proof