# Understanding operator product of three mixed states with three projector operators

I am recently studying the triangle scenario in the context of Bell nonlocality (for reference, see for instance this article). In it, we have three parties, commonly referred to as Alice, Bob and Charlie, with each pair sharing a source.

In the article cited above, they say that observed statistics are quantum compatible if they can be written as

$$P(a, b, c) = \mbox{tr}\left(M^T\rho_{AB}\otimes\rho_{BC}\otimes\rho_{AC}M\ \Pi_a\otimes\Pi_b\otimes\Pi_c\right)$$

Where $$a, b,c$$ are the outputs of each party, $$\rho_{AB},\rho_{BC},\rho_{AC}$$ are the bipartite density matrices, $$\Pi_a,\Pi_b,\Pi_c$$ are projectors and $$M$$ is a permutation matrix to align the underlying tensor structure of the states and measurements appropriately.

While I do understand where this expression comes from (basically it's Born's rule), I don't understand how this product of operators actually works, and what $$M$$ is doing.

By this I mean the following: as I understand it, any given state of the whole system must belong to the tensor product of the Hilbert spaces $$H_A \otimes H_B\otimes H_C$$. I know how to apply the operator $$\Pi_A\otimes\Pi_B\otimes\Pi_C$$ on one of this states, but I don't know how to apply the tensor product of density matrices $$\rho = \rho_{AB}\otimes\rho_{BC}\otimes\rho_{AC}$$ on it. For instance, $$\rho_{AB}$$ operates on $$H_A \otimes H_B$$, no? Thus wouldn't $$\rho$$ operate on

$$(H_A \otimes H_B) \otimes (H_B \otimes H_C) \otimes (H_A \otimes H_C)?$$

• I suspect that each space $A$, $B$, $C$ has two components (e.g. qubits). The $\rho$s only live on one component in each, while the $\Pi$ are measurements on both parts of the corresponding space. Commented Feb 22 at 19:08

Each party holds two quantum systems, which are parts of two bipartite systems. For example, the Hilbert space $$H_A$$ corresponding to the party $$A$$ is the tensor product $$H_A = H_{A1} \otimes H_{A2}$$ where $$H_{A1}$$ correspond to the system which is part of $$\rho_{AB}$$ and $$H_{A2}$$ correspond to the system which is part of $$\rho_{AC}$$.
We have three parties $$A, B, C$$ each holding a composite quantum system described by spaces $$H_A, H_B, H_C$$ respectively. Each of these spaces is a tensor product, corresponding to the fact that each party has hold parts of two independent bipartite systems: $$H_A = H_{A1} \otimes H_{A2}$$, $$H_B = H_{B1} \otimes H_{B2}$$, $$H_C = H_{C1} \otimes H_{C2}$$. So in total we have $$6$$ parts of out quantum systems.
The bipartite systems are described by density matrices $$\rho_{AB} \in \mathcal{D}(H_{A1} \otimes H_{B1})$$, $$\rho_{BC} \in \mathcal{D}(H_{B2} \otimes H_{C1})$$, and $$\rho_{AC} \in \mathcal{D}(H_{A2} \otimes H_{C2})$$ where $$\mathcal{D}(H)$$ denotes space of density matrices on $$H$$. These three density matrices form the density matrix of the whole triangle system $$\rho_{AB} \otimes \rho_{BC} \otimes \rho_{AC}$$ which lies in $$\mathcal{D}((H_{A1} \otimes H_{B1}) \otimes (H_{B2} \otimes H_{C1}) \otimes (H_{A1} \otimes H_{C2}))$$.
The order in this last tensor product arises from considering our system as consisting of three bipartite systems. To describe measurements, we instead need to view it as a tripartite system with parties $$A, B, C$$, that is, $$\mathcal{D}(H_A \otimes H_B \otimes H_C) = \mathcal{D}((H_{A1} \otimes H_{A2}) \otimes (H_{B1} \otimes H_{B2}) \otimes (H_{C1} \otimes H_{C2})).$$ To change from the first description to the second, we conjugate with the isometry $$M \colon (H_{A1} \otimes H_{A2}) \otimes (H_{B1} \otimes H_{B2}) \otimes (H_{C1} \otimes H_{C2}) \to (H_{A1} \otimes H_{B1}) \otimes (H_{B2} \otimes H_{C1}) \otimes (H_{A1} \otimes H_{C2})$$ which permutes the factors accordingly. Physically, it does nothing, it just changes how we view our $$6$$-partite system.
Now we can measure our system using local PVMs $$\{\Pi_a\}$$, $$\{\Pi_b\}$$, $$\{\Pi_c\}$$ which consists of projections in $$H_A$$, $$H_B$$, and $$H_C$$ respectively, so that $$\Pi_a \otimes \Pi_b \otimes \Pi_c$$ is a projection in $$H_A \otimes H_B \otimes H_C$$ that we can apply to our tripartite state $$M^T (\rho_{AB} \otimes \rho_{BC} \otimes \rho_{AC}) M$$.