Quantum Causal Models

A quantum causal model combines a proposed causal structure with quantum systems, channels, and interventions. Its purpose is to distinguish what merely correlates from what changes under an operation. Quantum common causes can generate correlations that no Bell-local classical latent variable reproduces under the same assumptions, so classical causal factorization needs a carefully stated quantum counterpart.

Conditioning is not intervention

In a classical directed acyclic model, a joint distribution factorizes as

$$p(x_1,\ldots,x_n)=\prod_i p(x_i\mid{\rm pa}_i),$$

where $\mathrm{pa}_i$ denotes the parent variables. An intervention fixing $X=x$ replaces its incoming mechanism; conditioning on the observed value $X=x$ usually does not.

For example, let $C$ be a fair bit and set $A=C$, $B=C$. Then $p(B=1\mid A=1)=1$. But actively setting $A=1$ while leaving $C$ unchanged gives $p(B=1\mid{\rm do}(A=1))=1/2$. The observational correlation arises from the common cause.

A quantum intervention must likewise be specified as a physical instrument or replacement channel, not merely a value inserted into a classical conditional formula.

A quantum common cause

Prepare a bipartite state $\rho_{AB}$ and send its parts to two laboratories. Their local measurements generate

$$p(a,b\mid x,y)= \operatorname{tr}[\rho_{AB}(E_{a|x}\otimes F_{b|y})].$$

A separable state gives a Bell-local model through its convex product decomposition. Some entangled states violate Bell inequalities for suitable measurements, but still cannot transmit a controllable message through local trace-preserving interventions.

The quantum common cause is a state with a nonclassical correlation structure; it is not equivalent to conditioning on a classical list of pre-existing outcomes for all settings. Allen and collaborators develop a quantum version of causal modeling based on channels and a quantum analogue of the common-cause principle.

A direct cause has a channel

If Alice sends a quantum output to Bob through $\mathcal E$, the joint model can have the form

$$p(a,b\mid x,y)= \operatorname{tr}\!\left[ F_{b|y}\mathcal E(\mathcal I_{a|x}(\rho))\right].$$

Now an intervention at Alice can change Bob’s input. For an identity channel, Alice can discard the old system and prepare $\lvert0\rangle$ or $\lvert1\rangle$; Bob’s $Z$ measurement distinguishes those choices. A source distributing an entangled state without a communication channel cannot reproduce this intervention behavior.

Yet one restricted observational data set may be compatible with either a direct cause or a common cause. Identifying structure therefore requires a sufficiently rich set of interventions and assumptions about hidden systems, memory, and laboratory boundaries.

Limits of causal discovery

Different quantum causal-model frameworks choose different representations and independence conditions. A diagram without its channel model and intervention semantics is incomplete. Statistical independence may arise accidentally even when a physical influence exists; conversely, postselection can create correlations that suggest a nonexistent direct connection.

Definite-order channel networks should also be distinguished from indefinite-causal-order processes. Extending the admissible process class changes the inference problem, not merely the labels on a classical graph.

Exercise

Alice and Bob share $\lvert\Phi^+\rangle$. Alice discards her qubit and prepares $\lvert0\rangle$, choosing this operation instead of doing nothing. What happens to Bob’s state? How does an identity channel from Alice’s output to Bob change the answer?

Solution

Without a communication channel, Bob remains in $I/2$ for either choice; Alice’s local trace-preserving operation cannot alter his marginal. If her output is instead sent through an identity channel and becomes Bob’s received system, Bob receives $\lvert0\rangle$. The two scenarios have different physical causal connections even if some passive measurement data coincide.

References

• Allen, J.-M. A., Barrett, J., Horsman, D. C., Lee, C. M., and Spekkens, R. W., “Quantum common causes and quantum causal models,” Physical Review X 7, 031021 (2017), arXiv:1609.09487 — a quantum common-cause framework.