Glossary - Quantum Causal AI

Definitions are deliberately short. Each lesson defines its terms again in context, with the assumptions that apply.

Amplitude: A complex number attached to a possible outcome of a quantum state. The probability of the outcome in an ideal measurement is the squared magnitude of its amplitude (the Born rule).
Average treatment effect (ATE): The mean difference between the outcomes a population would have under treatment and under no treatment, $E[Y(1) - Y(0)]$ .
Back-door criterion: A graphical condition on a set of variables $Z$ : no member of $Z$ is a descendant of $X$ , and $Z$ blocks every path from $X$ to $Y$ that starts with an arrow into $X$ . If it holds, adjusting for $Z$ identifies the effect of $X$ on $Y$ .
Barren plateau: A region of a variational circuit’s parameter space where gradients are exponentially small in the number of qubits, which makes training by gradient descent impractical.
Bayes’ rule: $P(H\mid E) = P(E\mid H)\,P(H) / P(E)$ . It updates the probability of a hypothesis $H$ after seeing evidence $E$ .
Bell inequality: A bound on correlations that every local hidden-variable model satisfies. Some measurements on entangled quantum states violate it.
Bloch sphere: A geometric picture of single-qubit states: pure states lie on the surface of a unit sphere, mixed states inside it.
Causal discovery: Inferring a causal graph, or a set of graphs that fit equally well, from data plus explicit assumptions such as the causal Markov condition and faithfulness.
Causal graph (DAG): A directed acyclic graph whose arrows represent direct causal influences among variables.
CHSH inequality: The Clauser–Horne–Shimony–Holt form of a Bell inequality, $|S| \le 2$ for local hidden-variable models. Quantum mechanics allows up to $2\sqrt{2}$ (Tsirelson’s bound).
Choi matrix: A matrix that represents a quantum channel completely. The channel is completely positive exactly when its Choi matrix is positive semidefinite.
Classical shadow: A compact classical description of a quantum state built from randomized measurements. It can estimate many properties of the state from relatively few measurements.
Collider: A variable with two arrows pointing into it on a path, as $X \to C \leftarrow Y$ . Conditioning on a collider can create an association between $X$ and $Y$ that is not causal.
Common cause: A variable that influences two others. An unmeasured common cause (a confounder) can create correlation without direct causation between the two.
Complete positivity: The property that a map stays positive when it acts on part of a larger system. Physical quantum channels are completely positive and trace preserving (CPTP).
Concurrence: A measure of entanglement for two qubits, from 0 (not entangled) to 1 (maximally entangled).
Confounding: Bias in an estimated causal effect caused by a common cause of treatment and outcome that the analysis does not account for.
Counterfactual: A statement about what would have happened to the same unit under a different action, given what actually happened.
Density matrix: A positive semidefinite matrix with trace one. It describes pure and mixed quantum states.
do-operator: Pearl’s notation $do(X = x)$ for an intervention that sets $X$ to $x$ by an external action, cutting the arrows into $X$ in the causal graph.
Entanglement: A joint quantum state is entangled when it cannot be written as a probabilistic mixture of product states.
Entropy: A measure of uncertainty. Shannon entropy applies to probability distributions; von Neumann entropy, $-\operatorname{Tr}\rho\log\rho$ , applies to density matrices.
Expectation value: The probability-weighted average of measurement outcomes, $\langle A \rangle = \operatorname{Tr}(\rho A)$ for a quantum observable $A$ .
Faithfulness: The assumption that every conditional independence in the data follows from the causal graph and is not due to an exact cancellation of effects.
Front-door criterion: A graphical condition that identifies a causal effect through a fully observed mediator even when the treatment and outcome share an unmeasured common cause.
Hermitian matrix: A matrix equal to its conjugate transpose. Its eigenvalues are real; quantum observables are represented by Hermitian operators.
Identification: Showing that a causal quantity can be computed from the distribution of observed data under stated assumptions.
Indefinite causal order: A situation, described by process matrices, in which the operations cannot be assigned a definite order, nor any probabilistic mixture of definite orders (causal nonseparability). A random but classical choice of order is not indefinite; the quantum switch with its control in superposition is the standard example.
Interference: The effect produced when amplitudes for different paths add, so that outcome probabilities depend on relative phase.
Intervention: An action that sets or changes a variable from outside the system, as opposed to passively observing it.
Kraus operators: Matrices $K_k$ that represent a quantum channel as $\rho \mapsto \sum_k K_k \rho K_k^\dagger$ , with $\sum_k K_k^\dagger K_k = I$ for trace preservation.
Parameter-shift rule: A rule that computes exact gradients of many variational quantum circuits from the circuit’s outputs at two shifted parameter values.
Partial trace: The operation that gives the state of a subsystem from the state of a larger joint system.
Process matrix: An operator that describes how local quantum operations in several laboratories are connected, without assuming a fixed causal order between them.
Quantum causal model: A causal model in which the nodes are quantum systems or quantum operations and the mechanisms are quantum channels, generalizing classical structural causal models.
Quantum channel: A completely positive, trace-preserving linear map that describes how a quantum state can change, including noise.
Quantum switch: A process in which a control qubit determines the order in which two operations act on a target; with the control in superposition, the order is indefinite.
Qubit: A two-level quantum system, the quantum counterpart of a bit.
Randomized controlled trial: An experiment in which treatment is assigned by a random mechanism, which balances measured and unmeasured characteristics between groups on average.
Structural causal model (SCM): A set of equations, one per variable, that determine each variable from its direct causes and an exogenous noise term, together with a distribution over the noise terms.
Superposition: A quantum state that is a linear combination of other states, such as $\alpha|0\rangle + \beta|1\rangle$ .
Variational quantum algorithm: A hybrid method in which a classical optimizer tunes the parameters of a quantum circuit to minimize a cost estimated from measurements.