Cross-Entropy Quantum Causal Influence

• Cross-entropy quantum causal influence is a measure that quantifies state-dependent and nonlocal causal effects in monitored quantum many-body systems.
• It employs a hybrid quantum–classical cross-entropy protocol to compare measurement distributions, avoiding the exponential overhead of post-selection.
• Analytic models such as BGUE and dual-unitary circuits demonstrate how XEQCI reveals exotic causal structures, including inverted light cones and an operational arrow of time.

Cross-entropy quantum causal influence (XEQCI) is a quantitative measure introduced to characterize causal structure in quantum many-body systems subject to measurements and post-selection, particularly in monitored dynamics. Unlike classical systems, where causal influence is dictated by the light cone and Lieb–Robinson bounds, XEQCI captures state-dependent and nonlocal causal phenomena unique to quantum systems, including spacelike causal influence and the emergence of unconventional structures such as inverted light cones. XEQCI is operationally accessible on current quantum hardware, avoids exponential post-selection overhead, and offers analytic tractability in key solvable models (Wang et al., 18 Jan 2026).

1. Formal Definition and Operational Content

Let an evolving quantum system interpolate between an initial state $\rho_i$ and a final state $\rho_f$, potentially involving non-unitary dynamics due to measurements or post-selection. The influence of a localized perturbation—a unitary $U_A$ inserted in region $A$—on measurements in region $B$ is probed. Measurement in $B$ is described by a POVM with Kraus operators $\{K_b\}$. The Born probability of observing outcome $b$ in the presence of $U_A$ is

$$\Pr(b \mid U_A) = \mathrm{Tr}\big[\rho_f\, \mathcal{T}(\cdots U_A \cdots)\, \rho_i\, \mathcal{T}^\dagger(\cdots U_A \cdots)\big] \equiv \varrho_B(U_A)(K_b,K_b^\dagger),$$

where $\rho_f$0 represents the intervening evolution on the Keldysh contour and $\rho_f$1 denotes the superdensity operator on $\rho_f$2.

The cross-entropy QCI for a fixed $\rho_f$3 is then defined as

$\rho_f$4

where $\rho_f$5 is the identity on $\rho_f$6. $\rho_f$7 when the two distributions coincide and approaches zero when they are orthogonal, directly quantifying the operational distinguishability of the two scenarios.

For circuits with probabilistic monitored dynamics, including mid-circuit measurements producing a record $\rho_f$8, the experimentally accessible XEQCI is

$\rho_f$9

where integration is over a unitary-design ensemble (e.g., Haar or random Clifford). The circuit-averaged cross-entropy, $U_A$0, further averages over random circuit realizations.

2. State Dependence and Nonlocal Causality

Classical relativistic causality restricts influence to regions inside the light cone, encoded quantitatively by Lieb–Robinson bounds. In unitary quantum circuits with pure initial states and no post-selection, the causal structure mirrors this classical intuition. However, in monitored quantum systems with projective or post-selected boundary conditions, XEQCI reveals two critical departures:

• State-Dependence: The causal structure is contingent on the initial and final boundary states. For instance, post-selecting a pure final state with a maximally mixed input ($U_A$1, $U_A$2 pure) inverts the causal structure, generating a backward-pointing light cone.
• Nonlocality: Quantum causal influence is not always locally detectable—a unitary $U_A$3 may affect only nonlocal observables across $U_A$4 even if no individual region $U_A$5 is locally sensitive.

XEQCI directly and quantitatively addresses both features by contrasting measurement distributions without presupposing a fixed time direction or geometric structure.

3. Measurement Protocols and Simulation Strategies

Direct estimation of nonlinear functionals such as XEQCI is challenging because they are not simple expectation values. Additionally, simulating non-unitary evolution by post-selection commonly incurs an exponential overhead. XEQCI circumvents these challenges with a hybrid quantum–classical cross-entropy protocol:

1. System Evolution: Prepare and evolve the system using a monitored circuit, inserting a random $U_A$6 at $U_A$7. Measure outcome $U_A$8 at $U_A$9 and record all other measurement outcomes $A$0, sampling $A$1.
2. Classical Simulation: Classically simulate the same circuit (without $A$2) to calculate $A$3; for Clifford circuits, stabilizer formalism provides efficient computation.
3. Cross-Entropy Accumulation: For each trial, compute $A$4; average $A$5 gives $A$6.

Each trial yields a bounded contribution, ensuring statistical error scales as $A$7. For Clifford and CSS-type circuits, both the state update and post-processing are polynomial-time, making the protocol practical for current experimental platforms.

4. Analytic Results in Solvable Quantum Models

XEQCI admits analytic computation in two paradigmatic models:

Brownian Gaussian Unitary Ensemble (BGUE)

In this model, a fully connected system with time-dependent Brownian random Hamiltonian $A$8 is considered. With $A$9 inserted at $B$0 and observables at $B$1, boundary conditions $B$2, $B$3, and standard random-matrix techniques, the averaged causal influence is

$B$4

where $B$5 is the second Rényi entropy and $B$6 depends on Hilbert space dimensions. For spatially disjoint $B$7 and $B$8, dependence remains through a different normalization, retaining the entropy-difference structure. Exponential decay with $B$9 characterizes separation dependence: $B$0 QCI thus flows from low to high boundary entropy, mapping a statistical arrow of time.

Dual-Unitary Brickwork Circuits

Each two-site gate $B$1 is unitary in both spatial and temporal directions. Prescribing regions with non–maximally mixed states in the bulk (region $B$2) while maximizing mixing on outer boundaries induces rich causal geometries. The nonzero-QCI region (the generalized "future light cone" $B$3) is determined by the intersection properties of 45° cones emanating from $B$4 and $B$5. Three distinct spatial/causal phases arise: zero influence (all cones blocked), two-cone propagation, and radially outward propagation, with "time direction" potentially rotating or becoming multivalued depending on inhomogeneities in the boundary data.

5. Emergence of Local Arrow of Time and Entropy Flow

Across both analytic models and Clifford-circuit numerics, a key operational result emerges: QCI, as measured by XEQCI, is exponentially enhanced when information propagates from regions of lower to higher entropy (as quantified by second Rényi purity or stabilizer rank). Influence in the entropy-decreasing direction is exponentially suppressed, yielding a precise, local, operational definition of an arrow of time in monitored quantum circuits.

6. Exotic Causal Structures: Inverted and Reflected Light-Cones

Numerical simulations in Clifford circuits synthesize several unconventional causal structures, directly reflected in color-maps of $B$6 as a function of spacetime coordinates:

• Standard Forward Light Cone: Emerges when $B$7 is pure, $B$8 is maximally mixed.
• Backward (Inverted) Light Cone: Occurs for maximally mixed $B$9 and pure $\{K_b\}$0.
• Bidirectional Cone: With both boundary states partially mixed, the causal structure extends forward and backward in time.
• Measurement-Induced Reflections: Mid-circuit measurements can "reflect" the causal region, allowing influence to double-back or circumvent dynamically-defined obstacles.

These findings underscore the fluidity and state-dependence of quantum causal structure under monitored dynamics.

7. Implementation on Quantum Hardware

XEQCI is experimentally attainable on present-day quantum hardware owing to its reliance on quantum–classical cross-entropy, which sidesteps the exponential costs associated with post-selection. Required capabilities include:

• Implementation of random or design-unity gates $\{K_b\}$1 on localized subsystems.
• Standard mid-circuit or final projective measurements in region $\{K_b\}$2.
• Synchronous readout of ancillary measurement records $\{K_b\}$3.

Platforms such as superconducting-qubit arrays (e.g., Google Sycamore, Rigetti) and neutral-atom arrays equipped for mid-circuit detection fulfill these requirements. Cross-entropy benchmarking—already realized in large-scale quantum processors—can be adapted to estimate XEQCI with suitable classical post-processing using stabilizer or tensor network simulations (Wang et al., 18 Jan 2026).

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Cross-entropy quantum causal influence thus provides a unified, operationally accessible framework for diagnosing and quantifying the complex, state-dependent, and nonlocal causal structures intrinsic to monitored quantum circuits. Analytic and numerical results tie XEQCI to entropy gradients and the emergent quantum arrow of time, while experimental feasibility ensures relevance for quantum information processing, hybrid architectures, and quantum simulation of exotic causal phenomena (Wang et al., 18 Jan 2026).