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Quantum causal modelling

Fabio Costa, Sally Shrapnel

arXiv:1512.07106v2quant-ph

TL;DR

Classical causal-modelling methods do not straightforwardly apply to quantum systems because they assume objective local properties, motivating a genuinely quantum framework. The paper formulates quantum mechanisms, interventions, causal Markov and faithfulness conditions using process matrices, enabling causal-structure discovery and extensions beyond definite causal order.

  • Problem

    Classical causal-modelling methods fail straightforwardly for quantum systems, while a general quantum framework with a quantum causal Markov condition had remained unavailable.

  • Method

    The framework models quantum events as local operations and defines quantum mechanisms, spatiotemporally localised interventions, and process-matrix causal structures.

  • Results

    The framework allows causal-structure discovery, recovers classical causal models as a limiting case, and extends concepts including faithfulness and direct versus indirect causes.

  • Takeaways & Limitations

    Quantum mechanics can receive a causal interpretation in which cause-effect relations are identified with correlations between controlled and observed events.

Abstract

from arXiv · show

Causal modelling provides a powerful set of tools for identifying causal structure from observed correlations. It is well known that such techniques fail for quantum systems, unless one introduces `spooky' hidden mechanisms. Whether one can produce a genuinely quantum framework in order to discover causal structure remains an open question. Here we introduce a new framework for quantum causal modelling that allows for the discovery of causal structure. We define quantum analogues for many of the core features of classical causal modelling techniques, including the Causal Markov Condition and Faithfulness. Based on the process matrix formalism, this framework naturally extends to generalised structures with indefinite causal order.

I. INTRODUCTION

Classical causal modelling explains observed correlations through autonomous mechanisms and interventions, but its assumptions conflict with quantum mechanics. The paper therefore develops a quantum framework that defines causal concepts directly within quantum theory.

  • Causal models explain observed correlations through mechanisms and distinguish causal relations from correlations via interventions.Mechanisms and interventions together define the causal Markov condition.
  • Classical causal modelling helps scientists predict, manipulate, explain, and extract causal information from observed data.
  • Quantum systems lack objective, locally observable and manipulable properties, making classical causal-modelling assumptions incompatible with quantum mechanics.
  • Classical analyses of quantum systems require hidden fine-tuned mechanisms such as action at a distance or retrocausality.Such mechanisms cannot be discovered or controlled within the interventionist picture.
  • The paper introduces quantum mechanisms, localised interventions, a quantum causal Markov condition, and quantum analogues of faithfulness and direct versus indirect causes.Its framework is based on quantum formalism and recovers classical causal models as a limiting case.
  • Quantum causal relations are identified with signalling: intervention on one event can modify the statistics of another event.The framework adopts a device-dependent description containing information about the physical devices producing events.

C. Causal models

Classical causal models represent mechanisms and interventions with directed acyclic graphs and conditional probabilities. The Markov condition tests compatibility with a causal structure, while latent variables and interventions address underdetermination and unobserved causes.

  • A classical causal model represents qualitative cause-effect structure with a directed acyclic graph whose vertices are random variables.
  • The DAG’s edges encode mechanisms, while each variable is specified by a conditional probability given its parents.These conditional probabilities represent autonomous causal mechanisms that interventions can modify.
  • The Markov condition makes the model’s generated probability distribution compatible with a causal structure and can be tested without interventions.It does not generally identify a unique DAG.
  • Latent variables can extend a model whose observed distribution violates the Markov condition, potentially restoring compatibility with a causal structure.In quantum settings, interpreting such hidden variables causally would require inaccessible interventions and can conflict with quantum predictions.
  • Even Faithfulness and Causal Sufficiency may leave causal structure underdetermined, making interventions necessary to uncover the correct structure.

III. QUANTUM CAUSAL MODELS

Quantum causal models replace classical random-variable events with local quantum operations between input and output spaces. Using completely positive maps and the process-matrix formalism, the framework represents quantum events in spatiotemporally local laboratories.

  • The process matrix formalism extends causally ordered quantum networks and supports finite-dimensional quantum causal modelling.
  • A quantum event is represented by a completely positive map from an input space to an output space.Input and output spaces correspond to the past and future boundaries of the event’s space-time region.
  • Complete positivity requires positivity to remain preserved when the map is extended with an arbitrary ancillary identity system.Input and output dimensions may differ because ancillas can be added or systems discarded.
  • Quantum operations are represented as matrices through the Choi-Jamiołkowski isomorphism, and completely positive maps are identified with these representations.Choi’s theorem characterizes complete positivity through positivity of the associated matrix.
  • Examples of quantum events include projective measurement-and-reset operations and unitary transformations occurring with unit probability.
  • A local laboratory is the space of potential quantum events, identified with the product of its input and output operator spaces.

B. Mechanisms

Quantum causal models represent causal influences with mechanisms connecting laboratory outputs to inputs, while interventions are represented by instruments. Process matrices encode the resulting event probabilities and causal relations.

  • Mechanisms: Quantum mechanisms map a local laboratory’s output space to another laboratory’s input space, with unitary maps representing reversible deterministic mechanisms.More general mechanisms can include environmental noise, while isometries provide the direct analogue of deterministic mechanisms.
  • Mechanisms: Connecting mechanisms and local laboratory events are both CPTP maps but use distinct representations in the formalism.The connecting-mechanism representation involves partial transposition on specified subsystems.
  • Interventions: Quantum interventions are choices of instruments, each comprising CP maps whose sum is a CPTP map.The trace-preserving condition ensures that the resulting probabilities sum to one.
  • Interventions: Fixing a POVM while varying prepared states breaks information flow through a laboratory, providing the quantum analogue of classical do interventions.Projective measurements in a fixed basis correspond to quantum idle interventions, but measurements necessarily disturb quantum systems.
  • Process matrices: For local laboratories, the generalized Born rule assigns probabilities to CP-map events using a positive process matrix that encodes causal relations.The process matrix is defined on the tensor product of all input and output spaces and represents information about the outside world available in the laboratories.
  • Causal relations: A laboratory represents a cause for another when it can signal to that laboratory; direct cause requires this influence to persist across possible instrument choices.The framework distinguishes direct from indirect causes using interventions and signalling between laboratories.

E. Examples

The examples show how process matrices recover ordinary quantum probability rules and represent causal connections between laboratories. In the two-laboratory example, dependence on the prepared state establishes a causal relation from B to A.

  • Single laboratory: For a single laboratory with only an input space, the process matrix is a density matrix and the generalized Born rule becomes the ordinary Born rule.The laboratory receives a quantum state from the environment and measures it using POVM elements.
  • Two laboratories: For two laboratories, the process matrix describes how the laboratories are connected, including an identity channel from B to A.The identity connection is represented by [[1]]BOAI.
  • Two laboratories: B represents a cause for A because the probability of nontrivial measurement outcomes at A depends on the state prepared by B.When B prepares the state ρ, the two-laboratory case reduces to the preceding single-laboratory example.

2. Common cause

The framework represents quantum causal structures with laboratories connected by quantum channels, allowing common causes, direct causes, and indirect causes to be distinguished within a DAG-based Markov model.

  • Common cause: A bipartite process matrix W^AB = ρ^AIBI ⊗ 1^AOB O describes laboratories A and B with no causal influence between them.The corresponding reduced statistics describe a shared bipartite state.
  • Common cause: Entangled states can produce Bell-inequality violations, with the common cause represented by a laboratory that prepares different states.The preparation laboratory is introduced as C.
  • Direct cause: A process containing an initial state and unitary channel U describes state evolution from A to B, where A can signal to B and therefore represents a cause.Signalling is possible through suitable choices of instruments at A.
  • Direct and indirect cause: Introducing an intermediate laboratory C can block the information flow, changing A's relation to B from direct cause to indirect cause.The distinction between direct and indirect cause depends on which variables are included in the model.
  • Markov quantum causal model: Each laboratory's output space tensor-factorises across outgoing edges, while incoming edges form a parent space connected to the laboratory input by a quantum channel.This represents generic causal influence rather than only undisturbed transfer through a quantum circuit.
  • Markov quantum causal model: A Markov quantum causal model is a DAG of local laboratories whose edges carry quantum systems and whose laboratory mechanisms are CPTP maps.Independent environmental noise is retained as an assumption, and the resulting process matrix factorises over the DAG.

G. Latent laboratories and non-Markovian models

Latent laboratories extend quantum causal models to unobserved events and non-Markovian evolution, while connecting mechanisms can also be represented as explicitly observed local events.

  • Latent laboratories: Latent laboratories represent unobserved environmental evolution or non-selective measurements with unknown outcomes.They are the quantum analogue of latent variables in classical causal models.
  • Latent laboratories: A Markovian causal explanation may include observed and latent laboratories arranged in a DAG, with CPTP maps and a partial trace over the latent laboratories.The resulting observed process is called a reduced process matrix.
  • Non-Markovian models: Fixing the latent laboratories' CPTP maps yields a reduced process matrix that fully describes the observed laboratories.Information about initial states and environmental evolution is encoded in the latent laboratories.
  • Non-Markovian models: Observed laboratories can therefore exhibit non-Markovian evolution, including initial system-environment correlations.The framework is closely related to formalisms for non-Markovian dynamics.
  • Observed connecting mechanisms: A connecting mechanism can be rewritten as a local event by adding a laboratory whose input and output spaces correspond to the mechanism's parent and target spaces.The original process is recovered by reducing the extended process matrix with the added laboratory's operation.
  • Observed connecting mechanisms: With control over all connecting mechanisms, the process matrix can describe only the wires of a quantum circuit, enabling arbitrary operations to replace its gates.This gives quantum circuits a causal interpretation.

IV. CAUSAL DISCOVERY

The causal-discovery section argues that the causal structure of a Markov quantum causal model can always be uniquely determined from experimental observations under assumptions similar to those used classically.

  • IV. CAUSAL DISCOVERY: The framework claims that experimental observations always uniquely determine the causal structure of a Markov quantum causal model.The required assumptions are described as similar to those imposed on classical causal models.

A. Process-matrix tomography

Process-matrix tomography reconstructs the process from informationally complete instruments or conventional process tomography, after which Hilbert-Schmidt structure can test causal compatibility.

  • A. Process-matrix tomography: A process matrix can be reconstructed using informationally complete instruments, analogously to informationally complete measurements.The process matrix can be treated formally as a density matrix with a different normalisation.
  • A. Process-matrix tomography: Conventional process tomography can reconstruct the CPTP map defined by a process matrix, whereas arbitrary testers generally require prior knowledge of causal order.That prior knowledge is unavailable in causal-discovery settings.
  • A. Process-matrix tomography: Full control of connecting mechanisms would permit tomography of the process matrix of wires, but this may be infeasible in practice.The paper notes that causal-structure information can be sought without full control.
  • A. Process-matrix tomography: Hilbert-Schmidt decompositions classify operator terms by subsystem support and relate term types to signalling and common-cause correlations.Terms of type A^OB^I allow signalling from A to B, while A^IB^I terms support common-cause correlations.
  • A. Process-matrix tomography: A process matrix is incompatible with a DAG if it contains a Hilbert-Schmidt term excluded by that DAG.Trace-preserving constraints eliminate certain term types, providing testable compatibility conditions.

C. Faithfulness

Faithfulness selects causal structures whose allowed quantum correlations are not fine-tuned. Under faithfulness, direct causal relations correspond exactly to DAG parent relations, enabling unambiguous discovery except when available data or latent laboratories impose limits.

  • Faithfulness: A process matrix may factorise over multiple DAGs, but only one may faithfully represent its causal relations.The other representations require specially selected mechanisms that cannot carry causal influence.
  • Faithfulness: Faithfulness requires non-vanishing Hilbert–Schmidt terms of every type compatible with the DAG factorisation.Terms implied by the DAG include terms linking each parent laboratory’s output to its child’s input.
  • Faithfulness: For a faithful causal model, a laboratory is a direct cause of another if and only if it is that laboratory’s parent in the DAG.The converse direction also holds for non-faithful models only in the stated screening-off sense.
  • Discovery of faithful causal structures: Two faithful MQCMs for the same process matrix have the same DAG, so faithful Hilbert–Schmidt terms uniquely identify causal structure.Complete identification requires an informationally complete instrument at every laboratory; measuring only a subset can leave multiple models compatible with the data.
  • Discovery of faithful causal structures: Fine-tuned causal models have measure zero under a non-singular measure, both for a fixed DAG and across a finite set of DAGs.Thus, absent additional information, faithful explanations are preferred because fine-tuned models receive vanishing probability.
  • Discovery of faithful causal structures: With latent laboratories, faithfulness may not single out a unique causal model, so further assumptions are needed among faithful explanations.This is the principal scope boundary for unique causal discovery in the presence of unobserved events.

V. CLASSICAL LIMIT

The classical limit restricts quantum laboratories to fixed-basis operations and further constrains how outputs depend on observed variables. Under these conditions, the resulting statistics are equivalent to those of a classical causal model with the same structure.

  • Classical limit: Quantum systems behave classically when accessible states and operations are restricted to a fixed pointer basis that factorises over separated systems.This transition may be enforced by decoherence, collapse models, or fuzzy measurements.
  • Classical limit: In the fixed-basis representation, x_j are observed variables, i_j are intervention variables, and z_j and output labels are latent variables.The local operation records x_j and sends systems labelled by the outgoing-edge outputs.
  • Classical limit: Recovering a causal model over observed variables requires restricting local operations so outgoing variables depend directly only on observed outcomes.Intervention and incoming-state variables may influence those outputs only indirectly.
  • Classical limit: Under these conditions, Markov quantum causal model statistics are equivalent to those of a classical causal model with the same causal structure.The corresponding classical DAG is isomorphic to the quantum DAG.
  • Classical interventions: Classical interventions are represented by diagonal quantum operations, with do interventions ignoring the input and preparing the chosen output value.Splitting classical nodes into input and output spaces reproduces single-node intervention graphs, or SWIGs.
  • Classical interventions: Classical causal tools can be applied to quantum models when the performed instruments are restricted to those compatible with a classical description.For example, conditional-independence consequences of the causal Markov condition continue to hold for appropriate quantum instruments.

VI. BEYOND DEFINITE AND ACYCLIC CAUSAL STRUCTURES

The framework extends quantum causal modelling beyond classical DAGs to mixtures and quantum superpositions of causal structures, including indefinite causal order. It also identifies open boundaries concerning general causal models, cyclic structures, and the scope of established results.

  • Quantum causal models discussed previously preserve the same DAG causal structure across the quantum-to-classical transition.
  • Generalised causal structures: Probabilistic mixtures of causal structures are represented by process matrices that factorise over their component DAGs.
  • Quantum-controlled causal structures: Quantum-controlled causal structures encode DAGs in control-system basis states, allowing coherent superpositions of causal structures.
  • Quantum-controlled causal structures: The quantum switch provides an example of a quantum-controlled causal structure incompatible with any definite causal order and useful for several tasks.
  • Quantum-controlled causal structures: Superpositions involving DAGs with the same partial order remain causally ordered and admit a Markovian causal explanation.
  • Open boundaries: The framework motivates extensions to cyclic and more general causal structures, but generalisations of the causal Markov condition and faithfulness remain unclear.
  • Conclusions: The work establishes a causal interpretation for quantum mechanics and suggests that quantum causal discovery can support further quantum machine-learning development.
  • Conclusions: The findings do not reproduce the full range of results available for classical causal models.

Appendix A: Proof of theorem 1

The appendix proves that faithfulness links parent relations in a DAG to signalling properties of the associated process matrix. It establishes both the presence of signalling for parents and its removal for non-parents under suitable interventions.

  • Faithfulness ensures that a parent relation L_k → L_h produces an O_kI_h term in the process matrix.
  • For arbitrary CPTP maps at other laboratories, the tensor product preserves the relevant term in the reduced process.
  • The surviving O_kI_h term permits signalling, establishing L_k as a direct cause of L_h for any instruments elsewhere.
  • If L_k is not a parent of L_h, every term nontrivial on O_k also involves an input associated with a different child of L_k.
  • Applying maximally noisy channels at all laboratories that L_k parents removes every O_kX term, preventing signalling from L_k in the reduced model.
  • The probabilities are equivalently generated by a diagonal process matrix and, using the quantum causal Markov condition, can be written as products of CPTP maps.
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