Witnessing causal nonseparability

Mateus Araújo; Cyril Branciard; Fabio Costa; Adrien Feix; Christina Giarmatzi; Časlav Brukner

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Witnessing causal nonseparability

Mateus Araújo, Cyril Branciard, Fabio Costa, Adrien Feix, Christina Giarmatzi, Časlav Brukner

arXiv:1506.03776v2Jun 11, 2015quant-ph

TL;DR

Quantum theory admits resources without a fixed causal order, but the relation between causal-inequality violations and physically realizable resources was unclear. The paper introduces causal witnesses, proves they detect every causally nonseparable process, and applies them to show that the quantum switch is causally nonseparable but violates no causal inequality.

Problem
The relation between device-independent causal-inequality violations and physically realizable resources such as the quantum switch was unclear.
Method
The paper develops causal witnesses and efficient semidefinite-programming conditions for detecting and characterizing causal nonseparability.
Results
The quantum switch corresponds to a causally nonseparable process, while a broad class of resources including it cannot violate any causal inequality.
Takeaways & Limitations
Causal witnesses provide a device-dependent way to verify causal nonseparability in a physically realizable quantum resource without requiring causal-inequality violation.
Takeaways & Limitations
The physical interpretation of process matrices is known only for resources that cannot violate causal inequalities, leaving characterization of physical process matrices open.

Abstract

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Our common understanding of the physical world deeply relies on the notion that events are ordered with respect to some time parameter, with past events serving as causes for future ones. Nonetheless, it was recently found that it is possible to formulate quantum mechanics without any reference to a global time or causal structure. The resulting framework includes new kinds of quantum resources that allow performing tasks - in particular, the violation of causal inequalities - which are impossible for events ordered according to a global causal order. However, no physical implementation of such resources is known. Here we show that a recently demonstrated resource for quantum computation - the quantum switch - is a genuine example of "indefinite causal order". We do this by introducing a new tool - the causal witness - which can detect the causal nonseparability of any quantum resource that is incompatible with a definite causal order. We show however that the quantum switch does not violate any causal nequality.

I. INTRODUCTION

The paper extends quantum-information frameworks beyond fixed causal order and introduces causal witnesses to detect causal nonseparability. It applies these tools to the physically realizable quantum switch, while showing that the switch cannot violate causal inequalities.

Motivation: The quantum switch controls the order of operations with a quantum degree of freedom and has been physically realized as a computational resource.Earlier work reported computational advantages over standard time-ordered protocols and an experimental proof of principle.
Motivation: Process matrices describe resources whose causal order need not be fixed, including causally separable mixtures and causally nonseparable processes.Causally separable resources are compatible with a definite order in each run, whereas causally nonseparable resources are incompatible with any definite order.
Research question: The paper addresses the unclear relation between device-independent causal-inequality violations and physically implementable resources such as the quantum switch.Causal inequalities impose device-independent constraints, whereas some computational tasks assume definite-dimensional quantum systems in each laboratory.
Contribution: A causal witness is a set of quantum operations whose expectation value is non-negative for causally separable resources and negative values certify causal nonseparability.The framework accommodates unitaries, channels, state preparations, and measurements.
Contribution: Every causally nonseparable process admits a causal witness, with causal separability characterized by conditions checkable efficiently through semidefinite programming.This provides necessary and sufficient conditions rather than only a one-sided detection test.
Results: The quantum switch is represented as a causally nonseparable process and can be detected using both a reformulated prior protocol and more efficient witnesses.The paper nevertheless proves that the quantum switch cannot violate any causal inequality.

2. Bipartite causally separable processes

This section formalizes causally ordered and causally separable process matrices, then develops causal witnesses as convex-geometric certificates of nonseparability. The witness conditions support efficient construction and characterization through semidefinite programming.

Bipartite ordered processes: A bipartite process compatible with A ≺ B is positive semidefinite, normalized, and satisfies the linear constraints defining that causal order; the analogous condition holds for B ≺ A.Non-signalling processes satisfy compatibility with both orders.
Bipartite causal separability: A bipartite process is causally separable when it is a convex combination of processes ordered as A ≺ B or B ≺ A.Processes that cannot be decomposed this way are causally nonseparable.
Tripartite extension: For tripartite processes with Charlie’s output dimension dCO = 1, only the orders A ≺ B ≺ C and B ≺ A ≺ C are relevant.Charlie cannot signal to the other parties, so every such process is compatible with Charlie being last.
Tripartite extension: The tripartite causally separable set is defined as convex combinations of processes compatible with those two relevant orders.The ordered-process conditions can be expressed using projectors onto the corresponding linear subspaces.
Application: The quantum resource studied later provides information-processing advantages over causally ordered processes and is shown to be causally nonseparable.This establishes the relevance of the formal tripartite definition to the quantum switch.
Scope: Generalizing causal separability to more parties with arbitrary output dimensions is nontrivial because operations may alter variables controlling future causal order.The paper leaves this broader generalization for future work.
Causal witnesses: A causal witness is a Hermitian operator whose expectation is non-negative on every causally separable process.The witness set is characterized as a closed convex cone, and every nonseparable process can be separated by a witness with negative expectation.
Causal witnesses: Witness construction is formulated through linear constraints on a convex cone, enabling semidefinite-programming methods to find witnesses.Operators from the orthogonal complement of the valid-process subspace can be added without changing expectation values on valid processes.

B. Finding causal witnesses

Causal witnesses detect whether a process matrix is causally nonseparable by minimizing its expectation value over a normalized witness cone using semidefinite programming. The framework also assigns robustness interpretations to witness values and supports experimental evaluation through decompositions into implementable operations.

Detecting causal nonseparability: SDP algorithms output a causal witness for a causally nonseparable process and an explicit causally ordered decomposition otherwise.A negative optimal expectation value certifies causal nonseparability; a zero value yields a decomposition from the dual SDP.
Detecting causal nonseparability: The witness optimization minimizes tr[S W] over a normalized cone, whose normalization prevents an unbounded value of −∞.Different compactness-enforcing normalizations produce different interpretations of the optimized expectation value.
Robustness measures: The generalized robustness −tr[S∗W] quantifies a process's resistance to worst-case noise and satisfies proposed axioms for causal nonseparability.It is defined through the smallest noise contribution that makes the process causally separable.
Robustness measures: Random robustness measures resistance to white noise, but it is not a proper causal-nonseparability measure because it is not monotonic under local operations.It remains useful for comparing witnesses when white noise is the appropriate noise model.
Experimental implementation: A witness expectation value can be measured experimentally by decomposing the witness into completely positive maps and combining the resulting probabilities.The decomposition uses Choi–Jamiołkowski representations of instruments with inputs x,y and outputs a,b; terms invisible on valid processes may be added without changing validity or trace values.

D. Example

The example process W_OCB is analyzed with causal witnesses and semidefinite programming, establishing causal nonseparability and relating witness bounds to causal-inequality bounds in this case.

W_OCB is a valid process matrix that allows violation of a causal inequality, implying that it is causally nonseparable.
The optimal witness satisfies -tr[S_OCB W_OCB] = R_r(W_OCB) = 2^-1 > 0, proving W_OCB is causally nonseparable.R_r(W_OCB) is the random robustness.
For λ ≥ 2^-1, W_OCB(λ) is causally separable, with an explicit decomposition obtained from the SDP solution.
The witness S_OCB can be measured experimentally by decomposing it into completely positive measure-and-prepare maps and combining their probabilities.
For this example, tr[S_OCB · W] ≥ 0 exactly when p_succ = tr[G_OCB · W] ≤ 3/4, and the noise thresholds for causal nonseparability and causal-inequality violation coincide.

IV. QUANTUM CONTROL OF CAUSAL ORDER

The quantum switch coherently controls whether A precedes B or B precedes A, and its process-matrix representation is shown to be causally nonseparable, including after reducing over the target system.

A. The quantum switch: The quantum switch uses a control qubit to determine whether U_A precedes U_B or U_B precedes U_A on a target qubit.
A. The quantum switch: With the control prepared in a superposition, the target undergoes the two unitaries in a superposition of orders.
A. The quantum switch: Discarding the control produces an equal mixture of the two causally ordered processes, whereas measuring the control enables a causally nonseparable process.
B. Process matrix representation of the quantum switch: The quantum switch is represented as a tripartite process matrix in which A and B apply arbitrary CP maps and C measures the resulting control-target state.
B. Process matrix representation of the quantum switch: The full switch is causally nonseparable because it coherently combines processes ordered A ≺ B ≺ C and B ≺ A ≺ C and is pure.
B. Process matrix representation of the quantum switch: The reduced switch is also causally nonseparable by causal-witness analysis, because the purity-based proof does not apply to this non-extremal process.

A. Optimal witness

The paper recasts Chiribella’s task as a causal witness for distinguishing commuting from anticommuting unitaries. The quantum switch succeeds perfectly, while finite witnesses trade measurement practicality against noise robustness.

Task: Alice and Bob apply unitaries, while Charlie measures whether those unitaries commute or anticommute.
Witness construction: The witness averages Charlie’s outcomes over commuting and anticommuting unitary pairs represented through Choi–Jamiołkowski operators.
Quantum-switch performance: 1 is the quantum switch’s success probability for any chosen commuting and anticommuting measures.
Quantum-switch performance: A causally separable circuit can also succeed with probability 1 for measures restricted to Pauli pairs, so the measures must be broadened.
Finite witness: Finite unitary sets make witness estimation feasible with finitely many measurements and allow measure optimization through semidefinite programs.
Noise robustness: 0.1507 worst-case-noise tolerance for the finite witness exceeds Chiribella’s 0.0766 but remains below the optimal witness’s 0.5454.

VI. CAUSAL INEQUALITIES

This section distinguishes device-independent causal inequalities from device-dependent causal witnesses. It proves that the quantum switch is causally nonseparable yet generates only causal correlations, so it violates no causal inequality.

Definitions: Causal inequalities constrain outcome distributions assuming an underlying causal order among parties.
Definitions: Causally ordered distributions may be combined using shared randomness, yielding convex mixtures of fixed-order correlations.
Device dependence: Causal inequalities are device-independent, whereas causal witnesses require trusted implementations of each party’s operations.
Quantum switch: Tracing out party C makes the quantum switch’s process causally separable, ruling out bipartite causal-inequality violations between A and B.
Quantum switch: Theorem 4 states that if C does not signal to the other parties and their marginal correlations are mixtures of causally ordered distributions, the full distribution is causal.
Conclusion: The quantum switch is causally nonseparable but generates only causal correlations, so it cannot violate any causal inequality.
Conclusion: Known physically interpretable causally nonseparable resources fall into this non-violating category, leaving physical implementation of causal-inequality violations open.

Appendix A: Details of the formalism

Appendix A develops the Choi–Jamiołkowski and process-matrix formalisms, then illustrates how process matrices represent states, channels, signalling, and pure processes.

1. Choi-Jamiołkowski isomorphism: The CJ isomorphism represents linear quantum maps as operators, with conventions chosen to identify non-signalling processes directly with quantum states.The appendix distinguishes pure and mixed CJ representations and relates complete positivity and trace conditions to properties of the CJ matrix.
1. Choi-Jamiołkowski isomorphism: Pure CJ vectors represent operations such as unitaries, measurement-preparations, and ideal non-demolition measurements.Every pure product CJ vector represents a measurement-preparation operation, while equal measurement and preparation states yield the ideal non-demolition case.
1. Choi-Jamiołkowski isomorphism: The CJ examples include state preparation, POVM outcome probabilities, and measurement followed by state preparation.These examples show how common laboratory operations are encoded within the same map representation.
2. Process matrices: A process matrix is an external quantum resource connecting local laboratories and determining the statistics of their operations.The process-matrix formalism generalizes the comb formalism for causally ordered quantum networks and uses the generalized Born rule for probabilities.
2. Process matrices: For a bipartite state process, the generalized Born rule becomes the probability of measuring POVM elements on a quantum state, and the process permits no signalling.Such a process is compatible with both causal orders because it allows signalling in neither direction.
2. Process matrices: For a channel process, A prepares a state and B measures it after the channel acts from A’s output to B’s input.The corresponding process matrix is related to the channel’s CJ representation by a transposition.
2. Process matrices: Reduced process matrices describe the remaining parties’ statistics after one party performs a CPTP map, including signalling induced by that choice.For a channel from A to B, the reduced process for B is the channel applied to A’s prepared state.
2. Process matrices: Pure process matrices are rank-one projectors, allowing probabilities for rank-one local operations to be calculated from probability amplitudes.The probability is obtained as the modulus square of the overall amplitude, and unitary sequences can be represented at the process level.

Appendix B: Valid process matrices

Appendix B derives a basis-independent characterization of valid process matrices using positivity, normalization, and projector constraints, including ancillary-system considerations.

Appendix B: Valid process matrices: The appendix replaces a basis-dependent characterization of valid process matrices with an equivalent basis-independent formulation.The derivation is given explicitly for bipartite and tripartite cases, with the N-partite case obtained by generalization.
Appendix B: Valid process matrices: Positivity of all probabilities is equivalent to W being positive semidefinite when ancillary systems are included.Without ancillary systems, one would obtain only positivity on pure tensors, which is a strictly larger class.
Appendix B: Valid process matrices: The ancillary-system argument establishes that non-negativity for the relevant CJ operators implies W ≥ 0.This connects complete positivity of local maps with the global positivity condition on the process matrix.
Appendix B: Valid process matrices: Normalization for all instruments is equivalent to requiring that every CPTP map has realization probability 1.The resulting constraints are expressed through partial-trace conditions on the local CJ matrices.
Appendix B: Valid process matrices: The valid-process subspace is the intersection of three commuting projector subspaces, so LV is their composition.This supplies an explicit projector used to test the linear validity constraints.
Appendix B: Valid process matrices: For bipartite processes, the constraints can be written as fixed-point equations under LA, LB, and LAB.These equations provide the projector-based form of validity used in the appendix.
Appendix B: Valid process matrices: A valid bipartite process matrix must be positive semidefinite, have trace dAOdBO, and satisfy W = LV(W).The projector LV encodes the required linear constraints on the process matrix.

2. Tripartite process matrices

The tripartite and N-partite extensions impose positivity, trace normalization, and families of commuting projector constraints defining the valid-process subspace.

2. Tripartite process matrices: A valid tripartite process matrix is positive semidefinite, has trace dAOdBOdCO, and satisfies the tripartite projector constraints.The constraints include single-party, pairwise, and three-party projector conditions.
2. Tripartite process matrices: The tripartite validity conditions are represented by the commuting maps LA, LB, LC, LAB, LAC, LBC, and LABC.Their composition yields the projector LV onto the valid-process subspace.
2. Tripartite process matrices: The N-partite generalization uses 2^N − 1 commuting projectors indexed by the non-empty subsets of the parties.The valid-process projector is obtained by composing all maps LX.
2. Tripartite process matrices: In the N-partite case, validity again requires positivity, trace normalization, and membership in the subspace defined by the composed projector LV.The subset-indexed projectors impose the generalized linear constraints.

Appendix C: Characterization of bipartite causal witnesses

Appendix C characterizes causally separable process matrices and causal witnesses through convex cones, duality, and semidefinite programs.

Appendix C: Characterization of bipartite causal witnesses: The appendix uses convex-cone duality and convex-hull relations to establish the witness characterization.These results rely on standard definitions of convex cones, dual cones, orthogonal complements, and closed-cone operations.
Appendix C: Characterization of bipartite causal witnesses: Causally separable bipartite processes are convex combinations of processes ordered A ≺ B and B ≺ A.The ordered components satisfy positivity and the corresponding output-trace constraints.
Appendix C: Characterization of bipartite causal witnesses: The causal-separability characterization is proved by showing that positive ordered components reconstruct valid causally ordered processes up to normalization.The converse direction verifies positivity, validity-subspace membership, and the required ordered constraints.
Appendix C: Characterization of bipartite causal witnesses: The set of causally separable processes becomes a convex cone after dropping the fixed trace normalization.Causal witnesses are elements of the dual cone under the Hilbert–Schmidt inner product.
Appendix C: Characterization of bipartite causal witnesses: The witness characterization uses the positive semidefinite cone together with linear subspaces enforcing the ordered-process constraints.The relevant subspaces include LBO, LAO, and LV.
Appendix C: Characterization of bipartite causal witnesses: For the ordered cones, duality yields conditions such as B_OS ≥ 0 or A_OS ≥ 0 for the corresponding output-trace maps.These conditions are derived by decomposing witness operators into positive and orthogonal components.
Appendix C: Characterization of bipartite causal witnesses: Every causal witness can be written as S = S_P + S_⊥ with B_OS_P ≥ 0, A_OS_P ≥ 0, and LV(S_⊥) = 0.The decomposition separates the constrained positive part from a component orthogonal to the valid-process subspace.
Appendix C: Characterization of bipartite causal witnesses: The witness and separability characterizations are rewritten as semidefinite programs to facilitate implementation.Using the ordered-process cone characterization yields the implementation forms of SDP problems (38) and (39).

Appendix E: Duality for conic problems

The appendix formulates the causal-separability robustness problems as dual semidefinite programs and verifies the conditions needed for efficient solvability. The primal and dual optima are linked to causal witnesses and decompositions of process matrices.

The two problems are semidefinite programs that are dual to each other, so their optimal solutions can be found efficiently and satisfy Equation (40).
The primal program minimizes tr Ω/dO, the worst-case noise required to make a process matrix W causally separable.The quadratic variable λeΩ is replaced by Ω = λeΩ, leaving the objective equal to tr Ω/dO.
The formulation embeds Wsep × W in a finite-dimensional space with a closed, convex, pointed cone having nonempty interior.This choice avoids empty interiors and non-pointed dual cones that would obstruct standard semidefinite-programming requirements.
The primal problem describes W using causally ordered components WA≺B and WB≺A, while the dual problem supplies a corresponding causal witness.
A value tr[Ω*]/dO = −tr[S*W] > 0 certifies causal nonseparability, whereas value 0 certifies causal separability and yields an ordered decomposition.

Appendix F: Measuring causal nonseparability

The appendix develops measures of causal nonseparability from causal witnesses and studies their behavior under mixing and local operations. Generalised robustness satisfies the desired properties, while random robustness fails monotonicity.

The proposed measure N(W) vanishes exactly when W is causally separable.
Generalised robustness does not increase when a process is composed with local CPTP maps.The proof uses the induced map on processes, duality, trace preservation, and witness normalization.
Generalised robustness and random robustness satisfy Discrimination and Convexity, but only generalised robustness satisfies Monotony.
The appendix represents composing local operations with W as an equivalent bipartite process obtained through a map preserving validity and causal separability.
Random robustness is not monotone under local operations because the dual map can increase a witness’s trace and violate its normalization condition.

Appendix G: Characterisation of tripartite causal witnesses

The appendix characterizes tripartite causal witnesses through the dual cone of tripartite causally separable processes and formulates an SDP for optimizing a discrimination task. The task constrains weights according to whether unitary pairs commute or anticommute.

The cone of tripartite causal witnesses S3C is the dual of the cone of tripartite causally separable processes with dCO = 1.
The optimization minimizes the maximal success probability psep_succ for causally separable processes, thereby increasing resistance to worst-case noise.
The discrimination problem is formulated as an SDP that optimizes weights q[·,·] subject to dO − Gfinite ∈ S3C.
The weights are constrained so the task remains guessing whether the unitaries commute or anticommute, with q{Ui,Uj} = 0 for non-anticommuting pairs.
The quantum switch’s success probability is always one in this task, providing the comparison against causally separable processes.