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Experimental Verification of an Indefinite Causal Order
Giulia Rubino, Lee A. Rozema, Adrien Feix, Mateus Araújo, Jonas M. Zeuner, Lorenzo M. Procopio, Časlav Brukner, Philip Walther
TL;DR
The experiment addresses how to demonstrate that a quantum process lacks a definite causal order. It measures a causal witness using a measurement inside a quantum SWITCH without destroying the superposition, finding causal non-separability by approximately seven standard deviations.
Problem
Previous experiments accessed the SWITCH’s causal non-separability only indirectly, while performing measurements inside it without destroying causal-order coherence had been assumed possible but lacked a proposed method.
Method
The experiment measures a causal witness and implements a polarization measurement through a PBS-coupled spatial probe inside a quantum SWITCH while preserving the superposition of causal orders.
Results
Approximately seven standard deviations separate the measured process from causal separability, demonstrating causal non-separability experimentally.
Takeaways & Limitations
Performing a measurement inside the quantum SWITCH increases the causal witness’s robustness to noise.
Takeaways & Limitations
The reported error bars include phase fluctuations and Poissonian counting errors but exclude systematic errors such as waveplate miscalibration.
Abstract
from arXiv · showhide
Investigating the role of causal order in quantum mechanics has recently revealed that the causal distribution of events may not be a-priori well-defined in quantum theory. While this has triggered a growing interest on the theoretical side, creating processes without a causal order is an experimental task. Here we report the first decisive demonstration of a process with an indefinite causal order. To do this, we quantify how incompatible our set-up is with a definite causal order by measuring a 'causal witness'. This mathematical object incorporates a series of measurements which are designed to yield a certain outcome only if the process under examination is not consistent with any well-defined causal order. In our experiment we perform a measurement in a superposition of causal orders - without destroying the coherence - to acquire information both inside and outside of a 'causally non-ordered process'. Using this information, we experimentally determine a causal witness, demonstrating by almost seven standard deviations that the experimentally implemented process does not have a definite causal order.
1 Theory
The theory develops causal witnesses to distinguish processes with definite causal order from causally non-separable processes. It represents causal relations using process matrices and evaluates them through controlled inputs, local operations, and final measurements.
- Operational evaluation: Evaluating a causal witness requires probing W with multiple input states, local operations, operation outcomes, and final detection measurements.The resulting probabilities are expressed using the Choi–Jamiołkowski isomorphism and weighted by witness coefficients.
- Process matrices and the SWITCH: A process matrix W describes the links between Alice and Bob, while the quantum SWITCH coherently superposes Alice-before-Bob and Bob-before-Alice orders.The relative amplitudes of the two orders are controlled by the control qubit.
- Causal separability: Causally separable processes are incoherent mixtures in which Alice acts first with probability p and Bob acts first with probability 1 − p.Each run therefore has a well-defined causal order, even though the mixture may contain both orders overall.
- Causal witnesses: A causal witness is a Hermitian operator whose negative expectation certifies that a process matrix is causally non-separable.Causally separable processes satisfy Tr(SW_sep) ≥ 0, whereas a causally non-separable process can yield Tr(SW) < 0.
- Causal witnesses: The causal nonseparability CNS(W) = −Tr(SW) measures how far a process is from causal separability and its tolerance to noise.Positive CNS certifies causal nonseparability; larger values indicate greater noise tolerance before the process becomes causally separable.
- Operational evaluation: The experimentally certifiable CNSexp(W) depends on the information gathered through the implemented local operations and equals CNS(W) only with full causal tomography in general.A complete operator set is required to fully assess the causal nonseparability.
2 Experiment
The experiment implements a quantum SWITCH with a path-encoded control qubit and polarization-encoded target, superposing a measurement-and-repreparation operation with a unitary operation. By estimating a causal witness across many settings and controlled noise conditions, it tests causal non-separability experimentally.
- Experimental setup: Alice’s projective measurement uses a polarizing beam splitter as a probe coupling, with delayed readout preserving coherence between the two causal orders.The PBS creates an additional spatial qubit carrying measurement information without immediately revealing when the measurement occurred.
- Experimental setup: The setup encodes control in photon path and target in polarization, so the path determines whether Alice’s measurement-repreparation precedes Bob’s unitary or occurs after it.A 50/50 beam splitter prepares the control in |+⟩, while waveplates and a polarizing beam splitter implement the operations.
- Witness estimation: The causal-witness expectation value is estimated term by term by injecting input states, applying Alice’s and Bob’s operations, and measuring the final outputs.The experiment uses three input states, |H⟩, |V⟩, and |+⟩, together with two control-qubit measurement operators.
- Error sources: The main statistical uncertainty arises from interferometer phase fluctuations and finite-count Poisson errors, while waveplate miscalibration is treated as a systematic deviation from the ideal SWITCH.The plotted error bars include phase and Poissonian errors but exclude systematic waveplate errors.
- Noise study: p_worst-case = 0.168 ± 0.001 is the measured lower bound on tolerated noise, whereas the studied dephasing model gives a noise tolerance of 0.342 before causal separability.Dephasing the path-encoded control reduces interferometer visibility and gradually converts the coherent order superposition into an incoherent mixture.
3 Discussion
The experiment preserves causal-order coherence while performing a measurement inside the quantum SWITCH, enabling direct causal non-separability testing. It demonstrates causal non-separability with a causal witness and realizes a superposition of non-unitary channel orders.
- 3 Discussion: A coherent measurement interacts with an ancilla first, then measures it after the SWITCH is finalized to erase ordering information.This avoids the coherence loss caused by standard measurements that reveal when the measurement occurred.
- 3 Discussion: The apparatus demonstrates causal non-separability by measuring a causal witness, rather than inferring indefinite order indirectly from computational performance.Earlier work accessed the SWITCH's causal non-separability only through a computational advantage.
- 3 Discussion: Performing a measurement inside the SWITCH increases the causal witness's robustness to noise.The experiment extends prior SWITCH implementations that used only unitary gates.
- 3 Discussion: The work is the first experimental realization of a quantum superposition of orders of non-unitary channels.The implemented superposition combines a unitary gate and a measurement operation.