Experimental verification of an indefinite causal order

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Science Advances  24 Mar 2017:
Vol. 3, no. 3, e1602589
DOI: 10.1126/sciadv.1602589
  • Fig. 1 The quantum SWITCH.

    Consider a situation wherein the order in which two parties Alice and Bob act on a target qubit |ψ〉T depends on the state of a control qubit in a basis {|0〉, |1〉}C. If the control qubit is in the state |0〉C, then the target qubit is sent first to Alice and then to Bob (A), whereas if the control qubit is in the state |1〉C, then it is sent first to Bob and then to Alice (B). Both of these situations have a definite causal order and are described by the process matrices WAB and WBA (Eq. 6). If the control qubit is prepared in a superposition state Embedded Image , then the entire network is placed into a controlled superposition of being used in the order Alice→Bob and in the order Bob→Alice (C). This situation has an indefinite causal order.

  • Fig. 2 A process matrix representation of Fig. 1.

    The process matrix W describes the “links” between Alice and Bob. For example, it could simply route the input state ρ(in) to Alice MA and then to Bob MB (solid line) or vice versa (dashed line). In the case of the quantum SWITCH, it creates a superposition of these two paths, conditioned on the state of a control qubit. The input state ρ(in), the two local operations MA and MB, and the final measurement D(out) must all be controllable and known a priori. The unknown process is represented by the process matrix (shaded gray area labeled W). A causal witness quantifies the causal nonseparability of W.

  • Fig. 3 Experimental setup.

    A sketch of our experiment to verify the causal nonseparability of the quantum SWITCH. We produce pairs of single photons using a type II SPDC source (not shown here). One of the photons is used as a trigger, and one is sent to the experiment. The experiment body consists of two MZIs, with loops in their arms. The qubit control, encoded in a path degree of freedom, dictates the order in which the operations MA and MB are applied to the target qubit (encoded in the same photon’s polarization). Alice implements a measurement and repreparation (MA), and Bob implements a unitary operation (MB). The state of the control qubit is measured after the photon exits the interferometers; that is, we check if the photon exits port 0 or port 1. Note that there are two interferometers, each corresponding to a different outcome for Alice: The yellow path means Alice measured the photon to be horizontally polarized (a logical 0), and the purple path means Alice found the photon to be vertically polarized (a logical 1). The first digit written on the detector labels this outcome. The second digit refers to the final measurement outcome, which, physically, corresponds to the photon exiting from either port 0 or port 1. In this diagram, port 0 (1) means the photon exits through a horizontally (vertically) drawn port. A half–wave plate at 0° was used in the reflected arm of the first beam splitter to compensate for the acquired additional phase. QWP, quarter–wave plate; HWP, half–wave plate; BS, beam splitter; PBS, polarizing beam splitter.

  • Fig. 4 Experimentally estimated probabilities.

    Each data point represents a probability p(a, d|x, y, z) in Eq. 12 for a = 0, 1 and d = 0, 1. The blue dots represent the experimental result, and the bars represent the theoretical prediction. The yellow (blue) bars refer to the external (internal) interferometer. The x axis is the measurement number, which labels a specific choice of its input state, measurement channel for Alice and Bob, and final measurement outcome. For our witness, it runs from 0 to 259, but we only show the first 44 here for brevity. Alice and Bob’s specific choice of operator is given in Table 1 and discussed in Materials and Methods. Additional information is in figs. S1 to S3.

  • Fig. 5 Expectation value of the causal witness [Embedded Image] in the presence of noise.

    Because the control qubit (initially in |+〉) is decohered, the superposition of causal orders becomes an incoherent mixture of causal orders. Hence, the causal nonseparability of the SWITCH is gradually lost. The plot shows the causal nonseparability of our experimentally implemented SWITCH because the visibility of the two interferometers is decreased (from right to left). The experimental data linearly decreases with visibility just as theory (dashed line) predicts. The gap between theory and experiment is attributed to systematic errors. The visibility (x axis) is a measure of the dephasing strength on the control qubit.

  • Fig. 6 Efficiency-corrected interferometer fringes out of the two interferometers.

    A plot of the coincidences between the herald and the two detectors at the output of each interferometer as the interferometer phase is varied.

  • Fig. 7 Determination of detection efficiency.

    Triggered coincidences detected in port 1 plotted against those detected in port 0 for both interferometers. Because the total number of photons exiting the interferometer should be constant, the relative collection/detection efficiency can be determined from the slope of this line.

  • Fig. 8 Schematic representation of a causal witness.

    In this two-dimensional representation, the causal witness is represented by the line (actually, a hyperplane) S. It separates the convex set of process matrices Embedded Image from a given causally nonseparable process matrix Wn−sep. Because the set of causally separable processes (Eq. 6) is convex, the separating hyperplane theorem (19) implies that one can always draw a hyperplane to separate it from any point outside the set (which corresponds to a causally nonseparable process). This hyperplane is the causal witness.

  • Table 1 List of operators performed by the two parties.

    The table shows Alice’s four measurement operators and her three repreparation operators, which Alice applies when her outcome is |0〉; when Alice’s outcome is |1〉, Alice performs the identity. Bob’s 10 unitary operators are shown in the third column.

    Alice’s measurement operatorsAlice’s repreparation operatorsBob’s unitary operators
    (1) Embedded Image(1) Embedded Image(1) Embedded Image(6) Embedded Image
    (2) Embedded Image(2) Embedded Image(2) Embedded Image(7) Embedded Image
    (3) Embedded Image(3) Embedded Image(3) Embedded Image(8) Embedded Image
    (4) Embedded Image(4) Embedded Image(9) Embedded Image
    (5) Embedded Image(10) Embedded Image
  • Table 2 Set of wave plate angles.

    A list of all of the wave plate angles used to perform the operators listed in Table 1. In our experiment, all combinations of these settings were used, which, together with our three input states, results in 360 measurement settings.

    Alice’s measurement
    Alice’s repreparation
    Bob’s unitary
    (1) 0°HWP, 0°QWP(1) 0°HWP, 0°QWP(1) 0°QWP, 0°HWP,
    (2) 22.5°HWP, 45°QWP(2) 22.5°HWP, 0°QWP(2) 0°QWP, 45°HWP,
    (3) 0°HWP, − 45°QWP(3) 0°HWP, − 45°QWP(3) 90°QWP, 45°HWP,
    (4) 45°HWP, 0°QWP(4) 90°QWP, 0°HWP,
    (5) 90°QWP, 0°HWP,
    (6) 90°QWP, 45°HWP,
    (7) 0°QWP, 0°HWP,
    (8) 45°QWP, 0°HWP,
    (9) 45°QWP, 45°HWP,
    (10) 45°QWP, 0°HWP,

Supplementary Materials

  • Supplementary material for this article is available at

    section A. Choi-Jamiołkowski isomorphism

    fig. S1. Experimentally estimated probabilities.

    fig. S2. Experimentally estimated probabilities.

    fig. S3. Experimentally estimated probabilities.

    table S1. List of all the experimental measurement settings and the corresponding coefficients.

    Reference (35)

  • Supplementary Materials

    This PDF file includes:

    • section A. Choi-Jamiołkowski isomorphism
    • fig. S1. Experimentally estimated probabilities.
    • fig. S2. Experimentally estimated probabilities.
    • fig. S3. Experimentally estimated probabilities.
    • table S1. List of all the experimental measurement settings and the corresponding coefficients.
    • Reference (35)

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