Fig. 1, This is a process matrix W that 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).

Is Cause and Effect a Fundamental Law of Physics?

Experimental Verification That Entanglement Can Cause the Temporal Order of Events to be Indeterminate.

May 9, 2021

Introduction

Most of us have assumed, up until now, that all events have a definite temporal order, and that a cause always has an effect, and an effect always has a cause. The notion of causality is an innate concept of our common sense, which defines the link between physical phenomena that temporally follow one another , with one phenomenon manifestly being the cause of the other. However, in quantum mechanics, causality is not explicitly defined or part of the calculations of the evolution of a wave function.

In addition, we have some ongoing mysteries regarding the nature and characteristics of entanglement. But, researchers have been doing experiments involving entangled particles to learn more about it. This research may soon illuminate the nature of entanglement, and its relationship to time, space, and the quantum foam. (See “Measuring the Quantum Foam”, by John Karpinsky in Medium.)

In Fig. 1, we give an example of a process matrix that shows two cases. Either Alice goes before Bob, or Bob goes before Alice. In this article, we will summarize the following paper by Rubino et al (1), in which they measure quantum particles, which they call qubits. This summary is needed for most of us because ,unless you are quantum Physicist, this paper is almost incomprehensible.

There has been a great deal of activity on this subject on the theoretical side, but this is the first definitive experimental work on the subject of causal order. To do this, they quantify the results by measuring a “causal witness.” This is a mathematical object that incorporates a series of measurements that are designed to yield a certain outcome only if the process under examination is not consistent with any well-defined causal order. In the experiment, they 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.” The experiment verified the indefinite causal order to an accuracy of better than 7 standard deviations.

This result can lead to disconcerting consequences, forcing one to question concepts that are commonly viewed as the main ingredients of our physical description of the world. However, these effects can be exploited to achieve improvements in computational of complexity and quantum communications. Recently, this computational improvement was experimentally demonstrated in the study of Procópio et al. (2)

In Procópio’s study, the absence of a causal order was inferred from the success of an algorithm rather than being directly measured . But this was one of the inspirations for this work, in which they expressly demonstrate the realization of a causally non-ordered process by measuring a so-called “causal witness”.

The demonstration of a “casual witness” is presented in the study by Araujo et al (3), which demonstrates a method of measuring whether a process is separable or not. This is also a key result that is used in this present study.

To make a result stronger (that is make the causal witness more robust to noise), they performed a superposition of the orders of a unitary gate and a measurement operation. In other words, a measurement was made inside a quantum process with an indefinite order of operations, (the quantum SWITCH). Performing a standard measurement inside the quantum SWITCH would destroy its coherence, because it would reveal the time in which the measurement is performed and would also reveal whether it is performed before or after operations. In other words, such a measurement would reveal the casual order between operations. However, in this study (1), the measurement of outcomes are read out only “at the end” of the process, thus preserving its coherence.

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) in figure 2. A causal witness quantifies the causal non-separability of W. In our usual understanding of causal relations, if we consider two events A and B, which are connected by a time like curve, then we will have one of two cases: Either A is in the past to B, or B is in the past of A. However, the application of the superposition principle to these causal relationships calls into question the interpretation of causal orders as a pre-existing property. The causal order can become genuinely indefinite. There are two qubits: the control qubit and the target qubit. The control qubit determines whether the target cubit is routed to A or to B.

Fig. 2 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. If the control qubit is prepared in a superposition state, then the entire network is placed into a controlled superposition of being used in the order Alice→Bob and in the order Bob→Alice. This situation has an indefinite causal order.

In this situation the causal order is not really a superposition. It is entangled with the state of the control cubit. From this simple example we can see that the causal order between events is not always a definite in quantum mechanics. One could, in the spirit of hidden variable theories, insist that there might none the less be a well defined casual order. However, such a claim requires, in general, a theory to be non-local and contextual because of the Bell (4) and Kochen-Specker (5) theorems.

In Procopio’s study, the quantum SWITCH is the first explicit example where in it was shown that quantum mechanics does not allow for well-defined causal order. The SWITCH was experimentally implemented in this study by superposing the order in which two unitary operations acted. That experiment confirmed that a cardinal a non-ordered quantum circuit can solve a specific computational problem more efficiently than an ordered quantum circuit. However, only an indirect evidence of indefinite causal order was observed through the demonstration of this computational advantage. But, the development of the SWITCH provided the tool needed to perform the experiment that provides definitive proof.

A causal witness is a carefully designed set of measurements, whose outcome will tell us if a given processes causally ordered or not. An intuitive way to introduce causal witnesses through the well-known concept of an entanglement witness as shown by Guhne et al (6). From this paper, it can be shown that if one measures an entanglement witness on a state and finds a negative value, then the state must be entangled. This concept was used to develop the causal witness.

A similar quantity was recently introduced to determine whether a process matrix W is causally separable or not. A process matrix (the counter part of the density matrix in the entanglement witness example) describes the causal relations between local laboratories. Consider two observers Alice and Bob who perform local operations Ma and Mb by local operations. We mean that the only connection that Alice and Bob have with the external world is given by the quantum state that they receive from it and the state that they return to it. The process matrix W then details how this quantum state moves between the two local Laboratories (Fig. 2). Hence, it is independent of the individual operations that Alice and Bob perform. In the case of the quantum switch, the process matrix first routes the input State to Alice and Bob in superposition controlled by the control qubit, and then connects Alice’s output to Bob’s input and vice versa, and finally coherently recombines their outputs. To make this work, we must probe the process with several different input states p(in). Then, for each input state, Alice and Bob implement several different known operations, and then, we perform a final measurement D(out) (Fig. 2). Alice and Bob are free to perform measurements Ma & Mb, respectively. Note that swapping the order of Alice and Bob is a simple as swapping the labels A and B. With the control qubit set to 0, the order is from A to B, if set to 1, the order is B to A. Now we construct the process matrix of the quantum SWITCH. We set the control bit to a superposition of 0 and 1 as shown in case c of Fig. 2. When this is done, there is no way to determine if the Target qubit went from A to B, or B to A. But how do we that anything was done at all? That is where the causal witness comes in. Causal witnesses are designed to distinguish between causally separable and causally non-separable (CNS) process matrices. For all process matrices, there exists a Hermitian operator S, called a causal witness. In this simplified description, I am not going to provide the actual equations, but just describe the results. Refer to Rubino et al (1) for the gory details. The operator S is positive for separable matrices, and negative for non-separable matrices. To implement a causal witness experimentally, we need to decompose it in terms of operations that we can realize in the laboratory: Preparation of states, applying quantum channels, and doing measurements. The measurements done of S also show the amount of worst case noise that can be tolerated while remaining CNS.

Experiment

To experimentally implement the quantum SWITCH, we need a control and a target qubit. In our experiment, we encode a control qubit in a path degree of freedom of a photon and a target qubit in the same photon’s polarization. This technique of using multiple degrees of freedom has enabled many previous quantum technologies such as Englert et al (7). For our present experiment, this is convenient because Bob has a unitary gate that can be implemented easily with three wave plates, whereas Alice can perform a projective measurement with wave plates and a polarizing beam splitter. Note that there are other proposals to coherently control the causal orders of events. In these proposals (as in ours), the target and control system and encoded in the same particle. In principle, it is also possible to use different particles. With photons, this could be done using a so-called controlled path gate or potentially by using a spin qubit to control the causal order acting on a photon.

In this experiment, the realization of a unitary channel (Bob’s channel) is straightforward, but a short remark is necessary concerning Alice’s measurement. It is clear that a polarizing beamsplitter enables one to distinguish the polarization of an incoming photon. However, a polarizing beam splitter gives rise to additional spatial modes (that is, there are two output paths after the polarizing beam splitter). These two spatial modes can be considered as a new spatial qubit. Then, the action of the polarizing beam splitter is to couple the polarizing qubit to this additional qubit. This is formally equivalent to a von Neumann system-probe coupling, which can model the interaction necessary for any projective measurement and has been used between path and polarization degrees of freedom. In this experiment, the polarization qubit is the system, and it is coupled via the polarizing beam splitter to an additional spatial qubit, which is the probe. We can read all the information about the system by measuring the probe (with a photon detector) at a later time. This solves a non-trivial problem of realizing a measurement operation inside a quantum switch. Most approaches to acquire information inside the switch would lead to distinguishing information about the order in which the operations were applied, destroying the quantum superposition. However, in our solution, because the probe qubit is not measured until the information about the order of application of the operations is erased, the entire process can remain coherent. This solution also works deterministically; that is, both of Alice’s outcomes are retained. It also allows Alice to implement a measurement-dependent re-preparation by placing different wave plates in each of the two outcome modes.

This information of the quantum switch draws inspiration from the study of Procópio et al (2), in which only orders of unitary operations were superimposed. Therefore, our experimental skeleton consists of a Mach-Zehnder (MZI) interferometer with a loop in each arm. However, because Alice’s measure-and-re-prepare channel adds an additional path degree of freedom, we need an extra interferometric loop.

Fig. 3 Experimental setup.A sketch of our experiment to verify the causal non-separability 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 re-preparation (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.

A diagram of our experimental apparatus is presented in figure 3. The first step is to set the state of the system qubit (encoded in the polarization) with a polarizer and a half-wave plate. Then, the photon impinges on a 50/50 beam splitter; this sets the state of the control qubit (encoded in the path degree of freedom). Depending on the state of the path qubit, the photon is sent to either Alice and then Bob or vice versa. As described above, the process is a projective measurement (a sequence of two wave plates and a polarizing beam splitter) and a corresponding re-preparation (a sequence of two wave plates in only one of the polarizing beam splitter outputs, and there is a unitary gate, a sequence of three wave plates. Because the polarizing beam splitter adds a second path qubit, this results in four path modes, including both the state of the control qubit and the outcome of the measure and re-prepare channel. Referring to Figure 3, the external (yellow) interferometer rises from the outcome H- also referred to as a logical 0 and the internal (purple) one arises from the outcome V- a logical 1. We finalize the SWITCH by erasing the information about the order of the gates. This can be done by applying a Hadamard gate to the control cubit. Because the control cubit is a path cubit, a Hadamard gate can be implemented with a 50/50 beam splitter. However in our experiment there are two path qubits (the control qubit and Alice’s ancilla measurement qubit). Thus, we must use two 50/50 beam splitters: one beam splitter to interfere the control cubit when Alice’s ancilla qubit is in the state 0 and one beam splitter when it is the state 1. Finally, each of the four outputs is coupled into single mode fibers, which are connected to single photon detectors (SPD). Then, detecting a photon in one of the four modes yields the result of both the measurement of the control cubit in the super position basis and Alice’s measurement (see the detector labels in figure 3).

We wish to evaluate the CNS of our quantum switch by experimentally estimating the expectation value of a causal witness S. Because the trace is linear, this can be done by implementing one term in the sum of S at a time. To estimate a single term, we injected an input state into the switch, Alice and Bob each perform an operation inside, and then we measured the outputs of the overall process. Because the control qubit measurement and Alice’s measurement are both single cubit projective measurements, there are a total of four possible outcomes. For each measurement setting, different input states are sent into the SWITCH, and the probabilities of each outcome are experimentally estimated by sending multiple copies of the same input state. To compute the final value of the causally non-separable (CNS) process matrices, the results of these measurements are weighted by the corresponding coefficients describing the input state and summed.

The number of terms in the sum for calculating S is determined by the specific witness we wish to evaluate. In general Alice and Bob must each implement a set of operators forming a basis over their channels. For Bob’s unitary channel, this requires 10 elements and for Alice’s measure and re-prepare channel, this requires 16. In our case, we formed Alice’s basis with four (non-commutative) projection operators and three unitary re-preparation operators when the outcome was H (horizontal polarization) and one operator (the identity operator) when the outcome was V (vertical polarization). This corresponds to 12 measure and re-prepare channels when the the outcome of Alice’s measurement is H and 4 when it is V, for a total of 16 measure and re-prepare operators. For Bob, we implement all 10 unitary’s.

Varying the input State can make CNS more robust for noise. Hence, for our experiment, we use three different input states H, V, and (entangled H and V). Finally, we implemented two different measurement operators D(out) on the control qubit (corresponding to the 2 outcomes of the projection onto (entangled H and V). For our experiment, the calculation of CNS translates into very complicated summation that is beyond the scope of this article, but the results of this summation is shown in figure 4.

The task is to experimentally estimate all of these probabilities to evaluate CNS. There are 1440 terms in this sum. However four outcomes (two from Alice’s measurement and two from the final detection) are collected simultaneously (experimentally this means the counts of for SPDs are collected in one setting). Therefore, we need 360 different experimental settings. However, for our witness of the 360 pre-factors, 101 are equal to zero; thus, there are actually only 259 relevant experimental settings.

With this in place, we can experimentally measure the CNS. Figure 4 shows some of the probabilities for the four outcomes; that is, for Alice, a =0, 1, and and our final measurement, d =0,1. In figure 4, the experimentally obtained values are denoted by blue dots, and the theoretical predictions are represented by bars. The experimental results match the predictions very well.

The main source of error is phase fluctuations in the two interferometers. Therefore, we performed a separate measurement to characterize this error. The error bars in Fig.4 represent both these phase errors and the Poissonian errors due to finite counts. These errors do not take into account systematic errors, such as wave plate mis-calibration, because these systematic errors represent a deviation of our experimental SWITCH from the ideal SWITCH.

Fig. 4. Experimentally estimated probabilities. Each data point represents a probability p(a, d|x, y, z) 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.

Conclusion

We can see from these experimental results that our common sense notion of causality does not match Quantum Mechanics. Surprise-surprise! What about QM does match our common sense? But this is a direct attack in a fundamental way on how our world seems to be built. We do have a previously known example of this issue in Special Relativity. Caltech Tutorial on Relativity. There, the effect can precede the cause depending of your frame of reference. But that is only for someone in a different frame of reference. In this case, the qubits are in the same frame of reference. Following are some other sources of information about causality that are beyond the scope of this article.

References

(1) “Experimental Verification of an Indefinite Causal Order.” Rubino et al, Science Advances 2017; 3:e1602589 24 March 2017 https://advances.sciencemag.org/content/advances/3/3/e1602589.full.pdf

(2) L. M. Procopio et al, Experimental superposition of orders of quantum gates. Nature Communications 6, 7913 (2015)

(3) M. Araujo et al, “Witnessing causal non-separability”. New Journal of Physics 17, 102001 (2015)

(4) J. S Bell, “On the Einstein Podolsky Rosen paradox”. Physics 1,195–200 (1964)

(5) S. Kochen, E. P. Specker, “The Problem of Hidden Variables in Quantum Mechanics” (Springer Netherlands, 1975), pp. 293–328

(6) O. Guhne, G. Toth, “Entanglement detection”. Phys. Rep. 474, 1–75 (2009)

(7) B. G. Englert, et al, “Universal unitary gate for single-photon two-qubit states”. Phys. Rev. A 63, 032303 (2001)

I am a retired industrial physicist interested in the fundamental physics of the universe.

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