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Causality in a quantum world

28 March 2018

The clever use of quantum superposition can create situations in which cause-and-effect relationships between events are not well-defined.

Falling dominoes
The typical rules of cause and effect don’t necessarily apply in the quantum realm. Credit: Marco Verch, CC BY 2.0

A neatly aligned row of dominoes stands before you. Satisfyingly, you gently tap the first tile with your finger to topple it. The domino falls and crashes into its neighbor, which falls in the same way, and creates a ripple effect that continues until all the dominoes have toppled.

Falling dominoes illustrate a deeply rooted concept of science and of everyday life: causation. Event B (the last domino falls) occurs because of event A (the first domino falls). B occurs only if A occurs, and the occurrence of A is independent of that of B.

But that simple causal structure of everyday life can break down in the quantum realm. Recent research reveals that causal relationships can be placed in quantum superposition states in which A influences B and B influences A. In other words, one cannot say if the toppling of the last quantum domino is either the result of the first domino’s fall or its cause. The emerging subject of indefinite causality in a quantum world may provide new insights into the theoretical foundations of quantum physics and general relativity.

Quantum switching

Causal relationships acquire an operational meaning when one considers communicating agents. By her free choice to push the first domino tile, one agent, conventionally called Alice, can send a signal to another agent, Bob, at the location of the last domino. The two agents can communicate by sending and receiving quantum bits through wires connecting the laboratories of the agents. An agent’s operation on a qubit defines an event. In a definite causal structure, in which two events are either time-, light- or spacelike separated, either Alice’s event A is the cause or the effect of event B in Bob’s laboratory, or the two events are independent of each other.

In a 2009 preprint Giulio Chiribella and coworkers laid out a proposal to consider the wires as quantum systems that can be brought into superposition. Such a setup would make it possible to coherently switch the order of operations applied to qubits. If the wire connects the output of Alice’s laboratory with the input of Bob’s, then operation A precedes operation B; if it connects the output of Bob’s laboratory with the input of Alice’s, then B precedes A  (see figure 1). Finally, by preparing the two-wire configurations in a quantum superposition state, one realizes a superposition of “A causing B” and “B causing A,” which we call a quantum switch. Such a setup is similar to some predator–prey relationships, in which predator numbers influence prey numbers, yet prey numbers also influence predator numbers. Following work that Ognyan Oreshkov, Fabio Costa, and I published in 2012, we now know that the quantum switch is just one example of an indefinite causal structure, in which it is not defined whether event A is a cause or an effect of event B, or whether the two are independent.

Figure 1: Indefinite causal structures
Figure 1. The order in which the boxes, A and B, are applied to the quantum state of a system depends on the wire configuration. By preparing the state of the wire in a quantum superposition of the two configurations, the circuit outputs a state in which both A comes before B and B comes before A.

Where else are we to look for events with a genuine indefinite causal structure in nature? That question has proved intensely difficult to address. In all our well-established theories, including quantum field theory in curved spacetime, the causal order between events is determined by the distribution of matter–energy on a spacelike hypersurface in their past light cone. Physicists, however, suspect that the notion of definite causal structures will be untenable in a theory in which gravity, and hence the metric field and spatiotemporal distances between events, are subject to quantum-mechanical laws. In 2005 Lucien Hardy suggested that ultimately, causal structure is both dynamic, as in general relativity, and indefinite, due to quantum theory.

Throwing in gravity

In a recent arXiv submission, my colleagues and I showed that by taking the matter–energy distribution on a spacelike surface and preparing it in a quantum state, one can realize an order of events like that of the quantum switch. The idea rests on gravitational time dilation—a difference of elapsed time as measured by observers located at varying distances from a gravitating mass. The closer the clock is to the source of gravitation, the slower time passes.

Suppose that initially the clocks in Alice’s and Bob’s laboratories, which are spatially separated in flat spacetime, are synchronized at 9:55am. Then, at 10:00am local time, both Alice (event A) and Bob (event B) apply an operation on a qubit. Because the two events are spacelike separated, Alice and Bob cannot exchange signals (see figure 2, left). Now we introduce a source of gravity, which we place closer to Bob’s laboratory than to Alice’s. Given sufficient time dilation, event A will end up in a causal past of event B, since the qubit on which Alice applies an operation can be transmitted to Bob’s laboratory such that his operation still occurs at 10:00am local time (but at 10:12am according to Alice’s clock; see figure 2, center). The two events have become timelike separated, and Alice can signal to Bob. We may reconsider the two events with the source of gravitation placed closer to Alice’s lab than to Bob’s. In that case, event B ends up in a causal past of event A (see figure 2, right). Finally, we can place the source of gravity in a superposition of the two locations, thus creating the superposition “A causes B” and “B causes A,” as in the quantum switch.

Figure 2: Influence of gravity
Figure 2. The ticks on the axes mark proper time intervals of five minutes as measured by the two clocks in the laboratories. Events are defined by application of unitaries A and B at 10:00am local time at the locations of the respective laboratories. The clocks are initially synchronized at 9:55am. In the configuration on the left, the two events are spacelike separated. In the center configuration, a mass is placed closer to Bob’s lab than to Alice’s, so event A ends up in a causal past of event B due to gravitational time dilation. In the configuration on the right, the mass is placed closer to Alice’s lab than to Bob’s, so event B ends up in a causal past of event A. If the state of the mass is in superposition of two spatial configurations, one realizes the quantum switch.

The main objective of quantum information research is to develop methods for processing information beyond what is classically possible. The concept of indefinite causal structure has already provided new solutions. In 2012 Chiribella and colleagues showed that algorithms that take advantage of the quantum switch can solve certain problems that cannot be solved via a quantum circuit with a fixed order of gates. One example is the task of distinguishing whether a pair of boxes, which represent two unitaries A and B, commute or anticommute―that is, whether AB = ±BA. A quantum switch algorithm queries each box only once, whereas realizing the task within the standard, causal circuit model would unavoidably require at least one of the unitaries to be used twice. The advantage of noncausal quantum computation scales with the size of the problem.

Researchers active in the field of quantum causality are bringing together concepts and ideas from quantum information, computer science, and general relativity to uncover quantum innovations to the very notion of causality and time. The methodology of quantum information has already opened new chapters in our understanding of fundamental physics, including the black hole information paradox and the AdS/CFT correspondence. Using that methodology to study the quirks of cause and effect could help scientists move closer to unifying quantum theory and gravity.

Časlav Brukner is professor of physics at the University of Vienna and director of the Institute for Quantum Optics and Quantum Information (IQOQI) in Vienna. He explores operational and information-theoretic approaches in quantum physics.

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