Chapter 12. Indefinite Causal Order: When ‘Before’ and ‘After’ Overlap

We are accustomed to the fixed causal order where “event A occurs next event B occurs” (\(A \prec B\)). This forms the basis of classical physics and our intuition. However, as we saw in Chapters 9 and 10, time may not be an absolute background, and at the boundary of quantum gravity and quantum information, it becomes possible that the causal structure itself can be quantum mechanically superposed.

Indefinite Causal Order (ICO) is a theoretical framework that describes this phenomenon. The Process Matrix (W) is a mathematical tool that expresses such generalized causal relationships, while the Causal Witness (S) is an experimental measure that demonstrates whether a given phenomenon cannot be explained as a simple probabilistic mixture of fixed orders, thus indicating a genuine quantum superposition of causal orders.

1. Fundamental Concepts

  • Process Matrix (W): When two agents, Alice (A) and Bob (B), perform local operations in their respective laboratories, the process matrix \(W\) is a mathematical object that describes the ‘background’ or ‘protocol’ connecting these two operations.

    • Traditional View: A signal goes from A to B (\(A \prec B\)) or from B to A (\(B \prec A\)). The temporal order is clear.
    • New View: \(W\) does not assume \(A \prec B\) or \(B \prec A\) in advance. \(W\) simply represents the most generalized ‘background spacetime’ that provides a consistent probability distribution for all possible local operations of A and B.

    💡 Detailed Explanation: Quantizing the Rules of the Stage Beyond the Stage Itself 🎭

    Standard quantum mechanics is akin to particles (actors) performing (evolving) on a fixed ‘stage’ of spacetime.

    The process matrix formalism takes a further step, considering that the causal order instructions themselves, such as “actor A appears before actor B,” can also become subjects of quantum superposition. The process matrix \(W\) corresponds to this generalized ‘quantum causal order instruction.’

  • Causal Separability: When a process matrix \(W\) can be expressed as a classical probabilistic mixture of fixed orders, it is said to be ‘causally separable.’ \[W = q W^{A\prec B} + (1-q) W^{B\prec A} \quad (0 \le q \le 1)\]

    • Meaning: This represents classical uncertainty, meaning “with probability q, A occurs before B, and with probability (1-q), B occurs before A, but we simply don’t know which” (ignorance). (See Appendix 2)

  • Indefinite Causal Order (ICO): This occurs when \(W\) cannot be expressed in the above causal separable form. It signifies that the two causal structures \(A \prec B\) and \(B \prec A\) are not in a classical probabilistic mixture, but instead in a genuine quantum mechanical superposition state.

  • Causal Witness (S): A special observable designed to experimentally determine whether a given process \(W\) is causally separable or has indefinite causality. > 💡 Detailed Explanation: The ‘Causal Witness’ of Causal Order 🕵️ > > The causal witness operates in the same way as the ‘entanglement witness’ learned in Chapter 2 for distinguishing quantum entanglement. > > 1. The witness \(S\) is designed such that its expectation value is always non-negative for all causal separable processes \(W_{\text{sep}}\): \(\mathrm{Tr}(S W_{\text{sep}}) \ge 0\). > 2. The expectation value of this \(S\) is calculated for our observed process \(W_{\text{obs}}\) measured through experiments. > 3. If a negative value is observed, i.e., \(\mathrm{Tr}(S W_{\text{obs}}) < 0\), it is mathematically proven that \(W_{\text{obs}}\) can never be a causal separable process. > > In other words, a negative value is a declaration that “this phenomenon can never be explained by a probabilistic mixture of fixed orders!” and serves as strong evidence for indefinite causal order.

  • Quantum Switch: The most famous and standard device for implementing indefinite causal order. It controls the order of applying two quantum channels (Chapter 5) \(\mathcal{A}\) and \(\mathcal{B}\) based on the state of the control qubit.

    • If the control qubit is \(|0\rangle_c\): \(\mathcal{B} \circ \mathcal{A}\) (apply A first, then B)
    • If the control qubit is \(|1\rangle_c\): \(\mathcal{A} \circ \mathcal{B}\) (apply B first, then A)
    • If the control qubit is in the superposition state \(\frac{|0\rangle_c + |1\rangle_c}{\sqrt{2}}\): a state where the two application orders are quantumly superposed. The process matrix \(W_{\text{switch}}\) of this device shows a negative value with respect to the causal witness.

2. Symbols and Key Relations

  • Choi Representation (Refer to Chapter 5): A mathematical tool (CJ Isomorphism) that maps a quantum channel (operation) \(\mathcal{E}\) to a positive matrix (state) \(M\). Alice’s operations \(\{ \mathcal{M}^A_{a|x} \}\) are represented as the matrix set \(\{ M^A_{a|x} \}\), and Bob’s operations \(\{ \mathcal{M}^B_{b|y} \}\) are represented as \(\{ M^B_{b|y} \}\).

  • Probability Formula: The joint probability that Alice performs operation \(a\) under setting \(x\) and Bob performs operation \(b\) under setting \(y\) is given by the linear inner product with the process matrix \(W\) as follows: \[p(a,b|x,y) = \mathrm{Tr}\left[ (M^A_{a|x} \otimes M^B_{b|y}) W \right]\] This formula shows that \(W\) contains all the causal connection information between Alice and Bob.

  • Physical Constraints on \(W\): For \(W\) to be a physically valid (giving probabilities between 0 and 1 for all operations) process matrix, it must satisfy the following:

    1. Positive Semi-Definite: \(W \ge 0\)
    2. Normalization Constraint: \(\sum_{a,b} p(a,b|x,y) = 1\) must hold for all possible (complete) operation choices \(\{M^A\}, \{M^B\}\). This is expressed as linear constraints on certain partial traces of \(W\).

  • Causal Witness \(S\): \(S\) is a Hermitian matrix.

    • Certification Condition: \(\mathrm{Tr}(S W_{\text{obs}}) < 0 \implies W_{\text{obs}}\) is indefinite causality.
    • Design: \(S\) can be constructed as a hyperplane separating the convex set of all causal separation processes and \(W_{\text{obs}}\) (optimizable as an SDP problem).

3. Easy Examples (Examples with Deeper Insight)

  • Example 1: Fixed Order \(A \prec B\)
    • \(W = W^{A\prec B}\). This \(W\) receives input from the past for A, sends output of A to input of B, and sends output of B to the future. The ‘backward signal’ channel from input of B to input of A is closed.
  • Example 2: Classical Random Order (Causal Separation)
    • Flip a coin; if heads, perform the experiment with order \(A \prec B\); if tails, perform it with order \(B \prec A\).
    • \(W = \frac{1}{2} W^{A\prec B} + \frac{1}{2} W^{B\prec A}\).
    • This \(W\) satisfies the definition of causal separation (classical ignorance in Appendix 2), and \(\mathrm{Tr}(S W) \ge 0\) holds for the causal witness \(S\).
  • Example 3: Quantum Switch (Indefinite Causality)
    • Implement \(W_{\text{switch}}\) using the superposition of a control qubit.
    • \(W_{\text{switch}}\) cannot be decomposed into a convex mixture like in Example 2 (contains coherence between the two orders).
    • It can be experimentally shown that \(\mathrm{Tr}(S W_{\text{switch}}) < 0\) for a properly designed witness \(S\), which is a clear evidence of indefinite causality.
  • Example 4: Effect of Noise
    • Mix a perfect quantum switch \(W_{\text{switch}}\) with white noise (completely random process) \(W_{\text{noise}}\).
    • \(W' = (1-q) W_{\text{switch}} + q W_{\text{noise}}\)
    • The witness value becomes \(\mathrm{Tr}(S W') = (1-q) \mathrm{Tr}(S W_{\text{switch}}) + q \mathrm{Tr}(S W_{\text{noise}})\).
    • Since \(\mathrm{Tr}(S W_{\text{switch}})\) is negative and \(\mathrm{Tr}(S W_{\text{noise}})\) is positive, once the noise \(q\) exceeds a certain threshold, \(\mathrm{Tr}(S W')\) becomes non-negative, making it impossible to certify indefinite causality anymore.

4. Practice Problems

  1. (Non-negativity of Probability): Show that the probability \(p(a,b|x,y) \ge 0\) when the fair matrix \(W \ge 0\) and the local operation Choi matrices \(M^A_{a|x}, M^B_{b|y} \ge 0\). (Hint: The trace \(\mathrm{Tr}(PQ)\) of the product of two positive semi-definite matrices \(P, Q\) is non-negative.)
  2. (Normalization Condition): Assuming Alice and Bob perform a complete POVM (measurement), i.e., \(\sum_a M^A_{a|x} = \mathbf{1}_{\text{out}}^A\), \(\sum_b M^B_{b|y} = \mathbf{1}_{\text{out}}^B\), derive the linear constraint imposed on \(W\) such that \(\sum_{a,b} p(a,b|x,y)=1\).
  3. (Causal Separation Condition): A fair process \(W^{A\prec B}\) following the order \(A \prec B\) must satisfy that “the result (B-output) of B cannot depend on the setting (A-input) of A”. Describe how this can be expressed as a condition on a specific partial trace of \(W^{A\prec B}\).
  4. (Noise Threshold Calculation): Suppose a quantum switch \(W_{\text{switch}}\) gives \(\mathrm{Tr}(S W_{\text{switch}}) = -0.5\) with respect to witness \(S\), and white noise \(W_{\text{noise}}\) gives \(\mathrm{Tr}(S W_{\text{noise}}) = 1.0\). Calculate the maximum noise threshold \(q\) such that the mixed process \(W' = (1-q) W_{\text{switch}} + q W_{\text{noise}}\) can be certified as having indefinite causality.
  5. (Work Advantage): Qualitatively explain why a quantum switch is more efficient than a fixed order (or its probabilistic mixture) in specific information processing tasks (e.g., determining whether two unitary channels commute or anticommute), relating this to the superposition of causal orders.

5. Explanation

  1. \(W\) and \(M^A \otimes M^B\) are both positive semidefinite matrices. The trace of the product of two positive semidefinite matrices \(P, Q\), \(\mathrm{Tr}(PQ)\), is always non-negative, even if \(PQ\) is not positive semidefinite. Therefore, \(p(a,b|x,y) = \mathrm{Tr}[(M^A \otimes M^B) W] \ge 0\).

  2. \(\sum_{a,b} p = \sum_{a,b} \mathrm{Tr}[(M^A_{a|x} \otimes M^B_{b|y}) W] = \mathrm{Tr}[(\sum_a M^A_{a|x}) \otimes (\sum_b M^B_{b|y}) W] = \mathrm{Tr}[(\mathbf{1}_{\text{out}}^A \otimes \mathbf{1}_{\text{out}}^B) W]\). Since this must equal 1 for all \(\{x, y\}\) choices, \(W\) must satisfy the normalization constraint \(\mathrm{Tr}[(\mathbf{1}_{\text{out}}^A \otimes \mathbf{1}_{\text{out}}^B) W] = 1\).

  3. This condition implies that when taking the partial trace over Alice’s input space (\(A_I\)), there should be no correlation between Bob’s output space (\(B_O\)) and Alice’s input space (\(A_I\)). That is, the reduced state of Bob’s output should not contain information about which input Alice selected.

  4. \(\mathrm{Tr}(S W') = (1-q) (-0.5) + q (1.0) = -0.5 + 0.5q + 1.0q = 1.5q - 0.5\). For the uncertainty principle to be certified, this value must be negative. \(1.5q - 0.5 < 0 \implies 1.5q < 0.5 \implies q < 1/3\). Therefore, the maximum threshold is \(q = 1/3\).

  5. With fixed order, one must test both \(A \to B\) and \(B \to A\) orders separately, requiring two experiments. The quantum switch uses the superposition of the control qubit to interfere both orders simultaneously, allowing information about the relationship (e.g., exchange or anti-exchange) of the two orders to be obtained with a single run. This utilizes the causal order itself as a resource.