Indefinite Causal Order

Indefinite causal order refers to processes that cannot be represented as an ordinary probabilistic mixture of the specified definite orders of local operations. The quantum switch is a central example of coherent order control. The precise claim depends on how local laboratories and operations are identified; it does not mean that all events in the apparatus lack a spacetime description.

A coherent switch of two operations

Let $U$ and $V$ act on a target $\lvert\psi\rangle$, and let a control qubit start in $\lvert+\rangle$. An ideal switch produces

$$\lvert\Psi_{\rm out}\rangle= \frac{\lvert0\rangle VU\lvert\psi\rangle+ \lvert1\rangle UV\lvert\psi\rangle}{\sqrt2}.$$

Measuring the control in the $X$ basis gives unnormalized target vectors $(VU\pm UV)\lvert\psi\rangle/2$. The control therefore probes interference between two orders, rather than merely recording which order was selected by a classical coin.

For commuting unitaries, $VU=UV$, the minus outcome never occurs. For anticommuting Pauli operations $U=X,V=Z$, the plus outcome never occurs. A classical mixture of the two order branches with its order label dephased gives equal $X$-basis control outcomes. This ideal calculation diagnoses order coherence; device-level certification must still characterize the allowed operations and side channels.

Process matrices

The process-matrix framework assigns probabilities to local quantum instruments without initially assuming a global order. In a fixed Choi convention, one writes schematically

$$p(a,b\mid x,y)= \operatorname{tr}\!\left[ W(M^A_{a|x}\otimes M^B_{b|y})\right].$$

A valid $W$ must give nonnegative normalized probabilities for all allowed local instruments. Positivity alone is insufficient; linear normalization constraints are essential. A causally separable bipartite process has the form

$$W=qW^{A\prec B}+(1-q)W^{B\prec A}, \qquad 0\le q\le1.$$

Causal nonseparability means no such decomposition exists. The quantum switch usually requires an explicit additional party or output receiving the control; forcing it into an incomplete two-party description can obscure the relevant order.

Two levels of certification

A causal witness is a functional nonnegative on the set of causally separable processes and negative on a selected process. Evaluating it relies on the specified quantum operations or trusted features of the implementation.

A causal inequality instead bounds observed classical input/output probabilities under causal-order assumptions, without the same detailed device characterization. These are different certification tasks. Oreshkov, Costa, and Brukner exhibited process-matrix correlations violating a causal inequality. Araújo and collaborators showed that the standard quantum switch is causally nonseparable while its ordinary classical-input/output scenario does not violate causal inequalities. Extra parties, assumptions, or resources define different tests and must be named.

Relation to spacetime and gravity

Laboratory switches can coherently route a target through operations on a fixed background. Their resource is not automatically a superposition of gravitational fields. Conversely, proposals involving quantum clocks or indefinite geometry introduce additional dynamical and localization questions.

There is no unrestricted ability to signal to one’s past in the switch circuit. An operation called $A$ may be delocalized over laboratory times conditional on a control. The identification of one abstract operation with those physical events is part of the experiment’s model.

Exercise

In the switch, suppose the control is dephased so that its off-diagonal coherence is multiplied by a real $v\in[0,1]$. For commuting $U,V$, find the two control probabilities.

Solution

The target factors out, leaving control state $\frac12\begin{pmatrix}1&v\\v&1\end{pmatrix}$. Thus $p(\pm)=(1\pm v)/2$. At $v=0$ the order branches are an incoherent mixture; at $v=1$ the ideal deterministic plus outcome is recovered.

References

• Araújo, M., Branciard, C., Costa, F., Feix, A., Giarmatzi, C., and Brukner, Č., “Witnessing causal nonseparability,” New Journal of Physics 17, 102001 (2015), arXiv:1506.03776 — switch witnesses and inequality distinction.
• Oreshkov, O., Costa, F., and Brukner, Č., “Quantum correlations with no causal order,” Nature Communications 3, 1092 (2012), doi:10.1038/ncomms2076 — process matrices and a causal-inequality example.