Indefinite Causal Order in a Quantum Switch

Quantum mechanics allows events to happen with no definite causal order: this can be verified by measuring a causal witness, in the same way that an entanglement witness verifies entanglement. Here, we realize a photonic quantum switch, where two operations $\widehat{A}$ and $\widehat{B}$ act in a quantum superposition of their two possible orders. The operations are on the transverse spatial mode of the photons; polarization coherently controls their order. Our implementation ensures that the operations cannot be distinguished by spatial or temporal position—further it allows qudit encoding in the target. We confirm our quantum switch has no definite causal order by constructing a causal witness and measuring its value to be 18 standard deviations beyond the definite-order bound.

Figure 1

The quantum switch. A control qubit determines the order in which two quantum operations $\widehat{A}$ and $\widehat{B}$ are applied to a target qubit, $|\psi {⟩}_t$. (a) When the control is $|0{⟩}_c$, $\widehat{A}$ is applied before $\widehat{B}$. (b) When the control $|1{⟩}_c$, $\widehat{B}$ is applied before $\widehat{A}$. When the control is in the superposition $(|0⟩+|1⟩{)}_c/\sqrt{2}$, there is a superposition of the two orders, yielding the output state $|\varphi ⟩=(\widehat{B}\widehat{A}|\psi {⟩}_t\otimes |0{⟩}_c+\widehat{A}\widehat{B}|\psi {⟩}_t\otimes |1{⟩}_c)/\sqrt{2}$. (c) In Refs. [12, 13], the control is the transverse position at which a photon passes through a set of wave plates—consequently, the operations are performed in distinct spatial locations depending on the order. (d) In our experiment, the control is polarization; hence, each operation takes place in a fixed spatial location, independent of the order. The yellow pulses are graphical representations of the difference in temporal characteristics: In (c), the pulses are orders-of-magnitude shorter than the experiment and its internal components; in (d), the pulses are orders-of-magnitude longer so that the operations are indistinguishable in time as well as space.

Figure 2

Experimental schematic. The control qubit is defined by polarization. The polarizing beam splitter PBS1 routes the photon into either events $A$ or $B$, which realize unitary operations $\widehat{A}$ or $\widehat{B}$ acting on the spatial mode of the photon. Event $C$ is a $\widehat{X}$ polarization measurement, determining the Stokes parameter of the photon in the diagonal/antidiagonal basis. Lenses $L1$ and $L2$ are used as a telescope to ensure mode matching.

Figure 3

Top view of the setup for realizing the unitary operations $\widehat{A}$ and $\widehat{B}$ using a set of special inverting prisms $R$ and pairs of cylindrical lenses $C$. The prisms rotate the incoming transverse mode, effectively implementing the rotation $R(\theta )=(\genfrac{}{}{0ex}{}{\cos 2\theta }{\sin 2\theta }\genfrac{}{}{0ex}{}{\sin 2\theta }{-\cos 2\theta })$ in the $\{|{\mathrm{HG}}_{10}⟩,|{\mathrm{HG}}_{01}⟩\}$ qubit subspace. The cylindrical lenses give a $\pi /2$ relative phase shift to Hermite-Gaussian components of the incoming photon, effectively implementing $C(\pi /2)=(\genfrac{}{}{0ex}{}{1}{0}\genfrac{}{}{0ex}{}{0}{i})$. The spherical lenses ($L$) are used for mode matching. The half-wave plates ($H$) and quarter-wave plates ($Q$) are used to correct polarization changes caused by reflections in the prisms, and $\phi$ represents a phase plate. The unitary operations of our interest are realized by varying the angles ${\theta }_1$ and ${\theta }_2$. For example, in the figure $R({\theta }_1)$ is rotated by 45°, and for $R({\theta }_2)$, the angle is set at 0°. With a 0° global phase, the above setup represents an $\widehat{X}$ operation which transforms an input Hermite-Gaussian ${\mathrm{HG}}_{10}$ beam to a Hermite-Gaussian ${\mathrm{HG}}_{01}$ (see Supplemental Material [24], which includes [30]).

Figure 4

Spatial transformations. The result of the unitaries acting on an input spatial mode of ${\mathrm{HG}}_{10}$ are also first-order spatial modes.

Figure 5

Stokes parameters $⟨\widehat{X}{⟩}_{\widehat{A},\widehat{B}}$ obtained by measuring the polarization of the output control qubit in the diagonal basis. The red bars show the ideal, theoretical values, and the blue bars are the experimentally measured values. The unitary combinations are defined by combining the unitary operations at the top arm ($\widehat{A}$) and the bottom ($\widehat{B}$) arm, maintaining the order $\widehat{I}$, $\widehat{X}$, $\widehat{Y}$, $\widehat{Z}$, $\widehat{P}=(\widehat{Y}+\widehat{Z})/\sqrt{2}$ and $\widehat{Q}=(\widehat{X}+\widehat{Z})/\sqrt{2}$. A Stokes parameter of $+1$ means the output is diagonally polarized light, $-1$ means it is antidiagonally polarized light. $1\sigma$ errors are too small to be visible in the plot.