Coherence of operations and interferometry

Quantum coherence is one of the key features that fuels applications for which quantum mechanics exceeds the power of classical physics. This explains the considerable efforts that were undertaken to quantify coherence via quantum resource theories. An application of the resulting framework to concrete technological tasks is however largely missing. Here, we address this problem and connect the ability of an operation to detect or create coherence to the performance of interferometric experiments.

Figure 1

Mach-Zehnder interferometer. Two light sources L.S. 0 and L.S. 1 illuminate the two input ports of a beam splitter. Its two output modes experience different phases, before they reach a second beam splitter, after which they are measured by the detectors D. 0 and D. 1.

Figure 2

Sketch of a multipath interferometer. The quantum channel ${\mathrm{\Theta }}_1$ distributes an incoherent state $\tau$ to $M$ different paths that can lead to different phases via, e.g., interactions with different samples. The paths are recombined by a second channel ${\mathrm{\Theta }}_2$, and its output is incoherently measured.

Figure 3

Schematic representation of the game played by Alice and Bob that is used to describe the role of the detection of coherence in interferometry.

Figure 4

Sketch of a potential method with which Bob might increase his chances in the guessing game represented in Fig. 3. He prepares a correlated state and hands only a subsystem to Alice.

Figure 5

Sketch of the game played by Alice and Bob that is used to describe the role of the creation of coherence in interferometry.

Figure 6

Schematic representation of the guessing game described in the main text.

Figure 7

Addition of a SWAP operation to the game in Fig. 6, which shows that $L_{\lambda ,\vec{\varphi }}\left(\mathrm{\Theta }\right)$ is not monotonic.

Figure 8

With $\vec{\varphi }$ and $\mathrm{\Theta }\left(p_1\right)$ as in the main text, where $p_1$ denotes the probability of applying the Hadamard gate, $M_{\lambda ,\vec{\varphi }}\left(\mathrm{\Theta }\right)$ is plotted for different choices of $\lambda \ge \mu$. One clearly sees that $M_{\lambda ,\vec{\varphi }}\left(\mathrm{\Theta }\right)$ is not faithful for $\lambda \ne \frac{1}{2}$.

Figure 9

Plot corresponding to Fig. 8 for $\lambda \le \mu$.