Quantum telescopy clock games

We consider the clock game–a task formulated in the framework of quantum information theory—that can be used to improve the existing schemes of quantum-enhanced telescopy. The problem of learning when a stellar photon reaches a telescope is translated into an abstract game, which we call the clock game. A winning strategy is provided that involves performing a quantum non-demolition measurement that verifies which stellar spatio-temporal modes are occupied by a photon without disturbing the phase information. We prove tight lower bounds on the entanglement cost needed to win the clock game, with the amount of necessary entangled bits equaling the number of time bins being distinguished. This lower bound on the entanglement cost applies to any telescopy protocol that aims to nondestructively extract the time bin information of an incident photon through local measurements, and our result implies that the protocol of Khabiboulline et al. [Phys. Rev. Lett. 123, 070504 (2019)] is optimal in terms of entanglement consumption. The full task of the phase extraction is also considered, and we show that the quantum Fisher information of the stellar phase can be achieved by local measurements and shared entanglement without the necessity of nonlinear optical operations. The optimal phase measurement is achieved asymptotically with increasing number of ancilla qubits, whereas a single qubit pair is required if nonlinear operations are allowed.

Figure 1

A general bipartite quantum game consists of questions and answers between a referee and two noncommunicating parties. However, the parties can use shared entanglement (wavy line) to coordinate their responses.

Figure 2

Schematic representation of the clock game. The referee delivers to Alice and Bob the phase encoded state $|{\mathrm{\Psi }}_{\varphi ,n}\rangle$ encoded in $2N$ qubits; each party receive half of them. They also receive an ancilla quantum state which they are free to specify. Both parties are allowed to manipulate the locally available quantum states to extract two pieces of classical information: integers $x$ and $y$. As a result, the qubits received from the referee are modified to the state ${\rho }^{AB}$. Alice and Bob send $(x,y,{\rho }^{AB})$ back to the referee.

Figure 3

(a) Circuit representation of the operations performed by Alice and Bob in the clock game. This procedure is later followed by sending the answer data $(x,y,{\rho }_{x,y}^{AB})$ back to the referee. (b) $C{Z}^n$ gate representation. (c) Symbol for Fourier basis measurement.

Figure 4

Circuit representation of the procedures performed by $K=3$ parties given that they have one entangled qudit state available. The measurements are performed in Fourier basis. Note that the procedure generalizes the scheme given in Fig. 3.

Figure 5

General scheme of quantum-enhanced long-baseline interferometry. The blue color indicates the path difference, which gives rise to the relative phase shift.

Figure 6

Circuit diagram of quantum-enhanced long-baseline interferometry with the photon arrival time-bin measurement.

Figure 7

The average Fisher information per ancilla photon ($y$ axis) as a function of ancilla photon number ($x$ axis). The phase angle $\mathrm{\Phi }$ is sampled uniformly over the interval $[0,2\pi )$.