Emulation of quantum measurements with mixtures of coherent states

We propose a methodology to emulate quantum phenomena arising from any nonclassical quantum state using only a finite set of mixtures of coherent states. This allows us to successfully reproduce well-known quantum effects using resources that can be much more feasibly generated in the laboratory. We present a simple procedure to experimentally carry out quantum-state emulation with coherent states, illustrate it emulating multiphoton NOON states with few phase-averaged coherent states, and demonstrate its capabilities in observing fundamental quantum mechanical effects, such as the Hong-Ou-Mandel effect, violating Bell inequalities and witnessing quantum nonclassicality.

Figure 1

The measurement of (a) a regular quantum statistical mixture and (b) a general emulation representation stated in (1). In both kinds of state preparation, the component ${\rho }_i$ is produced by the source with probability proportional to $|c_i|$. In the case of a statistical mixture state, estimation of the expectation value $\langle A\rangle$ of an observable $A$ requires no knowledge about how $\rho$ is prepared $\left({\rho }_i\right)$. In the emulation of $\rho$ that requires some negative $c_i$'s, classical information concerning state preparation is necessary: The observer measures either $A$ or $-A$ depending on the ancillary measurement of ${\rho }_{\mathrm{C}}$ with $B$.

Figure 2

Representation of the single-photon state in terms of phase-averaged coherent states: (a) decomposition coefficients, and (b) diagonal elements of the optimal linear combination of the coherent states, shown in different scales in the main plot and the inset.

Figure 3

Representation of the two-photon state in terms of phase-averaged coherent states: (a) decomposition coefficients, and (b) diagonal elements of the optimal linear combination of the coherent states, shown in different scales in the main plot and the inset.

Figure 4

The values of the nonclassicality witness operator $W$ [Eq. (28)] for the coherent states of the single-photon state representation depicted in Fig. 2. The dashed horizontal line shows the expectation value of the nonclassicality witness operator for the single-photon state.

Figure 5

Approximation of the single-photon nonclassicality witness by the four-detector measurement setup: (a) detection scheme, (b) Fock-basis decomposition coefficients of the POVM elements ${\mathrm{\Pi }}_m$, and (c) comparison of the ideal witness $W$ (gray bars) and the constructed witness $W_4$ (blue line). The inset in (c) shows the found decomposition coefficients $z_m$ in Eq. (35). The detection efficiency $\eta =0.8$ and the dark count rate $\varepsilon =0.001$ were used for the calculations.

Figure 6

(a) Scheme of observing the Hong-Ou-Mandel effect with single photons and (b) the setup for its “classical emulation.”. Gray dashed lines show how the phase-averaged coherent states can be generated by variable splitting of a reference coherent state with subsequent application of a random phase shift $\phi$ in one of the arms.

Figure 7

Scheme of interferometer for phase estimation using a two-photon NOON state.

Figure 8

Dependence of normalized coincidence rates $g_2\left(\theta \right)$ on the measured phase shift $\theta$. Solid lines indicate the dependence for interference of two single-photon states. Points and error bars show the values and the standard deviations for classically emulated single-photon states for $N={10}^8$ repetitions.

Figure 9

Detection scheme for classical emulation of two-mode entanglement and Bell-type measurements with phase-averaged probe coherent states. Alice and Bob prepare the same amplitudes and opposite phases of the coherent states.