Controlled generation of mixed spatial qudits with arbitrary degree of purity

We propose a method for preparing mixed quantum states of arbitrary dimension $D$ ($D\ge 2$) which are codified in the discretized transverse momentum and position of single photons, once they are sent through an aperture with $D$ slits. Following our previous technique we use a programmable single phase-only spatial light modulator (SLM) to define the aperture and set the complex transmission amplitude of each slit, allowing the independent control of the complex coefficients that define the quantum state. Since these SLMs give us the possibility to dynamically vary the complex coefficients of the state during the measurement time, we can generate not only pure states but also quantum states compatible with a mixture of pure quantum states. Therefore, by using these apertures varying on time according to a probability distribution, we have experimentally obtained $D$-dimensional quantum states with purities that depend on the parameters of the distribution through a clear analytical expression. This fact allows us to easily customize the states to be generated. Moreover, the method offers the possibility of working without changing the optical setup between pure and mixed states, or when the dimensionality of the states is increased. The obtained results show a quite good performance of our method at least up to dimension $D=11$, being the fidelity of the prepared states $F>0.98$ in every case.

Figure 1

Experimental setup. O is an expansor, ${\text{SF}}_{\text{i}}$ are spatial filters, $L_i$ are lenses with a focal distance $f,{\text{SLM}}_{\text{i}}$ are spatial light modulators, and D is a single pixel detector.

Figure 2

Fidelity occurrence for qudits states with $D=11$. There are represented 500 arbitrary states $|\psi \rangle ={\sum }_{\ell =0}^{D-1}{e}^{i{\varphi }_{\ell }}|\ell \rangle$. The mean fidelity is $\overline{F}=0.992$ and the standard deviation is $\sigma =0.003$.

Figure 3

Purity evolution for qubits as a function of the number of states composing the statistical mixture, and different probability distribution widths ${\mathrm{\Delta }}_1$. We note that after 250 pure states were used to generate the mixture, the purity behavior stabilizes and it achieves its final value.

Figure 4

Purity of qubits as a function of the probability distribution width ${\mathrm{\Delta }}_1$ for different relative amplitudes ${\beta }_0$ and ${\beta }_1$. The solid line represents the theoretical values according to Eq. (12).

Figure 5

Purity of $D=3$ qudits as a function of the probability distribution width ${\mathrm{\Delta }}_1$ for different fixed widths ${\mathrm{\Delta }}_2$. The relative amplitudes are ${\beta }_0={\beta }_1={\beta }_2$. The solid line represents the theoretical values according to Eq. (10).

Figure 6

Density matrices of mixed states in a $D=7$ Hilbert space. In the case of ${\mathrm{\Delta }}_{\ell }=2\pi$, panels (a) and (c) represent, respectively, the real and imaginary parts reconstructed after a mixture of 250 pure states. (b) and (d) are the corresponding theoretical matrices. When ${\mathrm{\Delta }}_{\ell }=\frac{2\pi }{7}\ell$, panels (e) and (g) show the real and imaginary parts, obtained after 250 pure states were mixed, while (f) and (h) are the corresponding theoretical matrices.

Figure 7

Density matrices for a $D=11$ mixed state, when ${\mathrm{\Delta }}_{\ell }=2\pi$. Panels (a) and (d) are, respectively, the real and imaginary parts reconstructed from the mixture of 250 pure states. Correspondingly, (b) and (e) are the numerically simulated results obtained from the mixture of the same 250 pure states. Panels (c) and (f) represent the theoretical matrices.

Figure 8

Density matrices for a $D=11$ mixed state, when ${\mathrm{\Delta }}_{\ell }=\frac{2\pi }{11}(11-\ell )$. Panels (a) and (d) are, respectively, the real and imaginary parts reconstructed from the mixture of 250 pure states. Correspondingly, (b) and (e) are the numerically simulated results obtained from the mixture of the same 250 pure states. Panels (c) and (f) represent the theoretical matrices.