Optimally cloned binary coherent states

Binary coherent state alphabets can be represented in a two-dimensional Hilbert space. We capitalize this formal connection between the otherwise distinct domains of qubits and continuous variable states to map binary phase-shift keyed coherent states onto the Bloch sphere and to derive their quantum-optimal clones. We analyze the Wigner function and the cumulants of the clones, and we conclude that optimal cloning of binary coherent states requires a nonlinearity above second order. We propose several practical and near-optimal cloning schemes and compare their cloning fidelity to the optimal cloner.

Figure 1

Illustrations of the BCS qubit basis states in terms of their Wigner functions. A perspective view is shown by the big illustration in the coordinate frame. The smaller illustrations depict, from left to right, a view along the $p$ quadrature, a view along the $x$ quadrature, and a view from the top.

Figure 2

(Left) Bloch sphere representation of the input BCS and the resulting clones. (Right) The state transformations of the optimal binary state cloning procedure can be decomposed into two parts. First, the lengths of the Bloch vector are reduced, i.e., the clones are in a mixed state. Second, the angle between the vectors is reduced, i.e., the mutual overlap of the clones increases.

Figure 3

(a)–(c) Wigner function of the optimally cloned BCS for a signal with mean photon number ${\left|\alpha \right|}^2=0.5$, i.e., $\theta =0.1884$. The distribution is viewed from the top, as well as along the $x$ and $p$ quadrature. (d)–(f) Illustration of the difference $W_{\mathrm{clone}}-W_{\mathrm{coherent}}$ between the optimally cloned state and the coherent signal viewed from the same perspectives.

Figure 4

(a) Cumulants along the $x$ quadrature and along the $p$ quadrature of the optimally cloned BCS. The dashed black lines indicate the coherent input state's mean amplitude ${\kappa }_1$ and variance ${\kappa }_2$. The clones are slightly squeezed along the $p$ quadrature. All odd-numbered cumulants vanish. The nonvanishing kurtosis (${\kappa }_4$, blue dash-dotted curve) shows that the clones exhibit non-Gaussian characteristics also along the $p$ quadrature. This is an interesting finding that points towards the complexity of the practical realization of the optimal cloning scheme. (b) Cumulants for the measure and prepare scheme featuring the preparation of states with the exact signal amplitude $\left|\alpha \right|$; see Sec. 5c. Along the $x$ quadrature the variation of the cumulants is more pronounced but the overall characteristics are very similar. Along the $p$ quadrature the only nonzero cumulant is the variance, which remains at the shot noise limit. (c) Cumulants for the measure and prepare scheme featuring the preparation of states with optimized amplitude $\left|\beta \right|$ and optimized squeezing parameter $r$.

Figure 5

Overview on the different considered cloning schemes. (a) Mere beam splitting. (b) Phase sensitive amplification. (c) Complete measurement and preparation. (d) Complete measurement, preparation, and phase-sensitive amplification. (e) Partial measurement and feedforward for regeneration. (f) Squeezing assisted partial measurement and feedforward for displacement of the phase-sensitively amplified remaining state. (g) Unambiguous state discrimination scheme.

Figure 6

Comparison of the cloning fidelity between different practical cloning approaches and the optimal cloner (dark gray). The measure and prepare schemes (three green curves with differently optimized prepared states; see main text for details) provide the lowest fidelity for small mean photon numbers. For higher mean photon numbers, the mere beam splitter (blue) performs worst. For small mean photon numbers, however, the fidelity of the beam splitting approach coincides with the tap-off measurement and feedforward scheme (red). Coinciding fidelities for small mean photon numbers are also provided by the cloner based on phase sensitive amplification followed by beam splitting (orange) and by the cloner based on the squeezed vacuum injected tap-off measurement followed by phase sensitive amplification and feedforward (light blue). The latter, however, branches to higher fidelity at mean photon numbers of about ${\left|\alpha \right|}^2⪆0.5$.

Figure 7

Difference $\mathrm{\Delta }\beta$ between the fidelity-optimized amplitude of the prepared clones $\beta$ and the signal's coherent amplitude $\alpha$ in the measure and prepare scheme. Due to the nonzero error probability in the discrimination of the states, the optimal value is always below the amplitude of the input state and the parameters only coincide in the classical limit $\alpha \gg 1$.

Figure 8

Phase-space illustration of the BCS cloning scheme with a squeezed vacuum input, phase-sensitive amplification, and a conditional displacement. (a) BCS alphabet at the input of the cloner. (b) Squeezed vacuum entering the open port of the tap-off beam splitter. (c) Signal after interference with the squeezed vacuum (qualitatively equal for both the transmitted and the reflected part). The signal amplitude is reduced, but the states are squeezed along the $x$ quadrature, which allows for a smaller error probability in the discrimination of the reflected signals compared to a plain vacuum input. (d) The coherent amplitude is increased via phase-sensitive amplification. Thereby the squeezing is shifted from the $x$ quadrature to the $p$ quadrature. (e) Subsequently, the amplitudes are further increased via coherent displacement. The displacement phase is conditioned on the outcome of the partial measurement on the tapped-off signal. (f) Finally, the signal is split symmetrically on a beam splitter to generate the clones.

Figure 9

Optimized parameters for the cloning scheme of Fig. 5. The corresponding $y$-axis labels are transmission $T$ (red curve), forward gain $g$ (green curve, opposite behavior compared to the transmission $T$), injected squeezing $r$ at the tap-off beam splitter (orange), and the squeezing associated with phase-sensitive amplification (purple curve). Up to a critical signal power of ${\left|\alpha \right|}^2\approx 0.56$ the beam splitter is perfectly transmissive (red curve, $T=1$) and the amplification of the coherent states is solely based on the squeezing in the phase-sensitive amplifier (purple curve). For higher signal powers, the measurement provides enough information about the state to compensate for the lost signal in the tap-off. The transmissivity instantly drops to a value as low as $T\approx 0.05$ and asymptotically decreases to $T=0$. Simultaneously the forward gain to the displacement stage jumps to $g\approx 0.95$ and increases to unity. The PSA squeezing parameter is just slightly above the squeezing parameter for the input to the free port of the tap-off beam splitter. Both tend to zero with increasing signal power.

Figure 10

Output fidelity and optimized parameters for the BCS cloning with an unambiguous state discrimination scheme. (Left) Overall output fidelity for differently prepared signal states in case of an inconclusive outcome. (Blue, lowest curve) Randomly preparing either of the input states, (green, middle curve) preparing a coherent state with optimized amplitude, and (red, highest curve) preparing a squeezed coherent state with optimized amplitude and squeezing parameter. (Right) The optimized parameters associated with the curves on the left and the fidelities for the separate inconclusive outcomes. (Dotted) Randomly preparing either of the input states. (Dashed) Optimized coherent state without squeezing. (Solid) Optimized squeezed coherent state. The fidelities (blue) of the states prepared at the event of an inconclusive outcome drop with increasing mean photon number from unity to 0.5. The coherent signal power of the prepared clones asymptotically coincides with the mean photon number of the signal states for all three strategies. For small mean photon numbers of the signal states, however, the optimized strategies initially remain at zero up to a value of ${\left|\alpha \right|}^2=0.5$ (optimized coherent state) and ${\left|\alpha \right|}^2\approx 1.2$ (optimized squeezed coherent state). The optimized squeezing parameter (green) initially increases up to $r\approx 0.75$ but drops to almost zero in exchange for the sudden onset of the coherent signal power at ${\left|\alpha \right|}^2\approx 1.2$.

Figure 11

Hilbert space representation of the coherent signal states and the different basis states. (Left) qubit basis: the signal states approach the ${45}^{\circ }$ diagonal in the limit of vanishing coherent amplitude $\left|\alpha \right|\mapsto 0$, and coincide with either of the basis states in the classical limit $\left|\alpha \right|\gg 1$. (Right) Cat-state basis: the situation is reversed. The signal states encompass an angle of ${45}^{\circ }$ with respect to the basis states in the classical limit and coincide with $|{\mathrm{\Psi }}_{+}\rangle$ for $\left|\alpha \right|\mapsto 0$.

Figure 12

Illustrations of the Wigner functions corresponding to the cat-state basis states. A perspective view is shown by the big illustration in the coordinate frame. The smaller illustrations depict, from left to right, a view along the $p$ quadrature, a view along the $x$ quadrature, and a view from the top.

Figure 13

(a) Sketch of a homodyne detector. The signal state $|\pm \alpha \rangle$ interferes with a bright local oscillator (LO) at a symmetric beam splitter and the emerging beams are detected by pin photodiodes. The difference signal yields a projected quadrature value where the quadrature phase is determined by the phase of the LO. (b) Sketch of the Kennedy receiver $\left(\right|\beta |=|\alpha \left|\right)$ and the optimized displacement receiver ($\left|\beta \right|$ optimized). The signal state is coherently displaced in phase space and is subsequently detected by a single photon detector (SPD). (c) Comparison of the error probability of the homodyne detector (red, dotted), the Kennedy receiver (green, dashed) [24], the optimized displacement receiver (purple, dash-dotted) [29], and the Helstrom bound [20, 21].