Preparation of quantum information encoded on three-photon decoherence-free states via cross-Kerr nonlinearities

Abstract

We present a scheme to encode quantum information (single logical qubit information) into three-photon decoherence-free states, which can conserve quantum information from collective decoherence, via nonlinearly optical gates (using cross-Kerr nonlinearities: XKNLs) and linearly optical devices. For the preparation of the decoherence-free state, the nonlinearly optical gates (multi-photon gates) consist of weak XKNLs, quantum bus (qubus) beams, and photon-number-resolving (PNR) measurement. Then, by using a linearly optical device, quantum information can be encoded on three-photon decoherence-free state prepared. Subsequently, by our analysis, we show that the nonlinearly optical gates using XKNLs, qubus beams, and PNR measurement are robust against the decoherence effect (photon loss and dephasing) in optical fibers. Consequently, our scheme can be experimentally implemented to efficiently generate three-photon decoherence-free state encoded quantum information, in practice.

Figure 1

Schematic plot of single logical qubit information into three-photon decoherence-free states: This scheme consists of two parts of generation of the three-photon decoherence-free state (part 1), and encoding process (part 2). In the generation of the three-photon decoherence-free state, the three (1st, 2nd, and 3rd) gates use the nonlinearly optical effects (XKNLs). The final gate in the encoding process uses XKNLs to produce the single logical qubit information having immunity (against collective decoherence between the system and the environment).

Figure 2

This plot represents the first gate via XKNLs, qubus beams, and PNR measurement for the interactions between two photons A and B. In the first gate, four conditional phase shifts are only positive in the qubus beams, including two linear phase shifts. After PNR measurement, according to the result of the measurement, feed-forward (phase shift Φⁿ and path switch S) is or is not operated on photon B.

Figure 3

This plot represents the second gate via XKNLs, qubus beams, and PNR measurement for the interactions between photons A, B, and C. In the second gate, six conditional phase shifts are only positive in the qubus beams, including two linear phase shifts. After PNR measurement, according to the result of the measurement, feed-forward (phase shift Φⁿ, path switch S, and two spin flippers) is or is not operated on photon C.

Figure 4

This plot represents the third gate via XKNLs, qubus beams, and PNR measurement for the interactions between two photons A and B. In the third gate, two conditional phase shifts are only positive, in the qubus beams, including one linear phase shift. After PNR measurement, according to the result of the measurement, feed-forward (two path switches S₁ and S₂) is or is not operated on photons A and B.

Figure 5

This plot represents the encoding process using a linearly optical device (arbitrary-BS) and nonlinearly optical gate (final gate: via XKNLs, qubus beams, and PNR measurement). For encoding the arbitrary quantum state into the three-photon decoherence-free states (single logical qubit information), arbitrary-BS, which can perform the reflection and transmission with the arbitrary probabilities, is applied to photon 2 in the linearly optical part. Then the final gate is used to arrange the paths of photon 2 into the merged single path.

Figure 6

Graph shows the error probability (P_err) according to the changing amplitude (α) of coherent state and magnitude (θ) of phase shift. Four nonlinearly optical gates (1st, 2nd, 3rd, and final) have the same error probability (${\mathrm{P}}_{\mathrm{err}}^{\mathrm{1st}}={\mathrm{P}}_{\mathrm{err}}^{\mathrm{2nd}}={\mathrm{P}}_{\mathrm{err}}^{\mathrm{3rd}}={\mathrm{P}}_{\mathrm{err}}^{\mathrm{Fin}}$), by PNR measurement. And the values of error probabilities calculated in terms of the parameters (α and θ) are listed on the table. Due to graph and table, the error probability is decreasing to 0 if we use the strong coherent state and the large phase shift.

Figure 7

Graphs of the absolute values of the coherent parameters in the output state ${\rho }_{\mathrm{AB}}^0$ of the first gate in our scheme (single logical qubit information) with respect to the amplitude of the coherent state α and the signal loss χ/γ in optical fibers, where fixed ${\mathrm{P}}_{\mathrm{err}}^{\mathrm{1st}}={10}^{-3}\left(\alpha \theta =\alpha \chi t\approx 2.5\right)$ with N = 10³.

Figure 8

Graphs of the absolute values of the coherent parameters in the output state ${\rho }_{\mathrm{ABC}}^1$ of the second gate in our scheme (single logical qubit information) according to the amplitude of the coherent state α and the signal loss χ/γ in optical fibers, where fixed ${\mathrm{P}}_{\mathrm{err}}^{\mathrm{2nd}}={10}^{-3}\left(\alpha \theta =\alpha \chi t\approx 2.5\right)$ with N = 10³.

Figure 9

Graph of the absolute value of the coherent parameter in output states ${\rho }_{\mathrm{ABC}}^2$ (third gate) and ${\rho }_{\mathrm{ABC}}^{\mathrm{E}}$ (final gate) in our scheme (single logical qubit information) with regard to the amplitude of the coherent state α and the signal loss χ/γ in optical fibers, where fixed ${\mathrm{P}}_{\mathrm{err}}^{\mathrm{3rd}}={\mathrm{P}}_{\mathrm{err}}^{\mathrm{Fin}}={10}^{-3}\left(\alpha \theta =\alpha \chi t\approx 2.5\right)$ with N = 10³.

Figure 10

Graph represents the error probabilities (${\mathrm{P}}_{\mathrm{err}}^{\mathrm{1st}}$,${\mathrm{P}}_{\mathrm{err}}^{\mathrm{2nd}}$,${\mathrm{P}}_{\mathrm{err}}^{\mathrm{3rd}}$, and ${\mathrm{P}}_{\mathrm{err}}^{\mathrm{Fin}}$) and photon loss rates (${\mathrm{\Lambda }}_t^4$, ${\mathrm{\Lambda }}_t^6$, and ${\mathrm{\Lambda }}_t^2$) in nonlinearly optical gates, according to the amplitude of coherent state α with αθ = 2.5 in optical fibers having 0.15 dB/km (χ/γ = 0.0303). In the range of weak ($\alpha :300~600$) and strong (α > 600) coherent states, the error probabilities are maintained to nearly zero when we consider only photon loss under the decoherence effect. The range of the weak coherent state (0 < α < 500) is expressed in the small graph.

Figure 11

For the quantification of dephasing, plot (A) represents an absolute value, $\left|{\left|\mathrm{OC}\right|}^2\right|$, of the coherent terms in ${\rho }_{\mathrm{AB}}^0$ of the first gate, and plots (B) and (C) show the absolute value of the coherent parameter in weak (B) and strong (C) amplitudes of the coherent state for αθ = 2.5 (fixed ${\mathrm{P}}_{\mathrm{err}}^{\mathrm{1st}}={10}^{-3}$) and N = 10³, regarding the signal loss χ/γ in optical fibers. Also, we can recalculate the error probability of the first gate for photon loss with dephasing and χ/γ = 0.0303 in the range of the weak (B-1) and strong (C-1) amplitude of the coherent state. In the weak and strong ranges of the coherent state (probe beam), the error probability and the absolute value of coherent parameter are listed in the table, when photon loss and dephasing are simultaneously applied to the first gate.