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

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.

For the immunity against collective decoherence, Kempe et al. 24 proposed three-photon decoherence-free states carrying the logical qubits =  { } 0 , 1 j 1 or 2 L j Lj , as follows: ) / 2. In quantum information processing schemes, we can preserve quantum information encoding into logical qubits (single logical qubit information) from the unwanted effect of collective decoherence (due to evolution of symmetrical interaction), as follows: Lj Lj Lj 2 2 For reliable performance of quantum information processing, we design an optical scheme to generate three-photon (logical qubits) states, in Eq. 1, and encode single logical qubit information, in Eq. 2 (arbitrary quantum state), assisted by nonlinearly optical gates (using XKNLs, qubus beams, and PNR measurements) and linearly optical devices.
We introduce the XKNL effect (XKNL's Hamiltonian: H Kerr = ħχN 1 N p , where N i and χ are the photon-number operator and the strength of nonlinearity) in Kerr medium. The interaction of Kerr (XKNL) between the photon state, n 1 , and the coherent state, α P (photon-probe system), is given by  Figure 1 shows an optical scheme to generate three-photon decoherence-free states (logical qubit as in Eq. 1) and encode quantum information (single logical qubit information as in Eq. 2) using XKNLs, qubus beams, and PNR measurements (nonlinearly optical gates) and linearly optical devices. For the protection of quantum information, this scheme can produce the single logical qubit information (α β + 0 1 L L ) from the initial state (product state of three-photon: . First, as shown in Fig. 1 Then, the nonlinearly optical gate (1st gate) will be applied to the state, ψ 0 ABC .

st gate (photons A-B).
The first gate (in Fig. 2) consists of four conditional phase shifts θ (by XKNLs: positive), two linear phase shifts −θ (negative), qubus beams (two 50/50 BSs and PNR measurement), and feed-forward (phase shifter Φ n and path switch S). After the operation of the first gate to the state ψ 0 ABC , the output state ψ′ 0 ABC of premeasurement (PNR) is given by  ( 1) 2 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).  where α P is the probe beam (coherent state). The interaction of 50/50 BS in qubus beams is expressed as for α ∈ R. When we measure the qubus beam of path b using PNR measurement, if the result is 0 P b (no detection), we can acquire the state as . On the other hand, if the result is n P b (n ≠ 0), we can transform the resulting state of n P b to the resulting state of 0 P b , , by feed-forward (phase shifter Φ n and path switch S), according to the result, n, as described in Fig. 2 Then, if the result of measurement (in the qubus beam of path b) is 0 P b (no detection), the state, ψ′ 1 ABC , of Eq. 7 is collapsed to On the other hand, if the result is n P b (n ≠ 0), we can transform the resulting state of n P b to the resulting state of 0 P b by feed-forward (phase shifter Φ n , path switch S, and two spin flippers), according to the result, as described in Fig. 3. Also, the error probability P err 2nd of the second gate is identical with the error probability P err 1st of the first gate as ). Subsequently, two 50/50 BSs (photon A and B), phase-flipper (PF), and PBS (photon C) in Fig. 1 are applied to the output state of the second gate, as follows: 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 Φ n , path switch S, and two spin flippers) is or is not operated on photon C.
Also, if we want the state ψ 2 ABC , in Eq. 8, to be the superposition state on the basis of { 0 , 1 } L 2 L 2 , as in Eq. 1, we can transform to apply the spin flippers on all paths (1, 2, 3, 4, and 5).

3rd gate (photons A-B).
For the merging of paths (1 and 2), the third gate consists of two conditional phase shifts θ (by XKNLs: positive), a linear phase shift −θ (negative), qubus beams (two 50/50 BSs and PNR measurement), and feed-forward (two path switches S 1 and S 2 ), as described in Fig. 4. After the operation of the third gate to the state ψ 2 ABC in Eq. 8, the output state ψ′ 2 ABC of premeasurement (PNR) is given by When we measure the qubus beam of path b using PNR measurement, if the result is 0 P b (no detection), we can show the state as . On the other hand, if the result is n P b (n ≠ 0), we can switch the path of state in the result n P b , , to the state of result 0 P b , , by feed-forward (path switches S 1 and S 2 ), according to the result n ≠ 0, as described in Fig. 4. The error probability P err 3rd of the third gate is also the same as the error probabilities of the first and second gates, as Consequently, the output state ψ 3 ABC (the superposition of logical qubits by three-photon decoherence-free states) from the scheme of generation of three-photon decoherence-free states is expressed as  After the state ψ E ABC , in Eq. 10, passes through the arbitrary-BS for encoding the arbitrary quantum information [random or prearranged values (α i and β i ) by arbitrary-BS] into the three-photon decoherence-free states (single logical qubit information), the state ψ E ABC encoded the arbitrary information is given by where the transmission rate (κ 1 ) and reflection rate (κ 2 ) of the arbitrary-BS can be adjusted for our purposes (communication, information transfer, computation, etc.), and |α 1 | 2 + |β 1 | 2 = |α 2 | 2 + |β 2 | 2 = 1. Then, after the operation of the final gate to the state ψ E ABC in Eq. 11, the output state ψ′ E ABC of premeasurement (PNR) is given by Here, also, the error probability P err Fin of the final gate will be calculated as after PNR measurement on path b. Subsequently, if the probe beam on path b is measured by PNR measurement, the final state (single logical qubit information on three-photon decoherence-free states in Eq. 2) is generated in accordance with the measurement outcome (PNR) with the error probability P err Fin (= = = P P P err 1st err 2nd err 3rd ), as follows: where the final state ϕ′ L1 ABC can be transformed to the state ϕ L 1 ABC by performing the rotation or unitary operations because of the setting parameters (κ 1 and κ 2 ) in the arbitrary-BS, and vice versa. Finally, in Fig. 6, we express the error probabilities of our gates in terms of the difference in the amplitude of coherent state and the magnitude of phase shift by XKNL.
The error probabilities of four nonlinearly optical gates (1st, 2nd, 3rd, and final) are identical to P err , as described in Fig. 6, due to the probability to measure 0 P b (no detection) in α θ ±i sin P b on path b. Figure 6 and the table (in Fig. 6) obviously represent that the error probability, P err , will be approaching 0 when both the amplitude (α) of coherent state and magnitude (θ) of phase shift increase (utilizing strong coherent state and large phase shift of XKNL). It is possible to efficiently produce the output states of nonlinearly optical gates by increasing the multiplying value (αθ) of parameters, α and θ of ≈ α − θ e P ( )/2 err 2 2 , in the ideal case (the assumption without decoherence effect). In the next section, to conveniently analyze the decoherence effect in gates (1st, 2nd, 3rd, and final), we take the value, < − P 10 err 3 , of error probability to fix the parameters (the amplitude of the coherent state and the magnitude of condition phase shift) as αθ = 2.5.
So far, we have designed a scheme to encode quantum information into three-photon decoherence-free states (single logical qubit information) using nonlinearly optical gates (XKNLs, qubus beams, and PNR measurement) and linearly optical devices (50/50 BSs, PBSs, single photon operators, and an arbitrary-BS) for the resistance against collective decoherence. However, due to the utilization of XKNLs in our scheme, the decoherence effect, which makes to evolve quantum pure state into a mixed state by photon loss and dephasing, occurs in nonlinearly optical gates (1st, 2nd, 3rd, and final) when experimentally implemented by our scheme in practical optical fibers 66,67 . Thus, we should analyze the nonlinearly optical gates using XKNLs, qubus beams, and PNR measurement under the decoherence effect (photon loss and dephasing) 55,56,62 .

Analysis of Nonlinearly Optical Gates Using XKNLs, Qubus Beams, and PNR Measurement Under Decoherence Effect
In our scheme to encode quantum information into three-photon decoherence-free states (single logical qubit information), the nonlinearly optical gates (1st, 2nd, 3rd, and final) utilize XKNLs, qubus beams, and PNR measurement. However, when the nonlinearly optical gates are operated in optical fibers 55,56,60,62,63 , the decoherence effect (as distinct from the collective decoherence [23][24][25] ) is induced by photon loss and dephasing (coherent parameters of photon-probe system), affecting the efficiency and performance of the nonlinearly optical gates 55 ) and dephasing (decreasing the fidelities of output states ψ 0 ABC , ψ 1 ABC , ψ 2 ABC , and ϕ ′ L1 ABC or ϕ L1 ABC ). The decoherence effect in the Kerr medium can be induced by solving the master equation (description of open quantum system) 68 . Therefore, we should apply the analysis (solving the master equation) 55  where γ, t (=θ/χ), and a + (a) are the energy decay rate, the interaction time, and creation (annihilation) operator. And the solution of master equation, Eq. 14, is given as ρ ρ = +t J L t ( ) exp[( )] (0). From the solution in Eq. 14, we can obtain the photon loss rate in the coherent state (probe beam) as Λ α t (Λ t = e −γt/2 ) after Kerr medium. Also, the interaction time t of decoherence, ∼ D t , and XKNL, ∼ X t , can be taken as arbitrarily small time Δt (=t/N) for a good approximation of analysis 55,56,62 . When the initial state is supposed as α α ⊗ H V (i.e. H: XKNL and V: no interaction), we can show the process model of the interaction of XKNL, ∼ X t , and decoherence, ∼ D t (photon loss and dephasing) 55,56,62 from Eq. 14 (the detailed description of this model: APPENDIX), as follows: N , and θ = χt = χNΔt = NΔθ for Δt (=t/N) and α ∈ R. In Eq. 15, the coefficient of the right hand of the equation is called the coherent parameter, which can quantify the amount of decoherence effect by the degree of dephasing, as described in 55,56,62 (APPENDIX). For the experimental implementation of the nonlinearly optical gates (1st, 2nd, 3rd, and final) in optical fibers, which require the length of about 3000 km to obtain the magnitude of phase shift, θ = π, of the XKNL 66,67 , we can analyze the performance and efficiency of the gates in pure silica-core fibers 67  1st gate (photons A-B) under the decoherence effect. According to decoherence effect, we should consider the photon loss and dephasing coherent parameters in our analysis when the first gate is realized in optical fiber 67 . Thus, the output state ψ ′ 0 ABC [photon (A and B)-probe system] of the first gate in Eq. 5 will evolve into the mixed state ρ AB 0 , as follows: where the bases of N , θ = χt = χNΔt = NΔθ for Δt (=t/N) and α ∈ R. These equations mean the coherent parameters (off-diagonal terms) in the output state ρ AB 0 (Eq. 16) of the first gate.In Fig. 7, we represent the absolute values of coherent parameters according to the amplitude, α, of the coherent state in the qubus beam and the signal loss rate, χ/γ, in optical fibers. Here, we fix the error probability as = − P 10 err 1st 3 for αθ = αχt = 2.5 with N = 10 3 . Thus, when fixed = − P 10 err 1st 3 , we can parameterize the coherent parameters (Eq. 17) in the amplitude of the coherent state, α, and the signal loss rate, χ/γ, by θ = χt = 2.5/α and γt = 2.5γ/χα for Δt = t/1000 and θ θ Δ = =  N /1000( 10 ) 3 . As described in Fig. 7, we conclude that if the amplitude of the coherent state increases, then the output state ρ AB 0 does not evolve into the mixed state by increasing the absolute values of the coherent parameters to one whole signal loss. Therefore, we can obtain a reliable performance of the first gate (high fidelity) to employ the strong coherent state under the decoherence effect. A-B-C) under the decoherence effect. For the analysis of photon loss and dephasing coherent parameters 55,56,62 , the output state ψ′ 1 ABC [photon (A, B, and C)-probe system] of the second gate in Eq. 7 should be revised to the density operator ρ ABC 1 , as follows: where the bases of ρ AB 1 are

2nd gate (photons
N , θ = χt = χNΔt = NΔθ for,Δt (=t/N) and α ∈ R. Also, the remains of coherent parameters (C, O, L, and K) of ρ ABC 1 are written in Eq. 17. Figure 8 shows the absolute values of the coherent parameters for α and χ/γ in optical fibers. In the second gate, we take the identical condition of the first gate, as = − P 10 err 2nd 3 for αθ = αχt = 2.5 with N = 10 3 . As described in Fig. 8, when we employ the strong coherent state (increasing α), the absolute values of the coherent parameters in ρ ABC 1 approach 1 for the reliable performance of the second gate (remaining the output state as quantum pure state) under the decoherence effect.

3rd gate (photons A-B) and final gate (photon B) under the decoherence effect. Two (3rd and
final) gates can be considered as being several types of merging path gates. Thus, by the process model 55,56,62 of Eqs 14 and 15 (APPENDIX), the coherent parameters of two output states (ρ ABC 2 and ρ ABC E ) are the same (but the bases of the two density matrices are different), and the two output states (ρ ABC 2 and ρ ABC E ) can be expressed from Eqs 9 and 12, as follows: where the coherent parameter (C) is given by Eq. 17, and photon loss rate as Λ t = e −γt/2 . When increasing the amplitude of the coherent state, the absolute value of the coherent parameter, C 2 , can be maintained to one in the requirement as = = − P P 10 err 3rd err Fin 3 for αθ = αχt = 2.5 with N = 10 3 , as shown in Fig. 9. Consequently, if we increase the amplitude of coherent state for the operations of nonlinearly optical gates (1st, 2nd, 3rd, and final), the reliable performance of those gates can be obtained to retain the absolute values of coherent parameters as one where fixed the error probabilities (≈10 −3 ) and the experimental parameters as αθ = αχt = 2.5 with N = 10 3 . This means to prevent the output states (ρ AB 0 , ρ ABC 1 , ρ ABC 2 , and ρ ABC E ) from evolving into mixed states by dephasing the coherent parameters under the decoherence effect.
Furthermore, the error probabilities (P err 1st , P err 2nd , P err 3rd , and P err Fin ) affected by photon loss can be calculated to confirm the efficiency of nonlinearly optical gates, as follows: where αθ = αχt = 2.5, Λ t = e −γt/2 , and γt = 2.5/(0.0303) · α in the pure silica-core fibers having the signal loss of 0.15 dB/km (χ/γ = 0.0303) 67 . In Fig. 10, we can show that the error probabilities (P err 1st , P err 2nd , P err 3rd , and P err Fin ) of nonlinearly optical gates converge to zero when increasing the amplitude of the coherent state, because the photon loss rates approach one ) under the decoherence effect in optical fibers 67 . Even if the amplitude of the coherent state is small, we can also obtain low error probabilities, as described in the small graph of Fig. 10 , and Λ ≈ .
. Namely, if we consider only photon loss induced by the decoherence effect in optical fibers, the high efficiency of nonlinearly optical gates from the error probabilities can be acquired by both the weak and the strong amplitude of the coherent state.
However, as mentioned above, photon loss and dephasing occur simultaneously when operating nonlinearly optical gates in optical fibers, and we could establish the effects from the process model in Eqs 14 and 15 55,56,62 . Figure 11 shows a single example (the first gate) to simultaneously take into account photon loss and dephasing by our analysis (the process model of the decoherence effect, Eqs 14 and 15) in optical fibers. When α > 400 in the weak range of the coherent state, it is possible to obtain the low error probability (<10 −2 ) [from (B-1) in Fig. 11] as strong range [from (C-1) in Fig. 11], as listed in the table in Fig. 11. While α ≪ 400 (extremely weak range), the error probability increases (decreasing the efficiency of the first gate). But, in view of the performance of the gate, due to dephasing the coherent parameter, the absolute value, OC 2 , of the coherent parameter (off-diagonal term in ρ AB 0 ) will decrease (evolving to mixed state) in the weak range, as described in the table in Fig. 11. This obviously means that we can acquire only a single merit of high efficiency (low error probability against photon loss) or high performance (approaching the absolute value of the coherent parameter to one against dephasing) in the weak range of amplitude of the coherent state. Fortunately, both high efficiency and high performance of the nonlinearly optical gate can be achieved by using the strong amplitude of the coherent state under the decoherence effect (photon loss and dephasing), as described in Fig. 11. Consequently, we can conclude that the nonlinearly optical gates (1st, 2nd, 3rd, and final), using XKNL, qubus beams, and PNR measurement can . Graph represents the error probabilities (P err 1st ,P err 2nd ,P err 3rd , and P err Fin ) and photon loss rates (Λ t 4 , Λ t 6 , and Λ 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 (α: 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. ) and N = 10 3 , 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. be robust against the decoherence effect to increase the coherent state (probe beam) for high efficiency (low error probability) and reliable performance (high fidelity) by our analysis.

Discussion and Conclusion
Recently, for the feasible XKNLs, researchers have variously proposed the experimental technologies in XKNLs with low error rate and high fidelity. The physical systems can realize to obtain the sufficient large strength of XKNL, such as electromagnetically induced transparency (EIT) 69,70 , circuit electromechanics 71 , an artificial atom 72 , and three-dimensional circuit quantum electrodynamic architecture 73 . Also, to acquire the strong phase shift for the photonic phase gate, Friedler et al. 74 showed the large nonlinear interactions between ultraslow-light pulses or two stopped light pulses 75 , which can be realized in the regime of EIT. However, in the traveling fields described by localized single-photon pulse, the large phase shifts are impossible by large absorption losses 76 . Fortunately, there exist nontrivial problems to overcome at a theoretical level (utilizing both EIT and long-range interaction) in order to exploit weak XKNL 70 . Thus, high fidelities, nonzero conditional phases, and high photon numbers are compatible under certain conditions in He et al. 77 . Furthermore, for the reliable quantum information processing, researches have improved the capability of detection from the increase of the absorption 78,79 , and proposed the method of utilizing the positive-operator-value measurement elements 5,62 for the reliable implementation of PNR measurement.
Therefore, in Sec. 2, we presented a scheme to encode quantum information (arbitrary information) into three-photon decoherence-free states (singe logical quantum information) via nonlinearly optical gates (1st, 2nd, 3rd, and final), which employ XKNLs, qubus beams, and PNR measurement, and linearly optical devices for immunity against collective decoherence. Then, for the high efficiency and reliable performance of the nonlinearly optical gates against photon loss and dephasing induced by the decoherence effect in optical fibers, we demonstrated the utilization of the strong amplitude of the coherent state (probe beam) by our analysis (Sec. 3) when operating nonlinearly optical gates (1st, 2nd, 3rd, and final) using XKNLs, qubus beams, and PNR measurement. Therefore, the advantages of our scheme are as follows: (1) To prevent quantum information transferred or operated in quantum information processing against collective decoherence between systems and the environment 23-25 , our scheme was designed to encode quantum information into three-photon decoherence-free states (logical qubits) using experimentally feasible nonlinear (XKNLs) and linear optics. The nonlinearly optical gates via XKNLs are critical components for generation of the three-photon decoherence-free states (logical qubits) and encoding quantum information in our scheme. Thus, for the robustness of the nonlinearly optical gates against decoherence effect, we also analyzed photon loss and dephasing 55,56,62 in optical fibers, besides collective decoherence [23][24][25] . (2) We showed that the nonlinearly optical gates designed by XKNLs, qubus beams, and PNR measurement should employ the strong amplitude of the coherent state to acquire the high efficiency (low error probability) and reliable performance (high fidelity) by our analysis due to the process model 55,56,62 of the master equation 68 because of simultaneous occurrence of photon loss and dephasing coherent parameters. Thus, when this scheme is experimentally realized, it will be immune to collective decoherence (nonunitary process between the system and the environment) and robust against the decoherence effect (photon loss and dephasing in the interaction of XKNLs). (3) In our scheme, the designed nonlinearly optical gates employ qubus beams (two 50:50 BSs) 11,43,47 , and the strategy of PNR measurement 7,11,31,34,55,56,62,64,65 for the controlled operation. If we use the displacement operator 7,55,56 or homodyne measurement 36,37,54-56 (instead of qubus beams or PNR measurement) for the same operation as nonlinearly optical gates (1st, 2nd, 3rd, and final), we should employ the conditional phase shift −θ (by XKNL: negative), as described in 62 . However, Kok 80 showed that it is generally not possible to change the sign of the conditional phase shift (−θ). Thus, our nonlinearly optical gates (1st, 2nd, 3rd, and final) using only positive XKNLs (θ) with qubus beams and PNR measurement are more experimentally feasible than other gates 7,33-37,55,56 that should to use the negative XKNL. (4) In particular, through our analysis in Sec. 3, we demonstrated that the nonlinearly optical gates in optical fibers can obtain high efficiency and reliable performance by increasing the amplitude of the coherent state (probe beam) under the decoherence effect (photon loss and dephasing). This also brings an experimental merit of the weak magnitude of an XKNL (θ) by due to increasing the amplitude of the coherent state in nonlinearly optical gates to obtain robustness against the decoherence effect (i.e., when α = 1 × 10 6 , the magnitude of XKNL is required to be θ = 2.5 × 10 −6 , because of the fixed condition αθ = 2.5 for < − P 10 err 3 ). Therefore, because of the difficulty of implementing a large magnitude in XKNLs 64,81,82 , our nonlinearly optical gates using XKNL can be realized in practice. (5) The probe beam of path a can be recycled for other nonlinearly optical gates, because PNR measurements are operated in the probe beam of path b in all nonlinearly optical gates (1st, 2nd, 3rd, and final). Namely, the unmeasured probe beam ( α α Λ Λ θ or cos t m P a t m P a where photon loss, Λ = γ − e t m mt/2 : photon loss rate according to nonlinearly optical gates) of path a can be utilized as the input probe beam (recycled) of other nonlinearly optical gates using qubus beams and XKNLs.
Consequently, we demonstrate that our scheme for generating single logical qubit information (into three-photon decoherence-free states) with the immunity (against collective decoherence between the system and the environment) can be experimentally implemented and be robust (against the decoherence effect in optical fibers) through our analysis of nonlinearly optical gates using XKNLs, qubus beams, and PNR measurements.