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Performance comparison between classical and quantum control for a simple quantum system
Physica A 387 (2008) 1056–1062 www.elsevier.com/locate/physa
Performance comparison between classical and quantum control for a simple quantum systemI Zairong Xi a,∗ , Guangsheng Jin b a Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science,
Chinese Academy of Sciences, Beijing, 100080, China b Institute of Physics, Chinese Academy of Sciences, Beijing, 100080, China
Received 23 June 2007; received in revised form 8 August 2007 Available online 12 October 2007
Abstract Bra´nczyk et al. pointed out that the quantum control scheme is superior to the classical control scheme for a simple quantum system using simulation [A.M. Bra´nczyk, P.E.M.F. Mendonca, A. Gilchrist, A.C. Doherty, S.D. Barlett, Quantum control theory of a single qubit, Physical Review A 75 (2007) 012329 or arXiv e-print quant-ph/0608037]. Here we rigorously prove the result. Furthermore we will show that any quantum operation does not universally “correct” the dephasing noise. c 2007 Elsevier B.V. All rights reserved.
Keywords: Quantum control; Quantum information; Quantum channel; Quantum noise; Classical control
1. Introduction Recently, Bra´nczyk et al. considered the classical and quantum control for a simple quantum system in Ref. [2]. They considered the following operational task: a qubit prepared in one of the two non-orthogonal states |ψ1 i or |ψ2 i (with overlap hψ1 |ψ2 i = cos θ for 0 ≤ θ ≤ π2 ) is transmitted along a quantum channel with the dephasing noise, i.e., with probability p a Pauli operator Z is applied to the system, and with probability 1 − p the system is unaltered, where Z |0i = |0i, Z |1i = −|1i and {|0i, |1i} be a basis for the qubit Hilbert space. The noise is thus described by a quantum operation, i.e., a completely-positive trace-preserving (CPTP) map E p , that acts on a single-qubit density matrix ρ as E p (ρ) = p Zρ Z + (1 − p)ρ.
(1)
Suppose the noisy channel is to be fully characterized, meaning that p is known and without loss of generality in the range 0 ≤ p ≤ 0.5. Two initial states to be oriented are chosen as
I This research is supported by the National Natural Science Foundation of China (No. 60774099, No. 60221301) and by the Chinese Academy of Sciences (KJCX3-SYW-S01). ∗ Corresponding author. E-mail address: [email protected] (Z. Xi).
c 2007 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2007.10.019
Z. Xi, G. Jin / Physica A 387 (2008) 1056–1062
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θ θ |ψ1 i = cos |+i + sin |−i, 2 2 (2) |ψ2 i = cos θ |+i − sin θ |−i, 2 2 √ where |±i = (|0i ± |1i)/ 2, in which their distinguishability is maintained under the dephasing noise by their trace distance. It is easy to see Z |+i = |−i, Z |−i = |+i. To quantify the performance of control scheme, the average fidelity is used to compare the noiseless input states |ψi i with the corrected output states ρi . Assuming an equal probability for sending either state |ψ1 i or |ψ2 i, the performance index is 1 1 F(|ψ1 i, ρ1 ) + F(|ψ2 i, ρ2 ) 2 2 1 1 = hψ1 |ρ1 |ψ1 i + hψ2 |ρ2 |ψ2 i, 2 2
FC =
(3)
where the fidelity between a pure state |ψi and a mixed state ρ is defined as F(|ψi, ρ) ≡ hψ|ρ|ψi. The fidelity F ranges from 0 to 1 and is a measure of how much the two states overlap each other (a fidelity of 0 means the states are orthogonal, whereas a fidelity of 1 means the states are identical). It has the following simple operational meaning when the input state is pure: the fidelity F(|ψi i, ρi ) is the probability that the state ρi will yield outcome |ψi i from the projective measurement {|ψi ihψi |, |ψi⊥ ihψi⊥ |}. They considered two kinds of control schemes, i.e., classical and quantum schemes, to “correct” the system without knowing which state was transmitted, i.e., undo the effect of the noise, through the use of a control scheme based on measurement and feedback. Classical control For the considered case, the optimal measurement given by Helstrom [3] is a projective measurement onto the basis {|0i, |1i} in terms of maximizing the average probability of a success, which successfully discriminates the states |ψ1 i and |ψ2 i with probability PHelstrom = 12 (1 + sin θ ). Then an entanglement-breaking trace-preserving (EBTP) scheme is used [4,8], which arises because the output system is unentangled with any other system, regardless of its input state. Through this scheme the optimal performance was proved to be q 1 1 FD R2 = + sin4 θ + cos2 θ (4) 2 2 when preparing states s s 2 1 sin θ 1 sin2 θ ± |0i + ∓ |1i |Ψ ± i = 2 2γ 2 2γ corresponds to measurement results 0, respectively, 1 where γ =
(5) p
cos2 θ + sin4 θ .
Do nothing Doing nothing to correct the states is also a classical control strategy. The average fidelity of this scheme is given by FN = 1 − p cos2 θ.
(6)
Random preparation Another control strategy would be to prepare the states randomly with probability 12 , i.e., 12 |ψ1 ihψ1 | + 12 |ψ2 ihψ2 |, which is not considered in Ref. [2]. Although trivial, this strategy is of interest for comparison with other schemes. This scheme also is not described by an EBTP map, but we will nonetheless refer to it as “classical”. The average fidelity of this scheme is given by FE =
1 (1 + cos2 θ ). 2
(7)
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Quantum control Bra´nczyk et al. showed that the above EBTP optimal classical control strategies can be outperformed by using a strategy based on quantum concepts. The quantum control scheme operates by performing a weak measurement of the system and then correcting it based on the results of the measurement. Notice α α (8) Z α = e−iα Z /2 = cos I − i sin Z , 2 2 the dephasing noise (1) can be represented as 1 1 Ď Z α ρ Z αĎ + Z −α ρ Z −α 2 2 α 2 α Zρ Z + cos2 ρ, = sin 2 2
Eα (ρ) =
(9)
which is the preferred ensemble. The weak measurement is chosen χ χ M0 = cos |+iih+i| + sin |−iih−i|, 2 2 χ χ M1 = sin |+iih+i| + cos |−iih−i|, 2 2 r √ sin4 θ where χ = arcsin 4 , r x = (1 − 2 p) cos θ , and |±ii = (|0i ± i|1i)/ 2 being the eigenstates of 2 2 2 2 (1−r x ) cos θ +(1−r x ) sin θ
Y . Based on the measurement result, the feedback control is performed on the quantum system to be a unitary rotation about the x-axis, Z ±η where tan η =
1 , (1 − 2 p) cos θ tan χ
which is a rotation about the x-axis applied to the state and returns the state back to the x z-plane. Then the maximum fidelity has been attained, s 4 1 sin θ . FQC = 1 + cos2 θ + 2 1 − r x2 In conclusion, the resulting weak measurement followed by feedback is thus described by a quantum operation (a CPTP map) C QC acting on a single qubit, given by C QC (ρ) = (Z +η M0 )ρ(Z +η M0 )Ď + (Z −η M1 )ρ(Z −η M1 )Ď . 2. Comparisons between quantum and classical schemes Bra´nczyk et al. pointed out that the quantum control scheme outperforms the classical control scheme for the simple quantum system using simulation [2]. Now we rigorously prove the result. 4θ Notice that sin ≥ sin4 θ, which implies that 1−r 2 x
FQC ≥ FD R2 . Because cos2 θ ≤ cos θ ≤
p
cos2 θ + sin4 θ we can get
FQC ≥ FD R2 ≥ FE . It is interesting to note that FD R2 ≤ 1 which implies that there exists point 0 < p ∗ ≤ 1 such that FD R2 ≤ FN when 0 ≤ p ≤ p ∗ . That is, for some small error probability, the classical scheme is inferior to doing nothing.
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The difficulty focuses on the comparison between the quantum scheme FQC and the doing nothing FN , i.e., proving s 4 1 sin θ 1 − p cos2 θ ≤ 1 + cos2 θ + . 2 1 − r x2 Now notice that 2r x cos θ sin θ = 2( 2
q
1 − r x2 cos θ )
≤ (1 − r x2 ) cos2 θ +
!
rx p
sin θ 2
1 − r x2
r x2 sin4 θ. 1 − r x2
Then r x2 cos2 θ + sin4 θ + 2r x cos θ sin2 θ ≤ cos2 θ +
1 sin4 θ, 1 − r x2
i.e., (r x cos θ + sin2 θ )2 ≤ cos2 θ +
1 sin4 θ. 1 − r x2
Notice that r x = (1 − 2 p) cos θ, so s (1 − 2 p) cos θ + sin θ = r x cos θ + sin θ ≤ 2
2
2
cos2 θ +
1 sin4 θ , 1 − r x2
which implies FN ≤ FQC . Notice that sin4 θ + cos2 θ − cos4 θ = sin4 θ + cos2 θ sin2 θ ≥ 0, which implies that q sin4 θ + cos2 θ ≥ cos2 θ, i.e., FD R2 ≥ FE . So the above EBTP optimal classical control strategies outperform the random preparation scheme. Remark 1. In order to implement the best feedback scheme, it is always advantageous to acquire as much information about the system as possible for the control of classical systems. For example, the state feedback results are easier to obtain than the output feedback for nonlinear systems [1,5–7] because the state information should be recovered from the output. Then, a typical classical control of quantum system is characterized as follows. (1) Acquire information on the system by a generalized measurement, (or positive operator-valued measure (POVM)) which yields a classical probability distribution. (2) The controller design step in which the quantum system is re-prepared based on the classical measurement outcome. However, quantum state is not observable. In controlling quantum system, the state should not be directly measured. In this case the quantum control is to measure the orientation of the quantum state imposed by the dephase noise which is different from the classical control in classical mechanical systems. 3. No universal scheme In previous sections we rigorously proved that the quantum scheme was superior to the classical scheme for the simple quantum system which was pointed out by Bra´nczyk et al. with simulation [2]. However, we will show that the
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above quantum control scheme does not outperform the classical “do nothing” scheme if the transmitted pure state is completely unknown, i.e., the effect of the dephasing noise will not be “corrected” by the proposed approach, which is consistent with the “no cloning theorem”. Consider the quantum control scheme. To quantify the performance, the average fidelity will be used for the noiseless input pure states 1 cos ϑ sin ϑe−iφ I+ , (10) |ψϑ,φ ihψϑ,φ | = sin ϑeiφ − cos ϑ 2 the fidelity is F=
1 4π
Z
F(|ψϑ,φ ihψϑ,φ |, ρ)dΩ ,
(11)
where the fidelity between a pure state |ψθ,φ ihψθ,φ | and a mixed state ρ is defined as F(|ψϑ,φ ihψϑ,φ |, ρ) ≡ T r [|ψϑ,φ ihψϑ,φ |ρ] ≡ hψϑ,φ |ρ|ψϑ,φ i. Do nothing cos ϑ − sin ϑe−iφ I+ , − sin ϑeiφ − cos ϑ 1 cos ϑ (1 − 2 p) sin ϑe−iφ E p (|ψϑ,φ ihψϑ,φ |) = I+ . (1 − 2 p) sin ϑeiφ − cos ϑ 2 1 Z |ψϑ,φ ihψϑ,φ |Z = 2
If there is no operation then the fidelity is Z 1 FE G = F(|ψϑ,φ ihψϑ,φ |, E p (|ψϑ,φ ihψϑ,φ |))dΩ 4π Z π Z 2π 1 [1 − p sin2 ϑ] sin ϑdϑdφ = 4π ϑ=0 φ=0 = 1−
2 p. 3
Quantum control Notice the weak measurement χ χ M0 = cos |+iih+i| + sin |−iih−i| 2 2 χ χ χ χ sin + cos i sin − cos 1 2 2 2 = 2χ χ χ 2 −i sin − cos χ sin + cos 2 2 2 2 χ χ M1 = sin | + iih+i| + cos | − iih−i| 2 2 χ χ χ χ sin + cos −i sin − cos 1 2 2 2 = 2χ χ χ 2 i sin − cos χ sin + cos 2 2 2 2 and the rotation about the x-axis, −iφ/2 e 0 −iφ Z /2 Zφ = e = . 0 eiφ/2
(12)
,
,
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It is easy to check that Ď
M0 E p (|ψϑ,φ ihψϑ,φ |)M0 1 + cos ϑ sin χ + (1 − 2 p) sin ϑ sin φ 1 = 4 (1 − 2 p) sin ϑ cos φ sin χ + i cos χ + i(1 − 2 p) sin ϑ sin φ
(1 − 2 p) sin ϑ cos φ sin χ − i cos χ − i(1 − 2 p) sin ϑ sin φ 1 − cos ϑ sin χ + (1 − 2 p) sin ϑ sin φ cos χ
,
Ď
M1 E p (|ψϑ,φ ihψϑ,φ |)M1 1 + cos ϑ sin χ − (1 − 2 p) sin ϑ sin φ cos χ 1 = 4 (1 − 2 p) sin ϑ cos φ sin χ − i cos χ + i(1 − 2 p) sin ϑ sin φ
(1 − 2 p) sin ϑ cos φ sin χ + i cos χ − i(1 − 2 p) sin ϑ sin φ 1 − cos ϑ sin χ − (1 − 2 p) sin ϑ sin φ cos χ
,
Ď
Z η M0 E p (|ψϑ,φ ihψϑ,φ |)M0 Z ηĎ 1 + cos ϑ sin χ + (1 − 2 p) sin ϑ sin φ 1 = 4 eiη [(1 − 2 p) sin ϑ cos φ sin χ + i cos χ + i(1 − 2 p) sin ϑ sin φ] Ď
[(1 − 2 p) sin ϑ cos φ sin χ − i cos χ − i(1 − 2 p) sin ϑ sin φ]e−iη 1 − cos ϑ sin χ + (1 − 2 p) sin ϑ sin φ cos χ
,
Ď
Z −η M1 E p (|ψϑ,φ ihψϑ,φ |)M1 Z −η 1 + cos ϑ sin χ − (1 − 2 p) sin ϑ sin φ cos χ 1 = 4 e−iη [(1 − 2 p) sin ϑ cos φ sin χ − i cos χ + i(1 − 2 p) sin ϑ sin φ] Ď
Ď
[(1 − 2 p) sin ϑ cos φ sin χ + i cos χ − i(1 − 2 p) sin ϑ sin φ]eiη 1 − cos ϑ sin χ − (1 − 2 p) sin ϑ sin φ cos χ
,
Ď
Z η M0 E p (|ψϑ,φ ihψϑ,φ |)M0 Z ηĎ + Z −η M1 E p (|ψϑ,φ ihψϑ,φ |)M1 Z −η 1 + cos ϑ sin χ 1 (1 − 2 p) sin ϑ cos φ sin χ cos η − cos χ sin η + i(1 − 2 p) sin ϑ sin φ cos η = 2 (1 − 2 p) sin ϑ cos φ sin χ cos η − cos χ sin η − i(1 − 2 p) sin ϑ sin φ cos η 1 − cos ϑ sin χ
.
Now we calculate the fidelity of the corrected states with the original states Z π Z 2π 1 [1 + cos2 θ sin χ − sin θ cos φ cos χ sin η + (1 − 2 p) sin2 θ sin2 φ cos η FQC G = 8π θ=0 φ=0 + (1 − 2 p) sin2 θ cos2 φ sin χ cos η] sin θdθdφ 1 1 1 1 = + sin χ + (1 − 2 p) cos η + (1 − 2 p) sin χ cos η. 2 6 6 6
(13)
From (12) and (13) it is known that FQC G ≤ FE G and the equality is satisfied when χ = π2 and η = 0, i.e., when there is no quantum control. That is, the proposed control approach cannot improve the fidelity without knowing the transmitted states. How to improve the performance of the quantum channel with some control approach is a very interesting problem in the future. Remark 2. If the sent pure state is unknown no one pure state has superiority. Then the performance index is chosen as (11). The above discussions imply that the prior information about a quantum process is very important. The role of prior information in quantum system is an important and interesting problem. Remark 3. Here we declare that the quantum control scheme does not outperform the classical “do nothing” scheme if the transmitted pure state is completely unknown. There is really something curious about this result. Why does it happen? Under what conditions the quantum control will be superior to the classical scheme? In fact, if the prior information about the transmitted pure states is lacking, the above discussed control schemes are not good from the above discussions. However, when sufficient information is known, for example, the known transmitted probability for every known transmitted pure state, the quantum control scheme is better. This resembles the well-known “no clone theorem”, which states that there is no universal cloner, but it does not preclude the cloner for a known state. Acknowledgements The authors sincerely thank the Reviewers for their suggestions.
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References [1] Z. Aganovi´c, Z. Gaji´c, Linear Optimal Control of Bilinear Systems, in: Lecture Notes in Control and Information Sciences, vol. 206, SpringerVerlag London Limited, 1995. [2] A.M. Bra´nczyk, P.E.M.F. Mendonca, A. Gilchrist, A.C. Doherty, S.D. Barlett, Quantum control theory of a single qubit, Physical Review A 75 (2007) 012329 or http://arxiv.org/abs/quant-ph/0608037. [3] C.W. Helstrom, Quantum Detection and Estimation Theory, in: Mathematics in Science and Engineering, vol. 123, Academic Press, New York, 1976. [4] M. Horodecki, P.W. Shor, M.B. Ruskai, Entanglement breaking channels, Reviews in Mathematical Physics 15 (6) (2003) 629–641 or http:// arxiv.org/abs/quant-ph/0302031. [5] A. Isidori, Nonlinear Control Systems: An Introduction, 3rd edition, in: Communications and Control Engineering, Springer-Verlag London Limited, 1995. [6] M. Krsti´c, I. Kanellakopoulos, P.V. Kokotovi´c, Nonlinear and Adaptive Control Design, Wiley, New York, 1995. [7] R. Marino, P. Tomei, Nonlinear Control Design: Geometric, Adaptive and Robust, Prentice Hall, London, 1995. [8] M.B. Ruskai, Qubit entanglement breaking channels, Reviews in Mathematical Physics 15 (6) (2003) 643–662 or http://arxiv.org/abs/quant-ph/ 0302032.
Performance comparison between classical and quantum control for a simple quantum systemI Zairong Xi a,∗ , Guangsheng Jin b a Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science,
Chinese Academy of Sciences, Beijing, 100080, China b Institute of Physics, Chinese Academy of Sciences, Beijing, 100080, China
Received 23 June 2007; received in revised form 8 August 2007 Available online 12 October 2007
Abstract Bra´nczyk et al. pointed out that the quantum control scheme is superior to the classical control scheme for a simple quantum system using simulation [A.M. Bra´nczyk, P.E.M.F. Mendonca, A. Gilchrist, A.C. Doherty, S.D. Barlett, Quantum control theory of a single qubit, Physical Review A 75 (2007) 012329 or arXiv e-print quant-ph/0608037]. Here we rigorously prove the result. Furthermore we will show that any quantum operation does not universally “correct” the dephasing noise. c 2007 Elsevier B.V. All rights reserved.
Keywords: Quantum control; Quantum information; Quantum channel; Quantum noise; Classical control
1. Introduction Recently, Bra´nczyk et al. considered the classical and quantum control for a simple quantum system in Ref. [2]. They considered the following operational task: a qubit prepared in one of the two non-orthogonal states |ψ1 i or |ψ2 i (with overlap hψ1 |ψ2 i = cos θ for 0 ≤ θ ≤ π2 ) is transmitted along a quantum channel with the dephasing noise, i.e., with probability p a Pauli operator Z is applied to the system, and with probability 1 − p the system is unaltered, where Z |0i = |0i, Z |1i = −|1i and {|0i, |1i} be a basis for the qubit Hilbert space. The noise is thus described by a quantum operation, i.e., a completely-positive trace-preserving (CPTP) map E p , that acts on a single-qubit density matrix ρ as E p (ρ) = p Zρ Z + (1 − p)ρ.
(1)
Suppose the noisy channel is to be fully characterized, meaning that p is known and without loss of generality in the range 0 ≤ p ≤ 0.5. Two initial states to be oriented are chosen as
I This research is supported by the National Natural Science Foundation of China (No. 60774099, No. 60221301) and by the Chinese Academy of Sciences (KJCX3-SYW-S01). ∗ Corresponding author. E-mail address: [email protected] (Z. Xi).
c 2007 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2007.10.019
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θ θ |ψ1 i = cos |+i + sin |−i, 2 2 (2) |ψ2 i = cos θ |+i − sin θ |−i, 2 2 √ where |±i = (|0i ± |1i)/ 2, in which their distinguishability is maintained under the dephasing noise by their trace distance. It is easy to see Z |+i = |−i, Z |−i = |+i. To quantify the performance of control scheme, the average fidelity is used to compare the noiseless input states |ψi i with the corrected output states ρi . Assuming an equal probability for sending either state |ψ1 i or |ψ2 i, the performance index is 1 1 F(|ψ1 i, ρ1 ) + F(|ψ2 i, ρ2 ) 2 2 1 1 = hψ1 |ρ1 |ψ1 i + hψ2 |ρ2 |ψ2 i, 2 2
FC =
(3)
where the fidelity between a pure state |ψi and a mixed state ρ is defined as F(|ψi, ρ) ≡ hψ|ρ|ψi. The fidelity F ranges from 0 to 1 and is a measure of how much the two states overlap each other (a fidelity of 0 means the states are orthogonal, whereas a fidelity of 1 means the states are identical). It has the following simple operational meaning when the input state is pure: the fidelity F(|ψi i, ρi ) is the probability that the state ρi will yield outcome |ψi i from the projective measurement {|ψi ihψi |, |ψi⊥ ihψi⊥ |}. They considered two kinds of control schemes, i.e., classical and quantum schemes, to “correct” the system without knowing which state was transmitted, i.e., undo the effect of the noise, through the use of a control scheme based on measurement and feedback. Classical control For the considered case, the optimal measurement given by Helstrom [3] is a projective measurement onto the basis {|0i, |1i} in terms of maximizing the average probability of a success, which successfully discriminates the states |ψ1 i and |ψ2 i with probability PHelstrom = 12 (1 + sin θ ). Then an entanglement-breaking trace-preserving (EBTP) scheme is used [4,8], which arises because the output system is unentangled with any other system, regardless of its input state. Through this scheme the optimal performance was proved to be q 1 1 FD R2 = + sin4 θ + cos2 θ (4) 2 2 when preparing states s s 2 1 sin θ 1 sin2 θ ± |0i + ∓ |1i |Ψ ± i = 2 2γ 2 2γ corresponds to measurement results 0, respectively, 1 where γ =
(5) p
cos2 θ + sin4 θ .
Do nothing Doing nothing to correct the states is also a classical control strategy. The average fidelity of this scheme is given by FN = 1 − p cos2 θ.
(6)
Random preparation Another control strategy would be to prepare the states randomly with probability 12 , i.e., 12 |ψ1 ihψ1 | + 12 |ψ2 ihψ2 |, which is not considered in Ref. [2]. Although trivial, this strategy is of interest for comparison with other schemes. This scheme also is not described by an EBTP map, but we will nonetheless refer to it as “classical”. The average fidelity of this scheme is given by FE =
1 (1 + cos2 θ ). 2
(7)
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Quantum control Bra´nczyk et al. showed that the above EBTP optimal classical control strategies can be outperformed by using a strategy based on quantum concepts. The quantum control scheme operates by performing a weak measurement of the system and then correcting it based on the results of the measurement. Notice α α (8) Z α = e−iα Z /2 = cos I − i sin Z , 2 2 the dephasing noise (1) can be represented as 1 1 Ď Z α ρ Z αĎ + Z −α ρ Z −α 2 2 α 2 α Zρ Z + cos2 ρ, = sin 2 2
Eα (ρ) =
(9)
which is the preferred ensemble. The weak measurement is chosen χ χ M0 = cos |+iih+i| + sin |−iih−i|, 2 2 χ χ M1 = sin |+iih+i| + cos |−iih−i|, 2 2 r √ sin4 θ where χ = arcsin 4 , r x = (1 − 2 p) cos θ , and |±ii = (|0i ± i|1i)/ 2 being the eigenstates of 2 2 2 2 (1−r x ) cos θ +(1−r x ) sin θ
Y . Based on the measurement result, the feedback control is performed on the quantum system to be a unitary rotation about the x-axis, Z ±η where tan η =
1 , (1 − 2 p) cos θ tan χ
which is a rotation about the x-axis applied to the state and returns the state back to the x z-plane. Then the maximum fidelity has been attained, s 4 1 sin θ . FQC = 1 + cos2 θ + 2 1 − r x2 In conclusion, the resulting weak measurement followed by feedback is thus described by a quantum operation (a CPTP map) C QC acting on a single qubit, given by C QC (ρ) = (Z +η M0 )ρ(Z +η M0 )Ď + (Z −η M1 )ρ(Z −η M1 )Ď . 2. Comparisons between quantum and classical schemes Bra´nczyk et al. pointed out that the quantum control scheme outperforms the classical control scheme for the simple quantum system using simulation [2]. Now we rigorously prove the result. 4θ Notice that sin ≥ sin4 θ, which implies that 1−r 2 x
FQC ≥ FD R2 . Because cos2 θ ≤ cos θ ≤
p
cos2 θ + sin4 θ we can get
FQC ≥ FD R2 ≥ FE . It is interesting to note that FD R2 ≤ 1 which implies that there exists point 0 < p ∗ ≤ 1 such that FD R2 ≤ FN when 0 ≤ p ≤ p ∗ . That is, for some small error probability, the classical scheme is inferior to doing nothing.
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The difficulty focuses on the comparison between the quantum scheme FQC and the doing nothing FN , i.e., proving s 4 1 sin θ 1 − p cos2 θ ≤ 1 + cos2 θ + . 2 1 − r x2 Now notice that 2r x cos θ sin θ = 2( 2
q
1 − r x2 cos θ )
≤ (1 − r x2 ) cos2 θ +
!
rx p
sin θ 2
1 − r x2
r x2 sin4 θ. 1 − r x2
Then r x2 cos2 θ + sin4 θ + 2r x cos θ sin2 θ ≤ cos2 θ +
1 sin4 θ, 1 − r x2
i.e., (r x cos θ + sin2 θ )2 ≤ cos2 θ +
1 sin4 θ. 1 − r x2
Notice that r x = (1 − 2 p) cos θ, so s (1 − 2 p) cos θ + sin θ = r x cos θ + sin θ ≤ 2
2
2
cos2 θ +
1 sin4 θ , 1 − r x2
which implies FN ≤ FQC . Notice that sin4 θ + cos2 θ − cos4 θ = sin4 θ + cos2 θ sin2 θ ≥ 0, which implies that q sin4 θ + cos2 θ ≥ cos2 θ, i.e., FD R2 ≥ FE . So the above EBTP optimal classical control strategies outperform the random preparation scheme. Remark 1. In order to implement the best feedback scheme, it is always advantageous to acquire as much information about the system as possible for the control of classical systems. For example, the state feedback results are easier to obtain than the output feedback for nonlinear systems [1,5–7] because the state information should be recovered from the output. Then, a typical classical control of quantum system is characterized as follows. (1) Acquire information on the system by a generalized measurement, (or positive operator-valued measure (POVM)) which yields a classical probability distribution. (2) The controller design step in which the quantum system is re-prepared based on the classical measurement outcome. However, quantum state is not observable. In controlling quantum system, the state should not be directly measured. In this case the quantum control is to measure the orientation of the quantum state imposed by the dephase noise which is different from the classical control in classical mechanical systems. 3. No universal scheme In previous sections we rigorously proved that the quantum scheme was superior to the classical scheme for the simple quantum system which was pointed out by Bra´nczyk et al. with simulation [2]. However, we will show that the
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above quantum control scheme does not outperform the classical “do nothing” scheme if the transmitted pure state is completely unknown, i.e., the effect of the dephasing noise will not be “corrected” by the proposed approach, which is consistent with the “no cloning theorem”. Consider the quantum control scheme. To quantify the performance, the average fidelity will be used for the noiseless input pure states 1 cos ϑ sin ϑe−iφ I+ , (10) |ψϑ,φ ihψϑ,φ | = sin ϑeiφ − cos ϑ 2 the fidelity is F=
1 4π
Z
F(|ψϑ,φ ihψϑ,φ |, ρ)dΩ ,
(11)
where the fidelity between a pure state |ψθ,φ ihψθ,φ | and a mixed state ρ is defined as F(|ψϑ,φ ihψϑ,φ |, ρ) ≡ T r [|ψϑ,φ ihψϑ,φ |ρ] ≡ hψϑ,φ |ρ|ψϑ,φ i. Do nothing cos ϑ − sin ϑe−iφ I+ , − sin ϑeiφ − cos ϑ 1 cos ϑ (1 − 2 p) sin ϑe−iφ E p (|ψϑ,φ ihψϑ,φ |) = I+ . (1 − 2 p) sin ϑeiφ − cos ϑ 2 1 Z |ψϑ,φ ihψϑ,φ |Z = 2
If there is no operation then the fidelity is Z 1 FE G = F(|ψϑ,φ ihψϑ,φ |, E p (|ψϑ,φ ihψϑ,φ |))dΩ 4π Z π Z 2π 1 [1 − p sin2 ϑ] sin ϑdϑdφ = 4π ϑ=0 φ=0 = 1−
2 p. 3
Quantum control Notice the weak measurement χ χ M0 = cos |+iih+i| + sin |−iih−i| 2 2 χ χ χ χ sin + cos i sin − cos 1 2 2 2 = 2χ χ χ 2 −i sin − cos χ sin + cos 2 2 2 2 χ χ M1 = sin | + iih+i| + cos | − iih−i| 2 2 χ χ χ χ sin + cos −i sin − cos 1 2 2 2 = 2χ χ χ 2 i sin − cos χ sin + cos 2 2 2 2 and the rotation about the x-axis, −iφ/2 e 0 −iφ Z /2 Zφ = e = . 0 eiφ/2
(12)
,
,
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It is easy to check that Ď
M0 E p (|ψϑ,φ ihψϑ,φ |)M0 1 + cos ϑ sin χ + (1 − 2 p) sin ϑ sin φ 1 = 4 (1 − 2 p) sin ϑ cos φ sin χ + i cos χ + i(1 − 2 p) sin ϑ sin φ
(1 − 2 p) sin ϑ cos φ sin χ − i cos χ − i(1 − 2 p) sin ϑ sin φ 1 − cos ϑ sin χ + (1 − 2 p) sin ϑ sin φ cos χ
,
Ď
M1 E p (|ψϑ,φ ihψϑ,φ |)M1 1 + cos ϑ sin χ − (1 − 2 p) sin ϑ sin φ cos χ 1 = 4 (1 − 2 p) sin ϑ cos φ sin χ − i cos χ + i(1 − 2 p) sin ϑ sin φ
(1 − 2 p) sin ϑ cos φ sin χ + i cos χ − i(1 − 2 p) sin ϑ sin φ 1 − cos ϑ sin χ − (1 − 2 p) sin ϑ sin φ cos χ
,
Ď
Z η M0 E p (|ψϑ,φ ihψϑ,φ |)M0 Z ηĎ 1 + cos ϑ sin χ + (1 − 2 p) sin ϑ sin φ 1 = 4 eiη [(1 − 2 p) sin ϑ cos φ sin χ + i cos χ + i(1 − 2 p) sin ϑ sin φ] Ď
[(1 − 2 p) sin ϑ cos φ sin χ − i cos χ − i(1 − 2 p) sin ϑ sin φ]e−iη 1 − cos ϑ sin χ + (1 − 2 p) sin ϑ sin φ cos χ
,
Ď
Z −η M1 E p (|ψϑ,φ ihψϑ,φ |)M1 Z −η 1 + cos ϑ sin χ − (1 − 2 p) sin ϑ sin φ cos χ 1 = 4 e−iη [(1 − 2 p) sin ϑ cos φ sin χ − i cos χ + i(1 − 2 p) sin ϑ sin φ] Ď
Ď
[(1 − 2 p) sin ϑ cos φ sin χ + i cos χ − i(1 − 2 p) sin ϑ sin φ]eiη 1 − cos ϑ sin χ − (1 − 2 p) sin ϑ sin φ cos χ
,
Ď
Z η M0 E p (|ψϑ,φ ihψϑ,φ |)M0 Z ηĎ + Z −η M1 E p (|ψϑ,φ ihψϑ,φ |)M1 Z −η 1 + cos ϑ sin χ 1 (1 − 2 p) sin ϑ cos φ sin χ cos η − cos χ sin η + i(1 − 2 p) sin ϑ sin φ cos η = 2 (1 − 2 p) sin ϑ cos φ sin χ cos η − cos χ sin η − i(1 − 2 p) sin ϑ sin φ cos η 1 − cos ϑ sin χ
.
Now we calculate the fidelity of the corrected states with the original states Z π Z 2π 1 [1 + cos2 θ sin χ − sin θ cos φ cos χ sin η + (1 − 2 p) sin2 θ sin2 φ cos η FQC G = 8π θ=0 φ=0 + (1 − 2 p) sin2 θ cos2 φ sin χ cos η] sin θdθdφ 1 1 1 1 = + sin χ + (1 − 2 p) cos η + (1 − 2 p) sin χ cos η. 2 6 6 6
(13)
From (12) and (13) it is known that FQC G ≤ FE G and the equality is satisfied when χ = π2 and η = 0, i.e., when there is no quantum control. That is, the proposed control approach cannot improve the fidelity without knowing the transmitted states. How to improve the performance of the quantum channel with some control approach is a very interesting problem in the future. Remark 2. If the sent pure state is unknown no one pure state has superiority. Then the performance index is chosen as (11). The above discussions imply that the prior information about a quantum process is very important. The role of prior information in quantum system is an important and interesting problem. Remark 3. Here we declare that the quantum control scheme does not outperform the classical “do nothing” scheme if the transmitted pure state is completely unknown. There is really something curious about this result. Why does it happen? Under what conditions the quantum control will be superior to the classical scheme? In fact, if the prior information about the transmitted pure states is lacking, the above discussed control schemes are not good from the above discussions. However, when sufficient information is known, for example, the known transmitted probability for every known transmitted pure state, the quantum control scheme is better. This resembles the well-known “no clone theorem”, which states that there is no universal cloner, but it does not preclude the cloner for a known state. Acknowledgements The authors sincerely thank the Reviewers for their suggestions.
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