2. Institute of Intelligent Machines, Chinese Academy of Sciences, 230031 Hefei, China;
3. Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA
During the last three decades,people have made
considerable achievements on the research of quantum information
processing
[1, 2, 3, 4, 5, 6]. Practical quantum information processing still
confronts with some important technical difficulties,such as the
laser noise,environmental decoherence effects and quantum
measurement. Quantum measurement is generally considered as a
deleterious factor toward the manipulation of quantum
dynamics[7, 8, 9, 10],which induces the disappearance of coherence of
the system's state irreversibly. However,recent studies have shown
that both the outcome and back-action from quantum measurements
could be used to control the quantum systems[11, 12, 13, 14]. In
standard closed-loop optimal quantum control,the outcomes of the
measurements are used by a learning algorithm to optimize the
coherent control field
Projective measurement is the most generally used quantum
observations. The manipulation of quantum dynamics driven by the
quantum projective measurements is explored theoretically
This paper explores some deep characteristics of the manipulation between eigenstates of a two-level system by optimal measurements. We consider a two-level system measured by a sequence of non-selective projection operators. Measurements are designed and performed instantaneously with a fixed frequency,so that the system in each interval can be seen as a closed quantum system. The measurement unavoidably affects the quantum state,so as to drive the system to the target state. We investigate three cases of the manipulations under the optimal measurements. The first is the optimal measurements itself,which is the ideal case. Then we further explore the effect of free evolution of the system to the manipulation. In the end,the performances of manipulation of the optimal measurements in the case with the external control fields are studied in order to compensate for the effect of the free evolution. Numerical simulations of the three cases are conducted to verify the enhanced effectiveness.
The rest of the paper is organized as follows: In Section Ⅱ,the concepts of optimal measurement-based control are presented. The cases with the free evolution and the external control fields are analyzed and discussed. The analytical solutions are derived for three cases. In Section Ⅲ,numerical simulations for three cases are conducted. The characteristics are discussed,and the optimal parameters of manipulations are obtained. A brief conclusion is in Section IV.
II. QUANTUM CONTROL BY OPTIMAL MEASUREMENTS A. Optimal Control with Projective MeasurementsConsider a two-level quantum system,with ρ being the density operator of the system. Suppose an instantaneous projective measurement be performed on the state,and the state density operator after the measurement will be ρ′:
ρ′=PρP+(I−P)ρ(I−P)=ρ−[P,[P,ρ]], | (1) |
where I is the identity operator and P is the projection operator. The instantaneous measurement is characterized by the set of projection operators {P,I−P} .
One can see from (1) that an important characteristic of the quantum measurements is to generally change the quantum state. However,on the other hand,quantum measurements can also be employed to actively control the quantum system dynamics by means of such a characteristic.
Our research objective is to manipulate the state in a two-level system by instantaneous projective measurements from the initial state ρ0=|0⟩⟨0| to the target state ρf=|1⟩⟨1| . The projection operators are specified by the operators
Pk=|ψk⟩⟨ψk|, | (2) |
|ψk⟩=cosαk2|0⟩+eiθksinαk2|1⟩, | (3) |
where |ψk⟩ is the basis vector of Pk ( k=1,...,m ), αk and θk are parameters of Pk and limited to the ranges: −π2≤αk2≤π2 , 0≤θk<π . Let ρk be the state after the k-th measurement and
ρk=ρk−1−[Pk,[Pk,ρk−1]], | (4) |
where ρk−1 is the state after the (k−1) -th measurement and ρm is the final state. Equation (4) is an iterative equation of the transfer. Given a set of projection operators {P1,...,Pm} ,one can easily calculate ρm. The fidelity between the initial state ρm and the target state ρf is defined as
F(ρm,ρf)=[tr(√√ρmρf√ρm)]2. | (5) |
The analytical solution of
F(ρm,ρf)
reaches maximum
value when the manipulation from
ρm
to
ρm
is
optimized
θ1=θ2=⋅⋅⋅=θk=0, | (6) |
θ1=θ2=⋅⋅⋅=θk=0, | (7) |
where θk and αk ( k=1,...,m ) are the parameters of the projection operators. Substituting (6)-(7) into (3) and (2),one can obtain the corresponding optimal projection operators,and can easily derive the corresponding fidelity as
Fm=12[1+(cosπm+1)m+1], | (8) |
where Fm is the optimal value of F(ρm,ρf) . Equation (8) is the expression for the the optimal fidelity Fm with the corresponding measurement time m .
It is easy to see from (8) that Fm increases monotonically with m increasing. The minimum value of Fm is 0.5 when m=1 . The maximum value of Fm approaches 100% when m tends to infinity. Therefore,the value of Fm is in the range of [0.5,1). In other words,when measuring infinite times to a quantum system,we can realize 100% transferring from an initial state to the target state,which is the so-called ``quantum anti-Zeno effect''.
B.Optimal Measurement-based Control with Free Evolution of the SystemIn fact,there is always free evolution in a quantum system. So a more realistic measurement-based control should consider the free evolution of the system itself. In this section we analyze the effect of free evolution to the optimal measurement-based control.
Suppose the instantaneous projective measurements change the state instantaneously and do not affect the characteristics of the system. The time interval of two measurements is fixed as Δt . The system dynamics with free evolution is
˙ρk(t)=−i[H0,ρk(t)], | (9) |
where, ρk(t) is the state in the interval between the k -th and (k+1) -th measurement, k=1,...,m , t∈[0,Δt] . H0 is the free Hamiltonian:
H0=σz=|100−1|, | (10) |
where σz is Pauli operator. The solution of (9) is
ρk(t)=e−iH0tρk(0)eiH0t. | (11) |
Since the quantum state changes with the free evolution,the iterative equation of the transfer is different from (4). Substituting (10)-(11) into (4),one can obtain the iterative equation with the free evolution:
ρk(0)=ρk−1(Δt)−[Pk,[Pk,ρk−1(Δt)]], | (12) |
in which ρk−1(Δt) is defined in (11).
When the optimal measurement control is performed in the case with free evolution,the optimal fidelity between ρm and ρf becomes:
Fm=12[1+Cm⋅m∏k=1(Cm−2sinαk⋅sinαk−1⋅sin2(Δt))]=12[1+Cm⋅m∏k=1(Cm−2Smk⋅Sm(k−1)⋅sin2(Δt))], | (13) |
where Cm , Smk and Sm(k−1) represent cos(πm+1) , sin(kπm+1) and sin((k−1)πm+1) ,respectively. From (13),one can see that Fm depends on m and Δt in the case with free evolution.
According to the symmetry of the trigonometric functions,it is easy to verify that when Δt>0,for any m there is the following inequality:
Cm⋅m∏k=1(Cm−2Smk⋅Sm(k−1)⋅sin2(Δt))<(Cm)m+1. | (14) |
It can be easily derived with (14) that the fidelity Fm in (13) is smaller than that in (8). In summary,the free evolution of the system may decrease the effectiveness of the optimal fidelity,which gives a disturbance of the optimal measurement.
C.Optimal Measurement-based Control with External Control FieldsIn this section we will analyze the manipulation effect of the optimal measurement-based control by introducing the external control fields.
The dynamics of a two-level quantum system with free evolution and the external control fields can be expressed as:
˙ρk(t)=−i[H,ρk(t)], | (15) |
where H=H0+uxHx+uyHy+uzHz is the system Hamiltonian. H0 is the free Hamiltonian, Hx , Hy and Hz are the external control Hamiltonian with ux , uy and uz representing the corresponding control fields. The Hamiltonians are
H0=σz,Hx=σx,Hy=σy,Hz=σz, | (16) |
where σx , σy and σz are Pauli operators.
We need to design the shape of the control fields ux , uy and / or uz in order to compensate for the effect of free evolution and increase the fidelity of optimal measurement control.
The solution of (15) is
ρk(t)=e−iHtρk(0)eiHt, | (17) |
in which ρk(0) is defined by the iteration equation of optimal measurement as:
ρk(0)=ρk−1(Δt)−[Pk,[Pk,ρk−1(Δt)]]. | (18) |
It should be pointed out that the state ρk−1(Δt) in (18) is calculated from (17),which is not the same as the state in (12).
Ⅲ. NUMERICAL SIMULATION AND THE CHARACTERISTIC ANALYSISIn this section,our tasks are to perform the simulation experiments,analyze the experimental results of manipulations, and determine the appropriate control fields designed. In the experiments,the initial and target states are both eigenstates, they are ρ0=|0⟩⟨0| and ρf=|1⟩⟨1| ,respectively. Suppose the manipulation is achieved by m times instantaneous projective measurements,and the projection operators {P1,P2,...,Pm} follow the optimal condition (2) and (3). The performance index of the fidelity is fixed as 95% .
A.Manipulation Between Eigenstates with Optimal MeasurementsThe first experiment is the manipulation between eigenstates with optimal measurements. The experimental results of optimal fidelities with m being 1,5,10,30,46,100,500 and 1000,respectively, are shown in Table I,from which one can see that:
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Table I OPTIMAL FIDELITY Fm OF THE STATE TRANSFER WITH DIFFERENT NUMBER OF MEASUREMENTS m |
1) When m ,the optimal fidelity Fm equals 50% . Because for one time's measurement,the state is just projected onto the maximum mixed state (Bloch vector →O=(0,0,0) ).
2) As m increases,the optimal fidelity reaches the given performance index of 95% for the first time when m=46 and F46=95.01% . So m=46 is the minimum measurement number needed to achieve the desired performance index.
In order to do better analyses,the transfer trajectory of the system state is drawn on the Bloch Sphere as shown in Fig. 1,in which Fig. 1(a) is the transfer trajectory of the system state with m=46 ,and Fig. 1(b) is the abstract graph of the trajectory on the x - z plane.
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Fig. 1 Transfer trajectory of the system state by projective measurements. |
The vector of the initial state ρ0 and the target state ρf are →s0=(0,0,1) and →sf=(0,0,−1) ,respectively. The red circle in Fig. 1 (a) represents the original state ρ0 ,and the red cross represents the final state ρm . One can see from Fig. 1 (a) that the trajectory is an approximate semi-circular,which is from the Bloch sphere's one pole to the other pole. The coordinate y of the states in Fig. 1 (a) is 0,so it is obvious that the vectors of the states can be plotted on x - z plane as shown in Fig. 1 (b),where →Pi represents the state ρi . The angle between →Pi and →Pi+1 is φ=π/47 ,which is defined in (7). Fig. 1 (b) indicates that each measurement rotates the preceding vector by the angle φ=π/47 clockwise and shortens its length by the factor cosφ . From Fig. 1 (b) one can easily derive the formula of optimal fidelity (7) by geometric analysis,which supplies an easier derivation method.
B.Manipulation Between Eigenstates with Free Evolution of the SystemIn this section,we will do the experiments of the manipulation between eigenstates with free evolution of the system. Let the time of measurement intervals be Δt . To analyze the effect of Δt on the performance,we investigate the Fm as the function of m with different values of Δt . The optimal fidelity Fm is calculated according to (13),and the experimental results are shown in Fig. 2,in which Δt is set as five different values: Δt=0 , Δt=0.015 , Δt=0.023 , Δt=0.05 and Δt=0.1 .
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Fig. 2 The function of the optimal fidelity Fm with different ∆ |
From Fig. 2 one can see that:
1) When Δt=0 ,the transfer is equivalent to the experiment without free evolution. The function Fm follows (8) and rises monotonically up to 1;
2) When Δt>0 ,the functions Fm are convex,and each function has a peak value F(o)m ,which is the maximum fidelity.
In order to make the optimal fidelity Fm with different Δt meet the given performance,it is more meaningful to investigate the maximum fidelity F(o)m . The experimental results of the maximum fidelities F(o)m with different values of Δt and the corresponding m are shown in Table Ⅱ,from which one can see that: 1) When Δt=0.023,the maximum fidelity is F(o)96=95.14% with m=96; 2) When Δt=0.024,the maximum fidelity is F(o)91=94.94% with m=91. It can be concluded that on one hand, with the increment of Δt,the maximum value of Fm decreases,as well as the corresponding measurement number m,on the other hand,when Δt≥0.024,the fidelity of optimal measurement control cannot reach 95% no matter how large the measurement number m is. So we choose Δt=0.023 as the optimal time of intervals,and in the later experiments of the paper Δt is fixed as 0.023.
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Table Ⅱ THE MAXIMUM FIDELITY F(o)m OF DIFFERENT Δt AND THE CORRESPONDING m |
In this section,we will analyze the manipulation between eigenstates by optimal measurement control with the external control fields. The external control fields can be seen as a compensation for the effect of the free evolution of the system as well as a enhancement of control effect. In Section Ⅱ-C,we design the external control fields with three components: ux, uy and/or uz. In this section,we investigate the effectiveness of different control fields in two cases,one is to use only one control field and another is to use two control fields,and compare the performances in these two cases.
1) Case 1. Only one external control field applied
For simplicity,we just choose a constant field in this case. Because the free Hamiltonian H0 is σz,we cannot only use uz on the system. We thought to use one of control fields ux or uy. As the analysis in Fig. 1 (b),the state is projected onto the x-z plane after each measurement, but the experimental results we did indicated that there is no evident compensation effectiveness by using control ux. Therefore,a constant uy control field is used to drive the state rotate on the x-z plane in the experiments. At the same time,the system is also affected by the measurements and free evolution.
The initial and target states in the experiments are the same as in Section Ⅱ-B,and the measurement interval time is set as Δt=0.023. The control Hamiltonian is H=σz+uyσy,and the system dynamics is:
ρk(t)=e−i(σ0+uyσy)Δtρk(0)ei(σ0+uyσy)Δt. | (19) |
Using (18) and (19) alternately with the given initial state ρ0 one can obtain the final state ρm.
Fig. 3 shows the curves of Fm of states manipulated by optimal measurements and an external control field in a two-level system, with the six different values of the control fields uy: uy=0,uy=0.1,uy=0.5,uy=1,uy=7.5 and uy=10,respectively,in which the solid lines represent the transitions of fidelity Fm which can reach the performance given, and the dotted line is for the opposite. We cut up the solid lines at Fm=0.95,and the cut-off points’ horizontal coordinates are the minimum measurement numbers of meeting the performance. As can be seen from Fig. 3 that the minimum measurement number decreases with the increment of uy,but when uy takes an oversized value,like the dotted line with uy=10,the maximum optimal fidelity F(o)m drops below 95%,and the minimum measurement number does not exist. We investigate the minimum measurement numbers for the performance given,the value of the minimum measurement number m and corresponding uy are as shown in Table Ⅲ. From Fig. 3 and Table Ⅲ,it can be seen that when uy≤7.5,the external control field uy effectively enhances the manipulation by optimal measurement-based control,but when uy>7.5,the experiment cannot meet the performance index. Based on the results and analysis,we select uy=7.5 as the optimal value of the control field.
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Fig. 3 The curve of the fidelity Fm with different control field uy |
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Table Ⅲ THE MINIMUM MEASUREMENT NUMBER OF MEETING THE PERFORMANCE |
To analyze the performances of three numerical experiments,we extract the optimal results of each experiment. The optimal results of minimum measurement number are as shown in Table IV, from which one can see that the minimum measurement number is increased from 46 to 75 as an effect of the free evolution,and is greatly reduced from 75 to 10 with the action of an appropriate external control field with uy=7.5. The results verify that the optimal quantum measurement control can manipulate the quantum states,and can become more effective by the aid of external control field.
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Table IV THE MINIMUM MEASUREMENT NUMBER OF MEETING THE PERFORMANCE |
2) Case 2. Two control fields ux plus uz applied
In this case,we consider a composite control field with two directions. In fact,there are three possible combinations: ux plus uy,ux plus uz,or uy plus uz. The experimental results indicated that the control field combination of ux plus uz has the best effectiveness on the population transfer in all three combinations. Therefore,the two control fields of ux plus uz are used here to compensate and promote the manipulation.
The initial and target states in the experiments are the same as in
Section Ⅲ-B with measurement interval time being Δt=0.023. Control Hamiltonian is H=σz+uxσx+uzσz. In this case,the control field uz is in
fact used to compensate the effect of the free evolution and is
fixed as uz=−1,which is equivalent to do a transformation and
obtain a system model in interaction picture
˜Pi=e−iuxσxmΔtPieiuxσxmΔt, | (20) |
where Pi is the projection operator determined by the optimal condition (2) and (3).
In this case the iteration equation of optimal measurement becomes:
ρk(0)=ρk−1(Δt)−[˜Pk,[˜Pk,ρk−1(Δt)]]. | (21) |
The system dynamics turns out to be:
ρk(t)=e−iHΔtρk(0)eiHΔt. | (22) |
The final state ρm can be obtained by (20) and (21).
The curves of Fm with different values of ux are shown in Fig. 4,in which ux=0 with solid line,ux=0.218 with the short dotted blue line,ux=0.435 with dashed-dot green line,and ux=0.870 with the short dotted red line,respectively. As can be seen from Fig. 4 that the behavior of the optimal fidelity Fm is fluctuant and periodic,whose period depends on the value of ux. The period decreases as the ux increases. The period which corresponds to the measurement number N equals 40 when ux=0.870,and the peak value of Fm in the first period cannot reach performance of 95%. However,the peak Fm in the second period (m=79) can reach 96.84%. Especially,the peak value of Fm increases as the number of period increases. Experiments show that the peak value of Fm turns to be 99.68% with m approaching 1000,which is much different from the results in the case with only one external control field used in Case 1 of Section Ⅲ-C,where the Fm is a convex curve without periodicity. The simulation result indicates that the periodic peak value of Fm can infinitely approach to 1 as long as the number of measurement is large enough.
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Fig. 4 The curve of optimal fidelity Fm with different control field ux |
By comparing and analyzing the results in Case 1 and Case 2,we
can conclude that the control field of uy is effective for the
reduction of measurement number to meet the performance. While the
control field of ux plus uz can reach the optimal fidelity
as high performance as possible,which is often used in the actual
experiments,for example in the manipulation of a double
quantum-dot charge qubit
This paper investigated the manipulation between eigenstates of a two-level quantum system by optimal measurements. The instantaneous projective measurements were used to drive the state from one eigenstate to another. The three cases manipulations were studied,and the analytical solutions were derived. Three kinds of numerical simulations of the manipulation were performed,in which characteristics of the optimal measurement-based control were analyzed,and the optimal parameters in different cases were obtained. The simulation experimental results verify the validity of the optimal measurement-based control,and indicate that the free evolution may affect the effectiveness of the manipulation, which can be compensated by the external control fields. Of course,the actual systems need to consider many other complex factors. The optimal measurement control and the external control fields proposed in this paper can be implemented by the open-loop control system. We believe that one can obtain better quantum state manipulation performance under advanced control methods and technologies.
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