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The impact of imperfect measurements of weak values on state tomography
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Physics Letters A ••• (••••) •••–•••
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The impact of imperfect measurements of weak values on state tomography Xi Chen a , Hong-Yi Dai b,c , Ming Zhang a,∗ a b c
College of Mechatronics Engineering and Automation, National University of Defense Technology, Changsha, 410073, China College of Science, National University of Defense Technology, Changsha, 410073, China Interdisciplinary Center for Quantum Information, National University of Defense Technology, Changsha, 410073, China
a r t i c l e
i n f o
Article history: Received 24 March 2017 Received in revised form 14 May 2017 Accepted 3 August 2017 Available online xxxx Communicated by A. Eisfeld Keywords: Basis tilting error Weak value measurement Quantum state tomography Information encoding
a b s t r a c t We investigated the impact of imperfect measurements of weak values on state tomography due to the error of detector basis tilting, as this impact is seldom discussed before. For a given error bound, the negative influence is evaluated with two tomography performance indices—fidelity and trace distance. The simulation based on a reported experiment suggests that the tilting error would lead to a worse tomography performance. However, when the target state is in the pattern of near 50% vector elements carrying phase π , the tomography performance is relatively insensitive to such error. These results provide a choice of a suitable state type for information encoding in the task of quantum information transferring. © 2017 Elsevier B.V. All rights reserved.
1. Introduction State tomography is the procedure of inferring the identity of the state-generating source with collected experiment data [1]. Conventionally, projective measurement is adopted and this process is indirect in that it involves a time consuming postprocessing step where the density matrix of the state must be globally reconstructed through a numerical search over the alternatives consistent with the measured projective slices. However, in contrast to the projective measurement, the state can be reconstructed directly by using the results of weak value measurement [2]. The weak value measurement was first proposed by Aharonov et al. [3] in 1988, and the established procedure is shown in Fig. 1. After its emergence, the weak value measurement has been extensively investigated [4], such as exploring the physical meaning of the weak value [5–11], considering new effects over the original procedure [12–15], realizing the procedure in different physical objects or manners [16–18], etc. Meanwhile its application in quantum information is also intriguing, such as state tomography [2,19–24], metrology [25–27], and as a window into nonclassical features of quantum mechanics [28–30], etc. For instance, Bin Ho and Imoto [5] generally expressed the relationship between the weak value and the spin-operator modular
*
Corresponding author. E-mail addresses: [email protected] (X. Chen), [email protected] (M. Zhang). http://dx.doi.org/10.1016/j.physleta.2017.08.011 0375-9601/© 2017 Elsevier B.V. All rights reserved.
value. Sokolovski showed the meaning of “anomalous weak values” in quantum and classical theories [9] and related this meaning to some paradoxes [28,29]. Ban [13] considered the effect of an external field applied after a measurement and analyzed its influence on the system response through postselection. Lee and Tsutsui [25] presented an inequality of uncertainty relations in metrology and analyzed its feature with the help of weak values. In addition, the state tomography by using weak value measurement was first experimentally realized by Lundeen et al. [2], and their work can reconstruct discrete-variable quantum pure states. Later, Fischbach and Freyberger [19] proposed to use the homodyne detection as an interferometer for the weak value measurement, thus continuous-variable quantum optical states were reconstructed. S.J. Wu [20] proposed an efficient weak value tomography scheme with no data discarded, and a mixed state was reconstructed as a density matrix. Malik et al. [21] performed the experiment of directly measuring a 27-dimensional orbitalangular-momentum state vector. Bamber and Lundeen [22] implemented the mixed state reconstruction into a Dirac distribution, as the Dirac distribution is the Fourier transformation of a density matrix. Mirhosseini et al. [23] experimentally measured a 19 200-dimensional state by exploiting the compressive measurement. Moreover, Maccone and Rusconi [24] compared the state tomography by using weak value measurement with using projective measurement through simulations, and found out that the former was much less precise than the latter, because the former had an inherent bias, which would introduce an unavoidable error in the reconstruction.
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Theoretically, the state |φ is expressed as a continuous wave function φ (x, y ) at planar positions (x, y ), however due to the sampling number limitation on measurement devices, it only can
Fig. 1. The procedure of weak value measurement.
T
be represented as a discrete column vector φ1 , · · · , φ j , · · · , φ N with dimension N, and all planar areas for the laser energy distribution are indexed with j = 1, · · · , N. In addition, there is φ j = j |φ, when considering | j as a column vector with 1 in the j-th row and 0 in the remaining rows, i.e. | j = [0, · · · , 1, · · · , 0]T . Step 2, performing weak coupling at a spatial light modula tor [23]. The unitary evolution of the joint system state is j = M j ⊗ R y (2θ) + I t − M j ⊗ I p |φ | V , where M j ⊗ R y (2θ) represents the weak coupling operation at the j-th area. Here M j = | j j | characterizes the action area and R y (2θ) = exp −iθ σ y performs a polarization rotation with angle θ , as σ y is a Pauli operator [31], moreover the angle θ describes the coupling strength at the action area. When 0 < |θ| 0.2 rad, the coupling can be considered weak, then the first order approximation can be applied as R y (2θ) = exp −iθ σ y = I p − iθ σ y , thus the joint state becomes | j = |φ | V − θ| j j |φ| H . Step 3, performing postselection on the target at a pinhole [23]. Here the postselected is a zero momentum state expressed as | p = √1 [1, 1, · · · , 1]T in the position representation. After the N
target postselection and under the first order approximation, the p |
Fig. 2. The experiment setup for measuring the weak value connected with the planar laser energy distribution. P: polarizer, L: lens, SLM: spatial light modular, WP: wave plate, PBS: polarized beam splitter, D1,D2: detectors, hWP: half-wave plate, qWP: quarter-wave plate.
However, previous papers about state tomography via weak value measurement are merely discussed under the error-free condition. As we know, practical operations are often imperfect and the error of detector basis tilting may exist in the last step of the procedure in Fig. 1. In order to understand the impact of imperfect measurements of weak values due to this kind of error, we examine the influence on the state tomography performance in this paper. Besides that, the state pattern insensitive to such error is also sought for, as these states are good candidates for information encoding in the task of quantum information transferring. The structure of the rest of this paper are as follows. In section 2, we review the process of state tomography via weak value measurement. In section 3, the mathematical description of the error of detector basis tilting is given and the influence on tomography performance is analyzed theoretically. In section 4, the influence due to such error is simulated based on the first reported experiment on directly measuring a quantum state [2]. In section 5, we seek for the state pattern insensitive to such error. The conclusion is summarized in section 6.
final probe state becomes |s j = p |φj = | V − θ j | H , where j = p| pj|φj|φ is the weak value connected with the laser energy distribution over planar positions in this scheme. Step 4, measuring the probe projectively and recording the readouts at two detectors behind either a half-wave plate or a quater-wave plate. The readouts corresponding to different setups in Fig. 2 are rh ( j ) = s j | D D |s j − s j | A A |s j = −2θ Re j and rq ( j ) = s j | R R |s j − s j | L L |s j = 2θ Im j , and the measurement bases are | D = √1 (| H + | V ), | A = √1 (| H − | V ), 2
|R =
√1
2
(| H + i| V ), | L =
√1
2
2
(| H − i| V ).
At last, the weak value in this scheme is obtained by rh ( j ) −2θ
rq ( j ) . 2θ
p| j p |φ j |φ
=
j =
+i· Weak values j at all j = 1, · · · , N areas can be used to reconstruct the target state |φ, since there exists j = √1
√1 N
NN
j =1
φj
· φ j = κ φ j , where κ =
N j =1 φ j
−1
. Af-
ter normalization the column vector | consisting of j changes into N ( ) | , where N ( ) ∈ C is the corresponding transform coefficient and in this particular case N ( ) = κ −1 , meaning that N ( ) | equals the target state |φ. The performance of state tomography can be evaluated with two equivalent indices—fidelity and trace distance [31]. When only considering two quantum pure states N ( ) | and |φ, the fidelity is defined as
2
2
F = N ( ) · φ| ,
2. The process of state tomography via weak value measurement
and the trace distance can be computed based on the fidelity as
Fig. 2 shows the experiment setup for measuring the weak value connected with the planar laser energy distribution [23]. The target is the laser attribute of the energy distribution over planar positions, and the probe is the laser polarization attribute. According to the general procedure in Fig. 1, the actual steps in this scheme are as follows. the initial probe polarization state |φ | V = Step1, preparing I t ⊗ P |φ |s , where |φ is the target state and |s is a nonpolarized probe state with two levels, and after a polarizer, a vertically polarized probe state | V is prepared. The state | V = [0, 1]T is one of two basis states for the probe, and the other is the horizontal polarization | H = [1, 0]T .
D=
1 − F 2.
(1)
These two indices F and D are both valued within [0, 1], moreover when D is closer to 0, F is reversely closer to 1. 3. Theoretical analysis of the error and its influence It is possible that the detector basis states | D , | A , | R and | L are disturbed by the error of tilting in practical experiments. Based on the parameterized form of a qubit, this kind of error changes the conventional probe bases | H and | V into
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ε
ε
2
2
| H e = cos | H + eiϕ sin | V ,
ε
ε
2
2
the trace distance D UB can be used. Here we derive F LB in detail, then D UB is obtained through Eq. (1). As we know, the lower fidelity bound F LB necessarily satisfies
| V e = cos | V + eiϕ sin | H , where 0 ε ε u π and 0 ϕ < 2π , here ε u is the upper bound to ε , thus the measurement bases in operation become
1 ε ε
| D e = √ (| H e + | V e ) = √ cos + eiϕ sin |H 2 2 2 2 1 ε ε
| V , + √ cos + eiϕ sin 2 2 2 1 1 ε ε
| A e = √ (| H e − | V e ) = √ cos − eiϕ sin |H 2 2 2 2 1 ε ε
| V , − √ cos − eiϕ sin 2 2 2 1 1 ε ε
|H | R e = √ (| H e + i| V e ) = √ cos + ieiϕ sin 2 2 2 2 1 ε ε
| V , + √ i cos + eiϕ sin 2 2 2 1 1 ε ε
| L e = √ (| H e − i| V e ) = √ cos − ieiϕ sin |H 2 2 2 2 1 ε ε
| V . − √ i cos − eiϕ sin 2 2 2 1
(2)
φ j +i φ j and j N j =1 φ j +i φ j
⎛
=⎝
ε, ϕ , θ, φ j
N
= 1, · · · , N. After normaliza-
(3)
⎞ ⎛ N ×⎝ σ2 φ j − i φ j ⎠ , j =1
where
sin ε cos ϕ
2θ sin ε cos ϕ
σ2 = i j cos ε + j −
2θ
+ +
(4)
ε , ϕ = arg min u F εu , 0, θ, φ j , F εu , ϕ m , θ, φ j ,
0ε ε 0ϕ <2π
εm , 0, θ, φ j , F εm , ϕ m , θ, φ j
.
To demonstrate the aforesaid theoretical analysis vividly we show below the simulation results in Table 1 on a randomly generated 2-dimensional target state
0.2573 − 0.6788i . −0.1103 + 0.6788i
(5)
The coupling strength in simulation is set as θ = 0.1 rad for applying the first order approximation regarding θ . Table 1 shows that the larger the tilting error bound ε u is, the sharper the difference is between the original target state |φ1 and the reconstructed state |e , thus the smaller the lower bound of the fidelity F LB is and the larger the upper bound of the trace distance D UB is. In other words, the tilting error bound ε u needs to be small if a faithful state tomography result is required. 4. Simulation on a reported experiment
j =1
σ1 = i j cos ε − j +
where j = 1, · · · tain θ and |φ. Since ε ∈ [0, ε u ] and ϕ ∈ [0, 2π ), it is reasonable that ε or ϕ is at either its range end point or the F ε, ϕ , θ, φ j extremum point within the range. Moreover since ε = 0 means no tilting, the making-sense end point for range [0, ε u ] is ε u . Similarly, since ϕ = 0 and ϕ = 2π describe the same tilting effect, the makingsense end point for range [0, 2π ) is 0. The internal extremum point for F ε , ϕ , θ, φ j must satisfy
|φ1 =
⎞ ⎞−1 ⎛ N σ1 σ2 ⎠ ⎝ σ1 φ j + i φ j ⎠
j =1
ϕ = ϕ with a cer-
ε , ϕ , θ, φ j ,
F
tion and taking out a global phase, the reconstructed state turns into N (e ) |e . The tomography fidelity can be expressed as a multivariable function
F
rqe ( j )
j =
Logically, this bound is attained at ε = ε and tain θ and a fixed target state |φ, i.e.
thus the extremum point ε m , ϕ m can be obtained by solving this equation numerically when θ and φ j j = 1, · · · , N are given. Therefore, we have
+i· −2θ 2θ sin ε cos ϕ i sin ε sin ϕ + , = j + i j cos ε − 2θ 2θ
where
ε, ϕ , θ, φ j .
∂2 F ε , ϕ , θ, φ j = 0, ∂ ε∂ ϕ
Therefore the reconstructed state |e consists of the following elements:
ej =
, N. Next, we show how to find (ε , ϕ ) under cer-
= −2θ j + sin ε cos ϕ , = 2θ j cos ε + sin ε sin ϕ .
rhe ( j )
F LB F
F LB = F
When this tilting error occurs in a practical experiment, j and |s j remain the same as in Sec.2, but the device readouts under the first order approximation regarding θ are changed into
rhe ( j ) rqe ( j )
3
i sin ε sin ϕ 2θ i sin ε sin ϕ 2θ
, ,
u and 0 ε ε π , 0 ϕ < 2π . Furthermore, the trace distance D ε , ϕ , θ, φ j can also be calculated according to Eq. (1) based on the above F ε , ϕ , θ, φ j . To quantify the negative influence on tomography performance, either the lower bound of the fidelity F LB or the upper bound of
In this section a simulation is performed based on a reported experiment [2]. The target state is a truncated Gaussian state with a flat phase profile. This state is essentially a planar energy distribution, however for simplicity on the experiment operation as well as on the theoretical analysis, the authors in Ref. [2] consider its energy distribution along the x-axis and neglect the y-axis distribution. Therefore this state, denoted as |φ2 , can be described with the following model. The magnitude of its wave function is
A=
1 2πσ
exp −
(x − μ)2 2σ 2
,
(6)
where μ = 0, σ = 28.2 mm and −21.5 mm x 21.5 mm, and the phase of the wave function is considered as zero since the state has a flat phase profile. The sampling interval on the state profile is 1 mm, hence the target state is represented as a column vector with dimension N = 43,
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Table 1 The simulation results for state |φ1 in Eq. (5).
εu
ε
ϕ
···
···
0.0001
5.1110 × 10−5
0.9950
0.0010
9.5332 × 10−4
0.9949
0.0100
0.0100
4.4905
0.1000
0.1000
0.0110
1.0000
1.0000
9.1465 × 10−7
0
a b
b
|e a 0.2573 − 0.6788i −0.1103 + 0.6788i 0.2573 − 0.6788i −0.1103 + 0.6789i 0.2569 − 0.6783i −0.1107 + 0.6795i 0.2588 − 0.6858i −0.1087 + 0.6715i 0.1858 − 0.6815i −0.1855 + 0.6831i −0.3744 − 0.3802i −0.7554 + 0.3802i
F LB
D UB
1
0
0.99999999
7.4716 × 10−5
0.99999903
0.0014
0.99989356
0.0146
0.98923494
0.1463
0.26739140
0.9636
The reconstructed state disturbed by the error. No tilting error disturbance.
outcomes would not reflex the original target state faithfully. In other words, one may have the illusion that the reconstructed state after the state tomography process is exactly the true target state, but as long as this tilting error exists, the reconstructed state is never the true state but a close one, yet the smaller the error is, the closer it would be. 5. Seeking for the insensitive state pattern
As shown in Eq. (3) the fidelity F ε , ϕ , θ, φ j is a multivariable function, hence there may exist a state pattern insensitive to the error of the detector basis tilting. When measuring the states belonging to this pattern, the accuracy demand on detectors can be relaxed, therefore these states are suitable candidates for quantum information transferring. We seek this pattern by observing the tomography performance of the following planar laser energy distribution states. These states have the same magnitude profile of a planar Gaussian distribution, i.e. Fig. 3. The wave functions of the reconstructed states |e1 , |e2 and the target state |φ2 in Eq. (7). In the simulation, the coupling strength is set as θ = 0.349 rad, and the tilting error over |e1 is considered as ε = 0.0100 rad and ϕ = 0.0015 rad, while that over |e2 is ε = 0.0010 rad and ϕ = 5.5832 rad.
⎡ ⎢ ⎢ ⎢ |φ2 = ⎢ ⎢ ⎢ ⎣
A1
⎤
.. ⎥ . ⎥ ⎥ Aj ⎥ ⎥, .. ⎥ . ⎦
(7)
A 43 where A j ( j = 1, · · · , 43) is calculated from Eq. (6) at the sampling interval point x j , e.g. x1 = −21 mm (x43 = 21 mm) corresponds to the left (right) end of the accessible range and x22 = 0 corresponds to the origin of the x-axis. In addition, the coupling strength in simulation is set the same as in the reported experiment [2], i.e. θ = 20◦ = 0.349 rad. Fig. 3 shows the simulation results. In this simulation, the tilting error over the state |e1 is ε = 0.0100 rad and ϕ = 0.0015 rad, and that over |e2 is ε = 0.0010 rad and ϕ = 5.5832 rad. It is clear in Fig. 3 that the two reconstructed states |e1 and |e2 are different from the target state |φ2 at both magnitude and phase of their wave functions. The fidelity between |e1 and |φ2 is 0.98118 and their trace distance is 0.1931, while the fidelity between |e2 and |φ2 is 0.99997 and their trace distance is 0.0077. The authors in Ref. [2] assume that there exists no detector basis tilting error during the experiment. However, the simulation results here indicate that if such error does exist, the experiment
A=
1 2πσ1 σ2
exp −
(x − μ1 )2 2σ12
−
( y − μ2 )2 2σ22
(8)
,
where μ1 = μ2 = 0, σ1 = 0.5, σ2 = 0.6 and (x, y ) represents the planar position with x ∈ [−1, 1] and y ∈ [−1, 1]. Through discrete sampling at 11 × 11 planar areas, the wave function of these states can be transformed into column vectors with dimension N = 121. As phase modulation is operable at a designated place on the profile, these states are also considered as having a certain relative phase ϕr modulated on some randomly determined elements of the state vector, in addition the ratio of such phase-modulated elements to the total vector elements is λ. Accordingly the state pattern considered here is parameterized with ϕr and λ. For example, the following state is one of the multiple states within a pattern of ϕr = 1 rad and λ = 30/121:
⎡
A 1 ei
⎤
⎢ .. ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ A 30 ei ⎥ ⎥ |φ3 = ⎢ ⎢ A 31 ⎥ . ⎢ ⎥ ⎢ . ⎥ ⎣ .. ⎦
(9)
A 121 The state |φ3 in Eq. (9) describes the profile of a laser with the phase modulated as ϕr = 1 rad at the indexed first 30 planar areas and zero phase at the others. The magnitude A j ( j = 1, · · · , 121) of each vector element in Eq. (9) is calculated from Eq.
(8) at
the sampling position xc( j ) , y r( j ) , where c( j ) = flr
j −1 11
+ 1 and
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Fig. 5. The fidelity histogram of uniformly distributed 10 000 states belonging to the pattern of near 50% vector elements carrying phase π . Multiple rectangles over the range of these fidelity results are shown in the histogram, and the height of each rectangle indicates the count number of elements in each fidelity value bin. Fig. 4. Seeking for the insensitive pattern by observing multiple F LB at a given ε u = 10−3 . Here one point in this figure, parameterized with ϕr and λ, corresponds to a pattern containing multiple states.
r( j ) = rmn
j −1 11
+ 1. Here the function flr( A ) rounds A to the
nearest integer less than or equal to A toward negative infinity and the function rmn( X /Y ) returns the remainder after division of X by Y , i.e. X − n ∗ Y where n = flr( X /Y ) and Y = 0, in addition both c( j ) and r( j ) are integers within [1, 11] as j = 1, · · · , 121. For example, (x1 , y 1 ) = (−1, 1) when j = 1, (x1 , y 6 ) = (−1, 0) when j = 6, (x1 , y 11 ) = (−1, −1) when j = 11, (x2 , y 1 ) = (−0.8, 1) when j = 12, (x6 , y 6 ) = (0, 0) when j = 61, and (x11 , y 11 ) = (1, −1) when j = 121. The simulation results are shown in Fig. 4 with a given error bound ε u = 10−3 . This figure shows that the highest fidelity of 0.9965 is attained at the point of (ϕr = π , λ = 50%), while the fidelity is only 0.9591 at an adjacent point (ϕr = 2.7053, λ = 60%). Therefore, it appears that the pattern of near 50% vector elements carrying phase π is more insensitive to the error of the detector basis tilting. However, the fidelity of this peak point is only 35 one candidate of 12 C60 + C61 states within the 121 121 ≈ 1.916 × 10 pattern. As the total number of state vector elements is 121 and the phase modulation percentage is λ = 50%, this candidate number represents the combination number of taking 60 or 61 items with half probability respectively from the total 121 items. Here Cnk = (n−nk!)!k! is the binomial coefficient, i.e. the number of combinations of n items taken k at a time. Next we calculate the fidelities of 10 000 states uniformly distributed in the state set associated with this pattern. Fig. 5 shows the histogram of these fidelity results. Clearly, these fidelity results are in the range of [0.9473, 1], moreover 99% states provide a fidelity higher than 0.965. Therefore, it is reliable to some extent that the state pattern of near 50% vector elements carrying phase π is really more insensitive than others to the error of the detector basis tilting. 6. Conclusion The impact of imperfect measurements of weak values on state tomography due to the error of detector basis tilting is investigated in this paper. Here only the weak value measurement is concerned since the state tomography procedure based on the projective measurement is essentially different. The comparison between using different measurement methods is also worthy of investigation, and the research results will be presented in other papers in the future. The mathematical description of the basis tilting error is given by Eq. (2). For a given error bound, the negative influence on state
tomography is evaluated with two performance indices—fidelity and trace distance. The fidelity result is shown in Eqs. (3) and (4), while the trace distance can be obtained from Eq. (1) under the above fidelity. From the simulation based on a reported experiment [2], the detector basis tilting error would lead to a worse performance on state tomography. Specifically, the original target state can not be reconstructed faithfully by using the error-disturbed experiment data, yet the smaller the error is, the closer the reconstructed state is to the original target. However, it should be underlined that when the target state is in the pattern of near 50% vector elements carrying phase π , the tomography performance is relatively insensitive to such error. The states belonging to this pattern are suitable candidates for quantum information transferring. Acknowledgements This work is supported by the Program for National Natural Science Foundation of P.R. China (No. 61673389, No. 61273202 and No. 61134008). References [1] S.T. Yong, Introduction to Quantum-State Estimation, World Scientific Publishing Company, Singapore, 2015. [2] J.S. Lundeen, B. Sutherland, A. Patel, C. Stewart, C. Bamber, Nature 474 (2011) 188. [3] Y. Aharonov, D.Z. Albert, L. Vaidman, Phys. Rev. Lett. 60 (1988) 1351–1354. [4] J. Dressel, M. Malik, F.M. Miatto, A.N. Jordan, R.W. Boyd, Rev. Mod. Phys. 86 (2014) 307. [5] L. Bin Ho, N. Imoto, Phys. Lett. A 380 (2016) 2129–2135. [6] L. Diosi, Phys. Rev. A 94 (2016) 010103. [7] M.J.W. Hall, A.K. Pati, J.D. Wu, Phys. Rev. A 93 (2016) 052118. [8] Y.X. Zhang, S.J. Wu, Z.B. Chen, Phys. Rev. A 93 (2016) 032128. [9] D. Sokolovski, Phys. Lett. A 379 (2015) 1097–1101. [10] B.L. Higgins, M.S. Palsson, G.Y. Xiang, H.M. Wiseman, G.J. Pryde, Phys. Rev. A 91 (2015) 012113. [11] Y. Turek, W. Maimaiti, Y. Shikano, C.P. Sun, M. Al-Amri, Phys. Rev. A 92 (2015) 022109. [12] S. Kanjilal, G. Muralidhara, D. Home, Phys. Rev. A 94 (2016) 052110. [13] M. Ban, Phys. Lett. A 379 (2015) 284–288. [14] B. de Lima Bernardo, W.S. Martins, S. Azevedo, A. Rosas, Phys. Rev. A 92 (2015) 012109. [15] S. Pang, T.A. Brun, Phys. Rev. A 92 (2015) 012120. [16] S. Sponar, T. Denkmayr, H. Geppert, H. Lemmel, A. Matzkin, J. Tollaksen, Y. Hasegawa, Phys. Rev. A 92 (2015) 062121. [17] L. Qin, P. Liang, X.Q. Li, Phys. Rev. A 92 (2015) 012119. [18] Q. Wang, F.W. Sun, Y.S. Zhang Jian-Li, Y.F. Huang, G.C. Guo, Phys. Rev. A 73 (2006) 023814. [19] J. Fischbach, M. Freyberger, Phys. Rev. A 86 (2012) 052110. [20] S.J. Wu, Sci. Rep. 3 (2013) 1193.
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[21] M. Malik, M. Mirhosseini, M.P.J. Lavery, J. Leach, M.J. Padgett, R.W. Boyd, Nat. Commun. 5 (2014) 3115. [22] C. Bamber, J.S. Lundeen, Phys. Rev. Lett. 112 (2014) 070405. [23] M. Mirhosseini, O.S. Magaña-Loaiza, S.M.H. Rafsanjani, R.W. Boyd, Phys. Rev. Lett. 113 (2014) 090402. [24] L. Maccone, C.C. Rusconi, Phys. Rev. A 89 (2014) 022122. [25] J. Lee, I. Tsutsui, Phys. Lett. A 380 (2016) 2045–2048. [26] L.J. Zhang, A. Datta, I.A. Walmsley, Phys. Rev. Lett. 114 (2015) 210801.
[27] X.Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.F. Li, G.C. Guo, Phys. Rev. Lett. 111 (2013) 033604. [28] D. Sokolovski, Phys. Lett. A 380 (2016) 1593–1599. [29] D. Sokolovski, Phys. Lett. A 381 (2017) 227–232. [30] T. Zimmermann, S. Mishra, B.R. Doran, D.F. Gordon, A.S. Landsman, Phys. Rev. Lett. 116 (2016) 233603. [31] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2010.
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Physics Letters A www.elsevier.com/locate/pla
The impact of imperfect measurements of weak values on state tomography Xi Chen a , Hong-Yi Dai b,c , Ming Zhang a,∗ a b c
College of Mechatronics Engineering and Automation, National University of Defense Technology, Changsha, 410073, China College of Science, National University of Defense Technology, Changsha, 410073, China Interdisciplinary Center for Quantum Information, National University of Defense Technology, Changsha, 410073, China
a r t i c l e
i n f o
Article history: Received 24 March 2017 Received in revised form 14 May 2017 Accepted 3 August 2017 Available online xxxx Communicated by A. Eisfeld Keywords: Basis tilting error Weak value measurement Quantum state tomography Information encoding
a b s t r a c t We investigated the impact of imperfect measurements of weak values on state tomography due to the error of detector basis tilting, as this impact is seldom discussed before. For a given error bound, the negative influence is evaluated with two tomography performance indices—fidelity and trace distance. The simulation based on a reported experiment suggests that the tilting error would lead to a worse tomography performance. However, when the target state is in the pattern of near 50% vector elements carrying phase π , the tomography performance is relatively insensitive to such error. These results provide a choice of a suitable state type for information encoding in the task of quantum information transferring. © 2017 Elsevier B.V. All rights reserved.
1. Introduction State tomography is the procedure of inferring the identity of the state-generating source with collected experiment data [1]. Conventionally, projective measurement is adopted and this process is indirect in that it involves a time consuming postprocessing step where the density matrix of the state must be globally reconstructed through a numerical search over the alternatives consistent with the measured projective slices. However, in contrast to the projective measurement, the state can be reconstructed directly by using the results of weak value measurement [2]. The weak value measurement was first proposed by Aharonov et al. [3] in 1988, and the established procedure is shown in Fig. 1. After its emergence, the weak value measurement has been extensively investigated [4], such as exploring the physical meaning of the weak value [5–11], considering new effects over the original procedure [12–15], realizing the procedure in different physical objects or manners [16–18], etc. Meanwhile its application in quantum information is also intriguing, such as state tomography [2,19–24], metrology [25–27], and as a window into nonclassical features of quantum mechanics [28–30], etc. For instance, Bin Ho and Imoto [5] generally expressed the relationship between the weak value and the spin-operator modular
*
Corresponding author. E-mail addresses: [email protected] (X. Chen), [email protected] (M. Zhang). http://dx.doi.org/10.1016/j.physleta.2017.08.011 0375-9601/© 2017 Elsevier B.V. All rights reserved.
value. Sokolovski showed the meaning of “anomalous weak values” in quantum and classical theories [9] and related this meaning to some paradoxes [28,29]. Ban [13] considered the effect of an external field applied after a measurement and analyzed its influence on the system response through postselection. Lee and Tsutsui [25] presented an inequality of uncertainty relations in metrology and analyzed its feature with the help of weak values. In addition, the state tomography by using weak value measurement was first experimentally realized by Lundeen et al. [2], and their work can reconstruct discrete-variable quantum pure states. Later, Fischbach and Freyberger [19] proposed to use the homodyne detection as an interferometer for the weak value measurement, thus continuous-variable quantum optical states were reconstructed. S.J. Wu [20] proposed an efficient weak value tomography scheme with no data discarded, and a mixed state was reconstructed as a density matrix. Malik et al. [21] performed the experiment of directly measuring a 27-dimensional orbitalangular-momentum state vector. Bamber and Lundeen [22] implemented the mixed state reconstruction into a Dirac distribution, as the Dirac distribution is the Fourier transformation of a density matrix. Mirhosseini et al. [23] experimentally measured a 19 200-dimensional state by exploiting the compressive measurement. Moreover, Maccone and Rusconi [24] compared the state tomography by using weak value measurement with using projective measurement through simulations, and found out that the former was much less precise than the latter, because the former had an inherent bias, which would introduce an unavoidable error in the reconstruction.
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Theoretically, the state |φ is expressed as a continuous wave function φ (x, y ) at planar positions (x, y ), however due to the sampling number limitation on measurement devices, it only can
Fig. 1. The procedure of weak value measurement.
T
be represented as a discrete column vector φ1 , · · · , φ j , · · · , φ N with dimension N, and all planar areas for the laser energy distribution are indexed with j = 1, · · · , N. In addition, there is φ j = j |φ, when considering | j as a column vector with 1 in the j-th row and 0 in the remaining rows, i.e. | j = [0, · · · , 1, · · · , 0]T . Step 2, performing weak coupling at a spatial light modula tor [23]. The unitary evolution of the joint system state is j = M j ⊗ R y (2θ) + I t − M j ⊗ I p |φ | V , where M j ⊗ R y (2θ) represents the weak coupling operation at the j-th area. Here M j = | j j | characterizes the action area and R y (2θ) = exp −iθ σ y performs a polarization rotation with angle θ , as σ y is a Pauli operator [31], moreover the angle θ describes the coupling strength at the action area. When 0 < |θ| 0.2 rad, the coupling can be considered weak, then the first order approximation can be applied as R y (2θ) = exp −iθ σ y = I p − iθ σ y , thus the joint state becomes | j = |φ | V − θ| j j |φ| H . Step 3, performing postselection on the target at a pinhole [23]. Here the postselected is a zero momentum state expressed as | p = √1 [1, 1, · · · , 1]T in the position representation. After the N
target postselection and under the first order approximation, the p |
Fig. 2. The experiment setup for measuring the weak value connected with the planar laser energy distribution. P: polarizer, L: lens, SLM: spatial light modular, WP: wave plate, PBS: polarized beam splitter, D1,D2: detectors, hWP: half-wave plate, qWP: quarter-wave plate.
However, previous papers about state tomography via weak value measurement are merely discussed under the error-free condition. As we know, practical operations are often imperfect and the error of detector basis tilting may exist in the last step of the procedure in Fig. 1. In order to understand the impact of imperfect measurements of weak values due to this kind of error, we examine the influence on the state tomography performance in this paper. Besides that, the state pattern insensitive to such error is also sought for, as these states are good candidates for information encoding in the task of quantum information transferring. The structure of the rest of this paper are as follows. In section 2, we review the process of state tomography via weak value measurement. In section 3, the mathematical description of the error of detector basis tilting is given and the influence on tomography performance is analyzed theoretically. In section 4, the influence due to such error is simulated based on the first reported experiment on directly measuring a quantum state [2]. In section 5, we seek for the state pattern insensitive to such error. The conclusion is summarized in section 6.
final probe state becomes |s j = p |φj = | V − θ j | H , where j = p| pj|φj|φ is the weak value connected with the laser energy distribution over planar positions in this scheme. Step 4, measuring the probe projectively and recording the readouts at two detectors behind either a half-wave plate or a quater-wave plate. The readouts corresponding to different setups in Fig. 2 are rh ( j ) = s j | D D |s j − s j | A A |s j = −2θ Re j and rq ( j ) = s j | R R |s j − s j | L L |s j = 2θ Im j , and the measurement bases are | D = √1 (| H + | V ), | A = √1 (| H − | V ), 2
|R =
√1
2
(| H + i| V ), | L =
√1
2
2
(| H − i| V ).
At last, the weak value in this scheme is obtained by rh ( j ) −2θ
rq ( j ) . 2θ
p| j p |φ j |φ
=
j =
+i· Weak values j at all j = 1, · · · , N areas can be used to reconstruct the target state |φ, since there exists j = √1
√1 N
NN
j =1
φj
· φ j = κ φ j , where κ =
N j =1 φ j
−1
. Af-
ter normalization the column vector | consisting of j changes into N ( ) | , where N ( ) ∈ C is the corresponding transform coefficient and in this particular case N ( ) = κ −1 , meaning that N ( ) | equals the target state |φ. The performance of state tomography can be evaluated with two equivalent indices—fidelity and trace distance [31]. When only considering two quantum pure states N ( ) | and |φ, the fidelity is defined as
2
2
F = N ( ) · φ| ,
2. The process of state tomography via weak value measurement
and the trace distance can be computed based on the fidelity as
Fig. 2 shows the experiment setup for measuring the weak value connected with the planar laser energy distribution [23]. The target is the laser attribute of the energy distribution over planar positions, and the probe is the laser polarization attribute. According to the general procedure in Fig. 1, the actual steps in this scheme are as follows. the initial probe polarization state |φ | V = Step1, preparing I t ⊗ P |φ |s , where |φ is the target state and |s is a nonpolarized probe state with two levels, and after a polarizer, a vertically polarized probe state | V is prepared. The state | V = [0, 1]T is one of two basis states for the probe, and the other is the horizontal polarization | H = [1, 0]T .
D=
1 − F 2.
(1)
These two indices F and D are both valued within [0, 1], moreover when D is closer to 0, F is reversely closer to 1. 3. Theoretical analysis of the error and its influence It is possible that the detector basis states | D , | A , | R and | L are disturbed by the error of tilting in practical experiments. Based on the parameterized form of a qubit, this kind of error changes the conventional probe bases | H and | V into
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ε
ε
2
2
| H e = cos | H + eiϕ sin | V ,
ε
ε
2
2
the trace distance D UB can be used. Here we derive F LB in detail, then D UB is obtained through Eq. (1). As we know, the lower fidelity bound F LB necessarily satisfies
| V e = cos | V + eiϕ sin | H , where 0 ε ε u π and 0 ϕ < 2π , here ε u is the upper bound to ε , thus the measurement bases in operation become
1 ε ε
| D e = √ (| H e + | V e ) = √ cos + eiϕ sin |H 2 2 2 2 1 ε ε
| V , + √ cos + eiϕ sin 2 2 2 1 1 ε ε
| A e = √ (| H e − | V e ) = √ cos − eiϕ sin |H 2 2 2 2 1 ε ε
| V , − √ cos − eiϕ sin 2 2 2 1 1 ε ε
|H | R e = √ (| H e + i| V e ) = √ cos + ieiϕ sin 2 2 2 2 1 ε ε
| V , + √ i cos + eiϕ sin 2 2 2 1 1 ε ε
| L e = √ (| H e − i| V e ) = √ cos − ieiϕ sin |H 2 2 2 2 1 ε ε
| V . − √ i cos − eiϕ sin 2 2 2 1
(2)
φ j +i φ j and j N j =1 φ j +i φ j
⎛
=⎝
ε, ϕ , θ, φ j
N
= 1, · · · , N. After normaliza-
(3)
⎞ ⎛ N ×⎝ σ2 φ j − i φ j ⎠ , j =1
where
sin ε cos ϕ
2θ sin ε cos ϕ
σ2 = i j cos ε + j −
2θ
+ +
(4)
ε , ϕ = arg min u F εu , 0, θ, φ j , F εu , ϕ m , θ, φ j ,
0ε ε 0ϕ <2π
εm , 0, θ, φ j , F εm , ϕ m , θ, φ j
.
To demonstrate the aforesaid theoretical analysis vividly we show below the simulation results in Table 1 on a randomly generated 2-dimensional target state
0.2573 − 0.6788i . −0.1103 + 0.6788i
(5)
The coupling strength in simulation is set as θ = 0.1 rad for applying the first order approximation regarding θ . Table 1 shows that the larger the tilting error bound ε u is, the sharper the difference is between the original target state |φ1 and the reconstructed state |e , thus the smaller the lower bound of the fidelity F LB is and the larger the upper bound of the trace distance D UB is. In other words, the tilting error bound ε u needs to be small if a faithful state tomography result is required. 4. Simulation on a reported experiment
j =1
σ1 = i j cos ε − j +
where j = 1, · · · tain θ and |φ. Since ε ∈ [0, ε u ] and ϕ ∈ [0, 2π ), it is reasonable that ε or ϕ is at either its range end point or the F ε, ϕ , θ, φ j extremum point within the range. Moreover since ε = 0 means no tilting, the making-sense end point for range [0, ε u ] is ε u . Similarly, since ϕ = 0 and ϕ = 2π describe the same tilting effect, the makingsense end point for range [0, 2π ) is 0. The internal extremum point for F ε , ϕ , θ, φ j must satisfy
|φ1 =
⎞ ⎞−1 ⎛ N σ1 σ2 ⎠ ⎝ σ1 φ j + i φ j ⎠
j =1
ϕ = ϕ with a cer-
ε , ϕ , θ, φ j ,
F
tion and taking out a global phase, the reconstructed state turns into N (e ) |e . The tomography fidelity can be expressed as a multivariable function
F
rqe ( j )
j =
Logically, this bound is attained at ε = ε and tain θ and a fixed target state |φ, i.e.
thus the extremum point ε m , ϕ m can be obtained by solving this equation numerically when θ and φ j j = 1, · · · , N are given. Therefore, we have
+i· −2θ 2θ sin ε cos ϕ i sin ε sin ϕ + , = j + i j cos ε − 2θ 2θ
where
ε, ϕ , θ, φ j .
∂2 F ε , ϕ , θ, φ j = 0, ∂ ε∂ ϕ
Therefore the reconstructed state |e consists of the following elements:
ej =
, N. Next, we show how to find (ε , ϕ ) under cer-
= −2θ j + sin ε cos ϕ , = 2θ j cos ε + sin ε sin ϕ .
rhe ( j )
F LB F
F LB = F
When this tilting error occurs in a practical experiment, j and |s j remain the same as in Sec.2, but the device readouts under the first order approximation regarding θ are changed into
rhe ( j ) rqe ( j )
3
i sin ε sin ϕ 2θ i sin ε sin ϕ 2θ
, ,
u and 0 ε ε π , 0 ϕ < 2π . Furthermore, the trace distance D ε , ϕ , θ, φ j can also be calculated according to Eq. (1) based on the above F ε , ϕ , θ, φ j . To quantify the negative influence on tomography performance, either the lower bound of the fidelity F LB or the upper bound of
In this section a simulation is performed based on a reported experiment [2]. The target state is a truncated Gaussian state with a flat phase profile. This state is essentially a planar energy distribution, however for simplicity on the experiment operation as well as on the theoretical analysis, the authors in Ref. [2] consider its energy distribution along the x-axis and neglect the y-axis distribution. Therefore this state, denoted as |φ2 , can be described with the following model. The magnitude of its wave function is
A=
1 2πσ
exp −
(x − μ)2 2σ 2
,
(6)
where μ = 0, σ = 28.2 mm and −21.5 mm x 21.5 mm, and the phase of the wave function is considered as zero since the state has a flat phase profile. The sampling interval on the state profile is 1 mm, hence the target state is represented as a column vector with dimension N = 43,
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Table 1 The simulation results for state |φ1 in Eq. (5).
εu
ε
ϕ
···
···
0.0001
5.1110 × 10−5
0.9950
0.0010
9.5332 × 10−4
0.9949
0.0100
0.0100
4.4905
0.1000
0.1000
0.0110
1.0000
1.0000
9.1465 × 10−7
0
a b
b
|e a 0.2573 − 0.6788i −0.1103 + 0.6788i 0.2573 − 0.6788i −0.1103 + 0.6789i 0.2569 − 0.6783i −0.1107 + 0.6795i 0.2588 − 0.6858i −0.1087 + 0.6715i 0.1858 − 0.6815i −0.1855 + 0.6831i −0.3744 − 0.3802i −0.7554 + 0.3802i
F LB
D UB
1
0
0.99999999
7.4716 × 10−5
0.99999903
0.0014
0.99989356
0.0146
0.98923494
0.1463
0.26739140
0.9636
The reconstructed state disturbed by the error. No tilting error disturbance.
outcomes would not reflex the original target state faithfully. In other words, one may have the illusion that the reconstructed state after the state tomography process is exactly the true target state, but as long as this tilting error exists, the reconstructed state is never the true state but a close one, yet the smaller the error is, the closer it would be. 5. Seeking for the insensitive state pattern
As shown in Eq. (3) the fidelity F ε , ϕ , θ, φ j is a multivariable function, hence there may exist a state pattern insensitive to the error of the detector basis tilting. When measuring the states belonging to this pattern, the accuracy demand on detectors can be relaxed, therefore these states are suitable candidates for quantum information transferring. We seek this pattern by observing the tomography performance of the following planar laser energy distribution states. These states have the same magnitude profile of a planar Gaussian distribution, i.e. Fig. 3. The wave functions of the reconstructed states |e1 , |e2 and the target state |φ2 in Eq. (7). In the simulation, the coupling strength is set as θ = 0.349 rad, and the tilting error over |e1 is considered as ε = 0.0100 rad and ϕ = 0.0015 rad, while that over |e2 is ε = 0.0010 rad and ϕ = 5.5832 rad.
⎡ ⎢ ⎢ ⎢ |φ2 = ⎢ ⎢ ⎢ ⎣
A1
⎤
.. ⎥ . ⎥ ⎥ Aj ⎥ ⎥, .. ⎥ . ⎦
(7)
A 43 where A j ( j = 1, · · · , 43) is calculated from Eq. (6) at the sampling interval point x j , e.g. x1 = −21 mm (x43 = 21 mm) corresponds to the left (right) end of the accessible range and x22 = 0 corresponds to the origin of the x-axis. In addition, the coupling strength in simulation is set the same as in the reported experiment [2], i.e. θ = 20◦ = 0.349 rad. Fig. 3 shows the simulation results. In this simulation, the tilting error over the state |e1 is ε = 0.0100 rad and ϕ = 0.0015 rad, and that over |e2 is ε = 0.0010 rad and ϕ = 5.5832 rad. It is clear in Fig. 3 that the two reconstructed states |e1 and |e2 are different from the target state |φ2 at both magnitude and phase of their wave functions. The fidelity between |e1 and |φ2 is 0.98118 and their trace distance is 0.1931, while the fidelity between |e2 and |φ2 is 0.99997 and their trace distance is 0.0077. The authors in Ref. [2] assume that there exists no detector basis tilting error during the experiment. However, the simulation results here indicate that if such error does exist, the experiment
A=
1 2πσ1 σ2
exp −
(x − μ1 )2 2σ12
−
( y − μ2 )2 2σ22
(8)
,
where μ1 = μ2 = 0, σ1 = 0.5, σ2 = 0.6 and (x, y ) represents the planar position with x ∈ [−1, 1] and y ∈ [−1, 1]. Through discrete sampling at 11 × 11 planar areas, the wave function of these states can be transformed into column vectors with dimension N = 121. As phase modulation is operable at a designated place on the profile, these states are also considered as having a certain relative phase ϕr modulated on some randomly determined elements of the state vector, in addition the ratio of such phase-modulated elements to the total vector elements is λ. Accordingly the state pattern considered here is parameterized with ϕr and λ. For example, the following state is one of the multiple states within a pattern of ϕr = 1 rad and λ = 30/121:
⎡
A 1 ei
⎤
⎢ .. ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ A 30 ei ⎥ ⎥ |φ3 = ⎢ ⎢ A 31 ⎥ . ⎢ ⎥ ⎢ . ⎥ ⎣ .. ⎦
(9)
A 121 The state |φ3 in Eq. (9) describes the profile of a laser with the phase modulated as ϕr = 1 rad at the indexed first 30 planar areas and zero phase at the others. The magnitude A j ( j = 1, · · · , 121) of each vector element in Eq. (9) is calculated from Eq.
(8) at
the sampling position xc( j ) , y r( j ) , where c( j ) = flr
j −1 11
+ 1 and
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Fig. 5. The fidelity histogram of uniformly distributed 10 000 states belonging to the pattern of near 50% vector elements carrying phase π . Multiple rectangles over the range of these fidelity results are shown in the histogram, and the height of each rectangle indicates the count number of elements in each fidelity value bin. Fig. 4. Seeking for the insensitive pattern by observing multiple F LB at a given ε u = 10−3 . Here one point in this figure, parameterized with ϕr and λ, corresponds to a pattern containing multiple states.
r( j ) = rmn
j −1 11
+ 1. Here the function flr( A ) rounds A to the
nearest integer less than or equal to A toward negative infinity and the function rmn( X /Y ) returns the remainder after division of X by Y , i.e. X − n ∗ Y where n = flr( X /Y ) and Y = 0, in addition both c( j ) and r( j ) are integers within [1, 11] as j = 1, · · · , 121. For example, (x1 , y 1 ) = (−1, 1) when j = 1, (x1 , y 6 ) = (−1, 0) when j = 6, (x1 , y 11 ) = (−1, −1) when j = 11, (x2 , y 1 ) = (−0.8, 1) when j = 12, (x6 , y 6 ) = (0, 0) when j = 61, and (x11 , y 11 ) = (1, −1) when j = 121. The simulation results are shown in Fig. 4 with a given error bound ε u = 10−3 . This figure shows that the highest fidelity of 0.9965 is attained at the point of (ϕr = π , λ = 50%), while the fidelity is only 0.9591 at an adjacent point (ϕr = 2.7053, λ = 60%). Therefore, it appears that the pattern of near 50% vector elements carrying phase π is more insensitive to the error of the detector basis tilting. However, the fidelity of this peak point is only 35 one candidate of 12 C60 + C61 states within the 121 121 ≈ 1.916 × 10 pattern. As the total number of state vector elements is 121 and the phase modulation percentage is λ = 50%, this candidate number represents the combination number of taking 60 or 61 items with half probability respectively from the total 121 items. Here Cnk = (n−nk!)!k! is the binomial coefficient, i.e. the number of combinations of n items taken k at a time. Next we calculate the fidelities of 10 000 states uniformly distributed in the state set associated with this pattern. Fig. 5 shows the histogram of these fidelity results. Clearly, these fidelity results are in the range of [0.9473, 1], moreover 99% states provide a fidelity higher than 0.965. Therefore, it is reliable to some extent that the state pattern of near 50% vector elements carrying phase π is really more insensitive than others to the error of the detector basis tilting. 6. Conclusion The impact of imperfect measurements of weak values on state tomography due to the error of detector basis tilting is investigated in this paper. Here only the weak value measurement is concerned since the state tomography procedure based on the projective measurement is essentially different. The comparison between using different measurement methods is also worthy of investigation, and the research results will be presented in other papers in the future. The mathematical description of the basis tilting error is given by Eq. (2). For a given error bound, the negative influence on state
tomography is evaluated with two performance indices—fidelity and trace distance. The fidelity result is shown in Eqs. (3) and (4), while the trace distance can be obtained from Eq. (1) under the above fidelity. From the simulation based on a reported experiment [2], the detector basis tilting error would lead to a worse performance on state tomography. Specifically, the original target state can not be reconstructed faithfully by using the error-disturbed experiment data, yet the smaller the error is, the closer the reconstructed state is to the original target. However, it should be underlined that when the target state is in the pattern of near 50% vector elements carrying phase π , the tomography performance is relatively insensitive to such error. The states belonging to this pattern are suitable candidates for quantum information transferring. Acknowledgements This work is supported by the Program for National Natural Science Foundation of P.R. China (No. 61673389, No. 61273202 and No. 61134008). References [1] S.T. Yong, Introduction to Quantum-State Estimation, World Scientific Publishing Company, Singapore, 2015. [2] J.S. Lundeen, B. Sutherland, A. Patel, C. Stewart, C. Bamber, Nature 474 (2011) 188. [3] Y. Aharonov, D.Z. Albert, L. Vaidman, Phys. Rev. Lett. 60 (1988) 1351–1354. [4] J. Dressel, M. Malik, F.M. Miatto, A.N. Jordan, R.W. Boyd, Rev. Mod. Phys. 86 (2014) 307. [5] L. Bin Ho, N. Imoto, Phys. Lett. A 380 (2016) 2129–2135. [6] L. Diosi, Phys. Rev. A 94 (2016) 010103. [7] M.J.W. Hall, A.K. Pati, J.D. Wu, Phys. Rev. A 93 (2016) 052118. [8] Y.X. Zhang, S.J. Wu, Z.B. Chen, Phys. Rev. A 93 (2016) 032128. [9] D. Sokolovski, Phys. Lett. A 379 (2015) 1097–1101. [10] B.L. Higgins, M.S. Palsson, G.Y. Xiang, H.M. Wiseman, G.J. Pryde, Phys. Rev. A 91 (2015) 012113. [11] Y. Turek, W. Maimaiti, Y. Shikano, C.P. Sun, M. Al-Amri, Phys. Rev. A 92 (2015) 022109. [12] S. Kanjilal, G. Muralidhara, D. Home, Phys. Rev. A 94 (2016) 052110. [13] M. Ban, Phys. Lett. A 379 (2015) 284–288. [14] B. de Lima Bernardo, W.S. Martins, S. Azevedo, A. Rosas, Phys. Rev. A 92 (2015) 012109. [15] S. Pang, T.A. Brun, Phys. Rev. A 92 (2015) 012120. [16] S. Sponar, T. Denkmayr, H. Geppert, H. Lemmel, A. Matzkin, J. Tollaksen, Y. Hasegawa, Phys. Rev. A 92 (2015) 062121. [17] L. Qin, P. Liang, X.Q. Li, Phys. Rev. A 92 (2015) 012119. [18] Q. Wang, F.W. Sun, Y.S. Zhang Jian-Li, Y.F. Huang, G.C. Guo, Phys. Rev. A 73 (2006) 023814. [19] J. Fischbach, M. Freyberger, Phys. Rev. A 86 (2012) 052110. [20] S.J. Wu, Sci. Rep. 3 (2013) 1193.
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[21] M. Malik, M. Mirhosseini, M.P.J. Lavery, J. Leach, M.J. Padgett, R.W. Boyd, Nat. Commun. 5 (2014) 3115. [22] C. Bamber, J.S. Lundeen, Phys. Rev. Lett. 112 (2014) 070405. [23] M. Mirhosseini, O.S. Magaña-Loaiza, S.M.H. Rafsanjani, R.W. Boyd, Phys. Rev. Lett. 113 (2014) 090402. [24] L. Maccone, C.C. Rusconi, Phys. Rev. A 89 (2014) 022122. [25] J. Lee, I. Tsutsui, Phys. Lett. A 380 (2016) 2045–2048. [26] L.J. Zhang, A. Datta, I.A. Walmsley, Phys. Rev. Lett. 114 (2015) 210801.
[27] X.Y. Xu, Y. Kedem, K. Sun, L. Vaidman, C.F. Li, G.C. Guo, Phys. Rev. Lett. 111 (2013) 033604. [28] D. Sokolovski, Phys. Lett. A 380 (2016) 1593–1599. [29] D. Sokolovski, Phys. Lett. A 381 (2017) 227–232. [30] T. Zimmermann, S. Mishra, B.R. Doran, D.F. Gordon, A.S. Landsman, Phys. Rev. Lett. 116 (2016) 233603. [31] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2010.