CN111008703A - Method for separating mixed state three bodies of four-quantum-bit W state and GHZ state in noise environment - Google Patents

Method for separating mixed state three bodies of four-quantum-bit W state and GHZ state in noise environment Download PDF

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CN111008703A
CN111008703A CN201911145608.2A CN201911145608A CN111008703A CN 111008703 A CN111008703 A CN 111008703A CN 201911145608 A CN201911145608 A CN 201911145608A CN 111008703 A CN111008703 A CN 111008703A
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陈�峰
蒋丽珍
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Abstract

The invention designs a method for separating mixed state triples of a four-quantum-bit W state and a GHZ state in a noise environment, and considers separability from the aspects of sufficiency and necessity. Since the method of constructing the entanglement witness can judge the lower bound of entanglement (the entangled states referred to in the present invention are all referred to as three-body inseparable states), the present invention uses it as a necessary condition for the criterion. In the derivation of sufficient conditions, the invention adopts a method for constructing a target state by using a trisomy separable state with parameters, and an upper bound of the separable state is found by using the method. The calculation result shows that under the conditions of a four-quantum W state and a four-quantum GHZ state, the three-body separable noise margin obtained by the method is very close to the result under the entanglement prover method. During the mixed state research process of the two methods, the three-body separable noise tolerance of the mixed state obtained by the method is more accurate than the result obtained by the entanglement witness method alone.

Description

Method for separating mixed state three bodies of four-quantum-bit W state and GHZ state in noise environment
Technical Field
The present invention relates to the field of quantum information processing. The invention designs a method for separating mixed state triples of a four-quantum-bit W state and a GHZ state in a noise environment, and the method can obtain sufficient conditions and necessary conditions for separating the mixed state triples of the four-quantum-bit W state and the GHZ state, thereby obtaining the noise tolerance.
Background
The entanglement witness approach is essentially an observable entanglement witness, which is noted as an entanglement witness
Figure BDA0002282087660000011
If it is
Figure BDA0002282087660000012
For all separable quantum states ρsAll satisfy
Figure BDA0002282087660000013
And at least one quantum state p is present such that
Figure BDA0002282087660000014
For such a
Figure BDA0002282087660000015
It is said to witness entanglement of rho. In practical applications, only one quantum state needs to be computed if an entanglement witness has already been obtained
Figure BDA0002282087660000016
Can determine whether the quantum state is entangled or separable.
The entanglement witness can also be simply understood as a hyperplane, and rho in one side of the hyperplane satisfies
Figure BDA0002282087660000017
The quantum states contained in the super-plane are all separable, and rho satisfies the condition in the other side of the super-plane
Figure BDA0002282087660000018
The internal quantum states are entangled. Understanding the entanglement witnesses as a hyperplane is not without evidence that if all separable states are grouped together, the set of them is convex and closed, and the entanglement witness, like the outer surface of the set, completely isolates the separable states from the entangled states. The following theorem can be understood in conjunction with the above description:
for an entangled state ρ, there is always an entanglement witness
Figure BDA0002282087660000019
Its entanglement can be seen.
The concept of the entanglement witness is not difficult to understand, and the use process is simple and convenient, but the idea of obtaining a suitable entanglement witness as the entanglement criterion is not easy. The separation of the multi-body quantum system generally has a plurality of types, the structure distribution of the system needs to be clear for constructing an accurate entanglement witness, and the construction process is relatively complex. Generally, constructing an entanglement witness first requires finding an optimal entanglement witness, and then adjusting on the basis of the optimal entanglement witness to obtain a matching entanglement witness.
The process of finding an entanglement witness is essentially to find an entanglement witness satisfying for all separable states
Figure BDA0002282087660000021
The process of (1).
Memory entanglement witness symbol
Figure BDA0002282087660000022
The form of (A) is as follows:
Figure BDA0002282087660000023
wherein
Figure BDA0002282087660000024
Referred to as a witness, is,
Figure BDA0002282087660000025
i is and
Figure BDA0002282087660000026
unit matrix of corresponding size, rhosRefers to a density matrix of quantum states. In quantum entanglement computations, qubits can each be represented by a bloch representation, which can be simply demonstrated:
assume that the density matrix for a qubit is as follows:
Figure BDA0002282087660000027
in order to satisfy the positive definite condition, it is required that a (1-a) ≧ a! a-2Equivalently rewriting the elements in ρ and unfolding the element r into the form of the sum of the real and imaginary partsFormula (iv) can be obtained:
Figure BDA0002282087660000028
wherein z is (2a-1) and x is 2rR，y＝-2rIX, Y, Z are respectively pauli matrices. The sum of the squares of x, y, z can be found:
Figure BDA0002282087660000029
order to
Figure BDA00022820876600000210
The qubit ρ can be expressed as:
Figure BDA00022820876600000211
taking the simplest two-body system as an example, the following introduces the concept of a property matrix, for a two-body separable system, the density matrix can be written as:
Figure BDA0002282087660000031
on this basis, the property matrix is defined as:
Figure BDA0002282087660000032
and because:
Figure BDA0002282087660000033
therefore, the method comprises the following steps:
Figure BDA0002282087660000034
by substituting the formula (9) into the formula (7), it is possible to obtain:
Figure BDA0002282087660000035
the density matrix of the final two-body system can be expressed as:
Figure BDA0002282087660000036
σ in formulae (7) to (11)0、σ1、σ2、σ3Each representing 4 pauli matrices I, X, Y, Z.
After introducing the concept of the characteristic matrix, taking a two-qubit system as an example, the derivation process of the optimal entanglement witness is introduced.
In a two-part system, witness operators
Figure BDA0002282087660000037
The definition is as follows:
Figure BDA0002282087660000038
wherein M islnIs an adjustable real ginseng. Then:
Figure BDA0002282087660000041
order to
Figure BDA0002282087660000042
It is easy to find out if one wants to find
Figure BDA0002282087660000043
The maximum value of (2) is obtained by simply obtaining the maximum value of S. For solving the maximum value of S, it is possible to separately order
Figure BDA0002282087660000044
Wherein (v)1,v2,v3)、(u1,u2,u3) Respectively, the basis vectors in spherical coordinates:
Figure BDA0002282087660000045
in a similar manner to that described above,
Figure BDA0002282087660000046
finally only to thetaA、θB、φA、φBThe maximum value of S can be obtained by traversing in angle, namely
Figure BDA0002282087660000051
Is measured. As can be seen from the formula (1),
Figure BDA0002282087660000052
knowing if one wants to get the optimal entanglement witness such that
Figure BDA0002282087660000053
In the required formula (16)
Figure BDA0002282087660000054
Namely, it is
Figure BDA0002282087660000055
This also explains the reason why Λ is thus defined in formula (1). The above demonstration has already been given
Figure BDA0002282087660000056
Thereby satisfying
Figure BDA0002282087660000057
The lambda of the condition can be determined. On the basis of the determination of lambda, only the proper one needs to be found
Figure BDA0002282087660000058
Operators for finding boundaries of all separable states in space, i.e. for a closed convex set formed by all separable statesFinding out the surface, and obtaining the entanglement witness according to the formula (1)
Figure BDA0002282087660000059
The process of finding the matching entanglement witnesses is essentially to find an entanglement witness for all quantum states satisfied by an entanglement witness operator on the basis of finding the optimal entanglement witness
Figure BDA00022820876600000510
The process of (1).
For the convenience of understanding the content of the study, the quantum state ρ' in a noisy environment is directly given:
Figure BDA00022820876600000511
wherein p is the proportion of the quantum state in the noise and quantum state mixed environment, and rho is an N multiplied by N matrix. When ordering
Figure BDA00022820876600000512
Then, it can be obtained according to the formula (1):
Figure BDA00022820876600000513
thereby advancing:
Figure BDA00022820876600000514
as can be seen from equation (19), when p is 1, it can be found through the above derivation process
Figure BDA00022820876600000517
If when p is greater than 1, then,
Figure BDA00022820876600000518
can thus define
Figure BDA00022820876600000515
For any quantum state, therefore, only by adjustment
Figure BDA00022820876600000516
So that L equals 1, the lower bound of the entangled state is obtained. This completes the structure of the entanglement witness.
Based on the above analysis, the present invention improves the entanglement witness method, both from the standpoint of sufficiency and necessity. Since the method of constructing the entanglement prover can judge the lower bound of entanglement (hereinafter, entangled states are all referred to as three-body inseparable states), it is used as a necessary condition of the criterion. In the derivation of sufficient conditions, the invention adopts a method for constructing a target state by using a trisomy separable state with parameters, and an upper bound of the separable state is found by using the method. Studies have shown that in the four-quantum W state and four-quantum GHZ state, the three-body separable noise margins of the two in the method of the present invention are much closer than in the entanglement witness method. During the mixed state research process of the two methods, the three-body separable noise tolerance of the mixed state in the method is more accurate than the result of the entanglement witness method alone.
Reference to the literature
[1]Nielsen M A,Chuang I L.Quantum computation and quantuminformation:10th anniversary edition[M].Cambridge University Press,2011.
[2]Charles H B,Gilles B,Claude C,et al.Teleporting an unknown quantumstate via dual classical and Einstein-Podolsky-Rosen channels[J].PhysicalReview Letters,1993,70(13):1895.
[3]Almut B,Berthold G E,Christian K,et al.Secure communication with apublicly known key[J].Acta Physica Polonica,2002,101(3):357.
[4]Jiang W,Ren X,Wu Y,et al.A sufficient and necessary condition for2n-1 orthogonal states to be locally distinguishable in a C-2 circle times C-n system[J].Journal of Physics A-Mathematical and Theoretical,2010,43(32):1610-1625.
[5]Hein M,Eisert J,Briegel H J.Multiparty entanglement in graphstates(20pages)[J].Physical Review A,2003,69(6):666-670.
[6]Kay A.Optimal detection of entanglement in Greenberger-Horne-Zeilinger states[J].Physical Review A,2011,83(2):4762-4767.
[7]Li M,Wang J,Fei S M,et al.Quantum separability criteria forarbitrary dimensional multipartite states[J].Physical Review A,2014,89(2):767-771.
[8]Chen X Y,Jiang L Z,Yu P,et al.Necessary and sufficient fullyseparable criterion and entanglement of three-qubit Greenberger-Horne-Zeilinger diagonal states[J].Quantum Information Processing,2015,14(7):2463-2476.
Disclosure of Invention
The invention designs a method for separating mixed state triples of a four-quantum-bit W state and a GHZ state in a noise environment, and the method can obtain sufficient conditions and necessary conditions for separating the mixed state triples of the four-quantum-bit W state and the GHZ state, thereby obtaining the noise tolerance.
A method for separating mixed state triples of a four-quantum-bit W state and a GHZ state in a noise environment comprises the following three processes:
s1) respectively constructing the optimal entanglement witnesses, searching respective matched entanglement witnesses on the basis, specifically analyzing different quantum states in the process of constructing the optimal entanglement witness, respectively solving parameters related to the construction process, then carrying out formula derivation and partial variable form replacement, and finally solving the optimal entanglement witness through MATLAB auxiliary calculation;
s2) finding a matching entanglement witness based on S1, the key to determining the matching entanglement witness is to find a suitable witness
Figure BDA0002282087660000071
In the calculation process, a progressive mode is adopted and MATLAB is utilized to carry out loop iteration, so that the optimal value is continuously reduced
Figure BDA0002282087660000072
Finally, a relatively accurate matching entanglement witness is solved and the lower bound of the entangled state is obtained;
s3) generalizing the three-body separable quantum state into a general form, that is, the quantum states in the form are all separable states, collecting and summarizing the characteristics and rules of the W state, the GHZ state and the mixed state of the four-quantum bit W state and the GHZ state in the process of constructing the entanglement witness, bringing the data into the general form of the three-body separable quantum state, and constructing the three-body separable upper bound of the quantum states as accurately as possible, and combining the lower bound of the entanglement state obtained in S2, the three-body separability criterion of the quantum states can be deduced.
Drawings
Fig. 1 shows that the maximum value of the mixing state Λ is theta at 50 degreesA、θBDistribution of the bottom.
FIG. 2 shows the maximum value of the mixing state Λ at 50 degrees theta, phiA-φBDistribution of the bottom.
FIG. 3 is a four quantum W state θA、θBThe maximum value of Λ in the case.
FIG. 4 is a four quantum W state θA、φA-φBThe maximum value of Λ in the case.
FIG. 5 is a four quantum GHZ state θA、θBThe maximum value of Λ in the case.
FIG. 6 is a diagram of four quantum GHZ states θ, φA-φBThe maximum value of Λ in the case.
FIG. 7 is a comprehensive sufficiency and necessity curve.
Detailed Description
The technical solution of the present invention is further described with reference to the following examples.
1. Computation of entanglement witnesses
The mixed state of the W state and the GHZ state of the four quanta bit is in the form as follows:
ρ＝p|W><W|+q|GHZ><GHZ|， (20)
wherein p is2+q 21. For convenience of calculation, the present invention may let p be cos Θ and q be sin ΘEquation (20)
Can be rewritten as:
ρ＝cosΘ|W><W|+sinΘ|GHZ><GHZ|， (21)
in order to make the necessary condition of separability of mixed state three-body more complete and accurate, said invention can respectively calculate every angle between 0 and 90 deg.
In the process of constructing the optimal entanglement prover for the mixed state, angles theta from 1 to 89 are respectively calculated, and the invention is described by taking an example that the angle theta is not particularly 50 degrees for explaining the calculation process.
When Θ is 50 degrees, equation (21) can be written as:
ρ＝0.6428|W><W|+0.7660|GHZ><GHZ|， (22)
after determining the expression of the density matrix, the mixed state feature matrix may be solved, and the result of the mixed state feature matrix is as follows:
Figure BDA0002282087660000091
Figure BDA0002282087660000092
Figure BDA0002282087660000093
Figure BDA0002282087660000094
Figure BDA0002282087660000095
Figure BDA0002282087660000096
Figure BDA0002282087660000097
Figure BDA0002282087660000098
from the formula (23), the mixed state R can be seenijklIn which R is removed0000There are 71 non-0 elements in total, correspondingly, MijklIn the structure, 71 non-0 elements and R should be providedijklAnd (7) corresponding. To simplify the calculation, the present invention utilizes the rotational symmetry of the quantum states to reduce the unknown parameters to 14. The specific correspondence is as follows:
TABLE 1MiAnd MijklCorresponding relationship of
Figure BDA0002282087660000101
At setting MiThen carrying out the next derivation on the basis of thetaA、θB、φA、φBAnd traversing to obtain the maximum value of the lambda. The maximum value of Λ is calculated to be 2.3218 when Θ is 50 degrees, and the distribution is shown in fig. 1 and fig. 2.
As can be seen from fig. 1 and 2, if Λ takes the maximum value, the condition is:
(1)θA＝θBin the figure at thetaA＝θBThe value Λ may be a maximum value when the value is around 0 degrees, 88 degrees, 98 degrees, or 180 degrees.
(2)φA-φB0 or pi.
In addition, phi is only taken in the calculation
Figure BDA0002282087660000102
The maximum value of Λ is obtained only for integer multiples of Λ.
Similar to the calculation of Θ at 50 degrees, all angles between 0 and 90 degrees can be solved for the maximum value of Λ by this method. In the traversal process of theta, the condition that the maximum value is taken by the lambda, except that the theta angle is different, meets the condition that theta is satisfiedA＝θB、φA-φBCondition 0 or pi.
2. Matching entanglement prover computation
At maximum value Λ and characteristic matrix
Figure BDA0002282087660000111
All in certain cases, as long as the suitable one is found
Figure BDA0002282087660000112
Can make it possible to
Figure BDA0002282087660000113
The critical point of separable state and entangled state is found. The mixed state of the four quantum W state and the GHZ state in a noisy environment can be written as follows:
Figure BDA0002282087660000114
wherein p is the proportion of W state in the system. Bring it into
Figure BDA0002282087660000115
In the calculation of (a), it is possible to obtain:
Figure BDA0002282087660000116
on the basis of which the
Figure BDA0002282087660000117
Bringing in, can obtain:
Figure BDA0002282087660000118
the process of solving for the matching witnesses is therefore equivalent to solving for the minimum of p. The same is true for the four quantum GHZ state, and if the proportion of the GHZ state in the whole system is q, the process of solving the matching entanglement witness is the process of solving the minimum value of q.
The invention has been pointed out in the foregoing
Figure BDA0002282087660000119
By MijklDetermining, looking for
Figure BDA00022820876600001110
Is equivalent to finding the appropriate MijklAnd M isijklAlso according to the characteristic matrix
Figure BDA00022820876600001111
A set of parameters. For MijklThe solution can be solved in an iterative manner. Specifically, a group M is randomly givenijklThen let the group MijklFluctuating over a large range and counting M taken each timeijklA p is obtained, and after a certain number of fluctuations, the minimum p value in the calculation and a group of M corresponding to the minimum p value can be obtainedijklThen at this MijklThen, the set M is given a range slightly smaller than the last timeijklContinuing to fluctuate, the minimum p value in this calculation and its corresponding M can also be foundijklIn this way, a group of M can be found continuouslyijklP can be made to approach its minimum indefinitely. This method, while effective, is cumbersome and requires a large number of round robin operations. In the calculation process herein, 15000 to 25000 cycles are generally required to solve a p-value.
3. Construction of target quantum states using trisomy separable states with parameters
For a triplet separable quantum state, it can always be written as follows:
Figure BDA0002282087660000121
using the bloch representation method, for the first qubit | ψ>1The following can be rewritten:
Figure BDA0002282087660000122
for the second qubit, it can be rewritten as:
Figure BDA0002282087660000123
in the calculation process of the optimal entanglement witness, the invention finds that the maximum value of Λ is always the maximum value
Figure BDA0002282087660000124
Or pi, theta1＝θ2. With this result, the parameters in equations (28) and (29) can be simplified, reducing the four angles to two, namely:
Figure BDA0002282087660000125
for the third part of the two qubit entanglement in equations (4-19), it is written as follows:
Figure BDA0002282087660000126
in the formula (31), α, β and gamma are matrixes respectively
Figure BDA0002282087660000127
The feature vector corresponding to the maximum feature value of (1).
The combination of formula (30) and formula (31) gives a representation of the triplet separable quantum states under the bloch representation:
Figure BDA0002282087660000128
Figure BDA0002282087660000131
developed in a pair formula (32) at
Figure BDA0002282087660000132
The following can be obtained:
Figure BDA0002282087660000133
in that
Figure BDA0002282087660000134
In case of | ψ>The calculation process is similar to the above and is not described in detail here. As can be seen from equation (33), if ρ ═ ψ is calculated><Psi |, it can be found that the constitutional three-body separable quantum state includes not only the quantum state W state but also the GHZ state and Dickie state. And then, the target quantum state can be obtained by eliminating the unnecessary quantum state, and the corresponding noise margin is obtained.
Example (b):
1. three-body separability of four-quantum W state in noise environment
S1) calculation of optimal entanglement witness
For a four qubit W state, it is of the form:
Figure BDA0002282087660000141
through calculation, lambda is at thetaA＝θB、φA-φBThe maximum value is 2.021 in the case of 0 or pi, and the Λ distribution is shown in fig. 3 and 4.
As can be seen from fig. 3 and 4, the maximum Λ for the four quantum W states is 2.2021. From the angle of theta as long as theta is satisfiedA＝θBThe condition Λ can be taken to be the maximum value, and considering that θ is combined with the angle of φ, Λ can be taken to be the maximum value in the following four cases:
(1)θA＝θB＝0，φA-φBtaking any value;
(2)θA＝θB＝pi，φA-φBtaking any value;
(3)φA-φB＝0，θA＝θBtaking any value;
(4)φA-φB＝pi，θA＝θ:any value is taken.
S2) computation of matching entanglement witnesses
p is M obtained after calculation of four quanta W stateijklThe following were used:
TABLE 2M when the minimum value of the four quanta W state L is takenijklValue taking
M1 M2 M3 M4 M5 M6 M7
-1 -1 -1 -1 0.2743 0.3838 0.3838
M8 M9 M10 M11 M12 M13 M14
0.4933 -0.2325 -0.4861 -0.4038 -0.2125 -0.6374 -0.6374
The minimum value of p obtained by this factor was 0.247656.
S3) calculation of sufficiency
By calculation, at θ1Take 39 degrees and theta2Taking that 106 degrees satisfies the construction condition, the upper bound mixture coefficient p of the four-qubit W-state triplet separable under the noise environment is calculated to be 0.2458, in other words, if the proportion of the four-qubit W-state in the noise environment is less than 0.2458, then it must be triplet separable.
2. Three-body separability of four-quantum GHZ state in noise environment
S1) calculation of optimal entanglement witness
The maximum value distribution of Λ is shown in fig. 5 and 6.
S2) computation of matching entanglement witnesses
M obtained after calculation of four quanta W stateijklThe following were used:
TABLE 3M when the minimum value of the four-quantum GHZ state L is takenijklValue taking
M1 M2 M3 M4 M5 M6 M7
0 0 0 0 0 0 0
M8 M9 M10 M11 M12 M13 M14
0 0 1 0 -0.5 0.5 0.5
The minimum value of q was 0.2 as determined by this index.
S3) calculation of sufficiency
In the final calculation, the method selects theta as 63 degrees to construct, and obtains a mixed coefficient q of a four-quantum-bit GHZ state as 0.1991, namely: when the proportion of the GHZ state in the noise environment is less than 0.1991, the system must be three-body separable.
3. Three-body separability of mixed state of W state and GHZ state of four quanta bits in noise environment
The conclusion is shown in fig. 7.
For a mixed state of a W state and a GHZ state in a noise environment, the proportion of the W state in a system is defined as p, the proportion of the GHZ in the system is defined as q, a point (p, q) is within a blue line, the system is definitely three-body separable, and if the point (p, q) is in an area above a black point, the system is definitely entangled. Due to the presence of computational errors, it is impossible to determine whether the system is entangled or three-body separable if the point (p, q) falls in the region between the blue line and the black point.

Claims (1)

1. The invention designs a method for separating mixed state triples of a four-quantum-bit W state and a GHZ state in a noise environment, and the method can obtain sufficient conditions and necessary conditions for separating the mixed state triples of the four-quantum-bit W state and the GHZ state, thereby obtaining the noise tolerance;
a method for separating mixed state triples of a four-quantum-bit W state and a GHZ state in a noise environment comprises the following three processes:
s1) respectively constructing the optimal entanglement witnesses, searching respective matched entanglement witnesses on the basis, specifically analyzing different quantum states in the process of constructing the optimal entanglement witness, respectively solving parameters related to the construction process, then carrying out formula derivation and partial variable form replacement, and finally solving the optimal entanglement witness through MATLAB auxiliary calculation;
s2) finding a matching entanglement witness based on S1, the key to determining the matching entanglement witness is to find a suitable witness
Figure FDA0002282087650000011
In the calculation process, a progressive mode is adopted and MATLAB is utilized to carry out loop iteration, so that the optimal value is continuously reduced
Figure FDA0002282087650000012
Finally, a relatively accurate matching entanglement witness is solved and the lower bound of the entangled state is obtained;
s3) generalizing the three-body separable quantum state into a general form, that is, the quantum states in the form are all separable states, collecting and summarizing the characteristics and rules of the W state, the GHZ state and the mixed state of the four-quantum bit W state and the GHZ state in the process of constructing the entanglement witness, bringing the data into the general form of the three-body separable quantum state, and constructing the three-body separable upper bound of the quantum states as accurately as possible, and combining the lower bound of the entanglement state obtained in S2, the three-body separability criterion of the quantum states can be deduced.
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CN111814362A (en) * 2020-08-28 2020-10-23 腾讯科技（深圳）有限公司 Quantum noise process analysis method and system, storage medium and terminal equipment
CN112580811A (en) * 2020-11-06 2021-03-30 南京邮电大学 Polarization mixed entangled state generation method

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN111814362A (en) * 2020-08-28 2020-10-23 腾讯科技（深圳）有限公司 Quantum noise process analysis method and system, storage medium and terminal equipment
CN112580811A (en) * 2020-11-06 2021-03-30 南京邮电大学 Polarization mixed entangled state generation method
CN112580811B (en) * 2020-11-06 2024-04-16 南京邮电大学 Polarization mixed entanglement state generation method

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