CN111555876B - Combined cycle remote state preparation method based on non-maximum entangled channel N-party control - Google Patents

Combined cycle remote state preparation method based on non-maximum entangled channel N-party control Download PDF

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CN111555876B
CN111555876B CN202010413770.4A CN202010413770A CN111555876B CN 111555876 B CN111555876 B CN 111555876B CN 202010413770 A CN202010413770 A CN 202010413770A CN 111555876 B CN111555876 B CN 111555876B
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CN111555876A (en
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张佳慧
锁珍
姜敏
陈虹
刘芹
许智航
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Suzhou University
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    • H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
    • H04L9/08—Key distribution or management, e.g. generation, sharing or updating, of cryptographic keys or passwords
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
    • H04L9/08—Key distribution or management, e.g. generation, sharing or updating, of cryptographic keys or passwords
    • H04L9/0816—Key establishment, i.e. cryptographic processes or cryptographic protocols whereby a shared secret becomes available to two or more parties, for subsequent use
    • H04L9/0852—Quantum cryptography
    • H04L9/0855—Quantum cryptography involving additional nodes, e.g. quantum relays, repeaters, intermediate nodes or remote nodes
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
    • H04L9/08—Key distribution or management, e.g. generation, sharing or updating, of cryptographic keys or passwords
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Abstract

The invention discloses a combined cycle remote state preparation method based on non-maximum entangled channel N-side control, which comprises the following steps: at N controllers D1，D2…DnUnder the control of (3), combining every two of Alice, Bob and Charlie to circularly prepare the respective required target states; the whole scheme consists of Alice, Bob, Charlie, N control parties and a non-maximum entanglement channel. Under the monitoring of N controllers, the non-maximum entangled state is used as a channel, Alice and Bob are in a Charlie joint preparation target state, Alice and Charlie are in a Bob joint preparation target state, and Bob and Charlie are in an Alice joint preparation target state. The advantages of the invention are as follows: only when the controllers agree with each other, the target information can be transmitted, compared with a single controller, the introduction of N controllers effectively improves the safety of the scheme, and the performance requirement on the use of the channel is reduced by using the non-maximum entangled state as the channel.

Description

Combined cycle remote state preparation method based on non-maximum entangled channel N-party control
Technical Field
The invention relates to the field of quantum state preparation, in particular to a combined cycle remote state preparation method based on non-maximum entangled channel N-party control.
Background
Quantum communication is an important research area in quantum informatics. The unit for quantum communication to transmit information is in a quantum state, and in the communication process, besides a traditional classical channel, a quantum channel between communication parties needs to be established, namely quantum entanglement between the communication parties. Quantum entanglement plays a significant role in quantum communication and quantum computation, and continuously creates miraculous traces in the classical information theory, such as quantum dense coding, quantum invisible state transfer, absolute safe quantum cryptography and the like.
Quantum remote state preparation (remote state preparation) is considered an important component of quantum communication. Since the quantum remote preparation protocol proposed by h.k.lo, attention has been paid to various fields. Quantum remote state fabrication was originally a solution to deliver pure quantum states using pre-shared quantum entanglement resources and some classical communication. RSP transfers a known quantum state between sender and receiver, Bob retrieves and restores the target state by performing unitary operations. So far, with a large number of related protocols proposed, quantum remote state fabrication has also been developed more deeply. In RSP, there is only one sender and receiver, and the sender can obtain all information of the preparation state, so that information leakage occurs. In order to improve the safety of quantum State Preparation, quantum associated Remote State Preparation (Joint Remote State Preparation) is proposed.
In JRSP, two or more senders share the state to be prepared, each sender possesses only partial information, and the receiver has no information about the state. When all senders collaborate, the receiver can restore the target state by performing some operation on the particles in his hand. For example, in 2007, Xiyan et al first implemented a JRSP scheme with multiple parties collaborating to prepare known quantum states. Since then, many kinds of JRSP scheme studies have been proposed in succession, such as joint remote state preparation of single particle state, Bell state, three particle GHZ state, four particle W state, etc. At present, joint preparation is basically performed in a scene of a maximum entangled channel, and actually, the maximum entangled channel is easily interfered by the environment and becomes a non-maximum entangled channel. The maximum entangled channel in an ideal environment has no generality.
Disclosure of Invention
The invention aims to provide a combined cycle remote preparation method based on non-maximum entangled channel N-party control.
In order to solve the technical problem, the invention provides a method for combined cycle remote state preparation based on non-maximum entangled channel N-party control. The method comprises the following steps: combining Alice, Bob and Charlie in pairs to circularly prepare the required target states; the whole scheme consists of Alice, Bob, Charlie, N control parties and a non-maximum entanglement channel. Under the monitoring of N controllers, a non-maximum entangled state is used as a channel, Alice and Bob are in a Charlie joint preparation target state, Alice and Charlie are in a Bob joint preparation target state, and Bob and Charlie are in an Alice joint preparation target state; the method comprises the following steps:
step 1: the seven-bit quantum entanglement channel comprises the following specific steps:
Figure BDA0002494275170000021
the H transform is performed on particle 7 while introducing N-1 particles, and the channel becomes the following form:
Figure BDA0002494275170000022
to pair
Figure BDA0002494275170000023
Particles of (2)
Figure BDA0002494275170000024
Performing a CNOT operation, and combining the whole channels into:
Figure BDA0002494275170000031
wherein the real coefficient satisfies | α2+|β|2＝1(|α|≥|β|)，|λ|2+|δ|2＝1(|λ|≥|δ|)，|μ|2+|ρ|2＝1(|μ|≥|ρ|)。
Step 2: the target states are as follows:
alice and Charlie want to prepare an arbitrary quantum state for Bob, in the form:
Figure BDA0002494275170000032
wherein α and b are amplitude coefficients and satisfy a normalization condition | a2+|b|2＝1，0≤θ1＜2π。
Bob and Alice want to prepare an arbitrary quantum state for Charlie, in the form:
Figure BDA0002494275170000033
wherein c and d are amplitude coefficients and satisfy a normalization condition | c2+|d|2＝1，0≤θ2＜2π。
Charlie and Bob want to prepare an arbitrary quantum state for Alice, in the form:
Figure BDA0002494275170000034
wherein f and g are amplitude coefficients and satisfy a normalization condition | f2+|g|2＝1，0≤θ2＜2π。
The expression of the non-maximally entangled channel shared by Alice, Bob, Charlie, the controller is as follows:
Figure BDA0002494275170000035
wherein the real coefficient satisfies | α2+|β|2＝1(|α|≥|β|)，|λ|2+|δ|2＝1(|λ|≥|δ|)，|μ|2+|ρ|2＝1(|μ|≥|ρ|)。
Alice has particles 1,2, Bob has particles 3,4, Charlie has particles 5,6, D1,D2....DnAre control particles.
Respectively introducing auxiliary particles into Alice, Bob and Charlie
Figure BDA0002494275170000036
And to particle pair (1, omega)1),(3,ω2) And (5, ω)3) Implementing CNOT operations, the system expression is as follows:
Figure BDA0002494275170000041
and step 3: POVM measurement and phase measurement are specifically as follows:
alice, Bob, Charlie each separate the particle ω1，ω2，ω3Sending the measurement result to Charlie, Alice and Bob, and simultaneously Alice, Bob and Charlie select proper measurement bases to respectively carry out POVM measurement on the particles 2,4 and 6, and according to the measurement result, Charlie, Alice and Bob carry out the POVM measurement on the particles omega1，ω2，ω3Performing the measurement;
and 4, step 4: control measurement and target state preparation
For the N controllers, the target states can only be jointly prepared if they all agree on the communication between Alice, Bob and Charlie. Therefore, if preparation is agreed, each control method needs to perform a single bit measurement, whose measurement basis is { | + >, | - >, expressed as follows:
Figure BDA0002494275170000042
Figure BDA0002494275170000043
the final system is written as:
Figure BDA0002494275170000044
alice, Bob, Charlie obtained 8 combinations of measurements, respectively. Wherein, according to the measurement results of N control parties, Charlie, Bob and Alice will receive different measurement informationAccording to the respective measurement result, executing corresponding unitary operation I ═ 0><0|+|1><1|，σX＝|0><1|+|1><0|，σz＝|0><0|-|1><1|，σy＝i(|0><1|-|1><0|) to recover the target state.
In step 3, taking Alice as an example, the following details are provided:
alice and Charlie for particles 2 and ω, respectively3The measurement is performed.
Alice selects a set of measurement bases Mi>(ii) a POVM measurements were performed for particle 2 by i ∈ (0,1) }:
Figure BDA0002494275170000051
Figure BDA0002494275170000052
the collapse state of the measurement result is | M0>Or | M1>The following are:
get
Figure BDA0002494275170000053
The optimal POVM measurement basis (given by the matrix) is as follows:
Figure BDA0002494275170000054
where I is the identity matrix and the value of x ensures P2The non-negative operator is a non-negative operator,
Figure BDA0002494275170000055
Figure BDA0002494275170000056
when the measurement result of the particle 2 is E0When the state of the particle 2 is | M0>(ii) a Alice publishes the successful result of the measurement, and Charlie selects a proper measurement basis to measure the particle omega3Measurement is based on：
Figure BDA0002494275170000057
Figure BDA0002494275170000058
When the measurement result of the particle 2 is E1When the state of the particle 2 is | M1>(ii) a Alice publishes the successful result of the measurement, and Charlie selects a proper measurement basis to measure the particle omega3Measurement, the measurement base is:
Figure BDA0002494275170000059
Figure BDA00024942751700000510
when the measurement result of the particle 2 is E2When this happens, no inference can be made and the measurement fails.
Bob and Alice for particles 4 and ω, respectively1The measurement is performed.
Bob selects a set of measurement bases { | μi>(ii) a POVM measurement is performed for particle 4 by i ∈ (0,1) }:
Figure BDA0002494275170000061
Figure BDA0002494275170000062
the collapse state of the measurement result is | mu0>Or | μ1>The following are:
get
Figure BDA0002494275170000063
Optimal POVM measurement basis (moment of inertia)The matrix is given) as follows:
Figure BDA0002494275170000064
wherein I is a unit matrix, and the value of x is ensured to be P'2The non-negative operator is a non-negative operator,
Figure BDA0002494275170000065
Figure BDA0002494275170000066
when the measurement result of particle 4 is E'0When the state of the particle 4 is | μ0>. Bob publishes the successful result of the measurement, and Alice selects a proper measurement basis to measure the particle omega1Measurement, the measurement base is:
Figure BDA0002494275170000067
Figure BDA0002494275170000068
when the measurement result of particle 4 is E'1When the state of the particle 4 is | μ1>. Bob publishes the successful result of the measurement, and Alice selects a proper measurement basis to measure the particle omega1Measurement, the measurement base is:
Figure BDA0002494275170000069
Figure BDA00024942751700000610
when the measurement result of particle 4 is E'2When this happens, no inference can be made and the measurement fails.
Charlie and Bob for particle 6 and omega, respectively2The measurement is performed.
Charlie selects a set of measurement bases { | Tj>(j ═ 0,1) } POVM measurement was performed on the particles 6:
Figure BDA0002494275170000071
Figure BDA0002494275170000072
the collapse state of the measurement result is | T0>Or | T1>The following are:
get
Figure BDA0002494275170000073
The optimal POVM measurement basis (given by the matrix) is as follows:
Figure BDA0002494275170000074
wherein I is an identity matrix and the value of x is guaranteed to be P'2The non-negative operator is a non-negative operator,
Figure BDA0002494275170000075
Figure BDA0002494275170000076
when the measurement result of the particles 6 is E "0When the state of the particle 6 is | T0>(ii) a Charlie discloses the results of successful measurements, Bob selects the appropriate measurement basis for the particle omega2The measurement was performed on the basis of:
Figure BDA0002494275170000077
Figure BDA0002494275170000078
when the measurement result of the particles 6 is E "1When the state of the particle 6 is | T1>(ii) a Charlie discloses the results of successful measurements, Bob selects the appropriate measurement basis for the particle omega2The measurement was performed on the basis of:
Figure BDA0002494275170000079
Figure BDA00024942751700000710
when the measurement result of the particles 6 is E "2When the measurement fails, no inference can be made
In step 4, the controller measures the following specifically:
according to the measurement results of Alice, Bob and Charlie, if the controllers agree with the preparation among the controllers, each controller selects the measurement bases { | + >, | - >) to perform single particle measurement:
Figure BDA0002494275170000081
Figure BDA0002494275170000082
alice, Bob, Charlie respectively obtain 8 combinations of measurement results, wherein 4 combinations of results are invalid combinations, and the unitary operation corresponding to the specific measurement result is shown in table 1. According to the measurement results of the N control parties, Charlie, Bob and Alice receive different measurement information, and execute corresponding unitary operation I ═ 0 according to the respective measurement results><0|+|1><1|，σX＝|0><1|+|1><0|，σz＝|0><0|-|1><1|，σy＝i(|0><1|-|1><0|) to recover the target state. "is specifically shown in Table 1:
Figure BDA0002494275170000083
TABLE 1 measurement results and recovery of performed unitary operations for Bob, Charlie, Alice
As shown in Table 1, if Alice and Charlie have the measurement results of
Figure BDA0002494275170000084
Bob receives measurements that depend on N controlling parties. If Bob receives m ' 0's and (n-m) 1's, the particle measurements of the controlling party are analyzed as follows:
Figure BDA0002494275170000085
Figure BDA0002494275170000086
wherein, |0>nAlways has a positive expansion, |1>nThe expansion of (1) has positive and negative. To |1>nThe following discussion is expanded:
in the first case: if m is an even number, n is an odd number, or n is an even number, m is an odd number;
|1>nhas a negative expansion, Bob receives the measurement result of the control party as (| 0)>-|1>) I.e. | ->Bob obtains the information about the target state as
Figure BDA0002494275170000091
At this time, Bob only needs to execute on the obtained state
Figure BDA0002494275170000092
The target state can be obtained by operation.
In the second case: if m and n are both even numbers, or m and n are both odd numbers;
|1>nhas a positive expansion, Bob receives the measurement result of the control party as (| 0)>+|1>) I is | +>Bob obtains the information about the target state as
Figure BDA0002494275170000093
At this time, the information obtained by Bob is the target state.
In addition, the analysis process of Charlie and Alice for obtaining measurement information and recovering the target state is the same as Bob.
In step 3, the POVM measurement success probability analysis of the particle 2 is as follows:
POVM measurement result is E0Probability of (c):
Figure BDA0002494275170000094
measurement result is E1Probability of (c):
Figure BDA0002494275170000095
measurement result is E2Probability of (c):
Figure BDA0002494275170000096
it follows therefrom that the probability of success of the POVM measurement of the particle 2 is
Figure BDA0002494275170000097
The probability of failure is
Figure BDA0002494275170000098
In step 3, the probability analysis of successful POVM measurement of the particles 4 is as follows:
POVM measurement result is E'0Probability of (c):
Figure BDA0002494275170000099
measurement result is E1The probability of `:
Figure BDA00024942751700000910
measurement result was E'2Probability of (c):
Figure BDA00024942751700000911
it follows therefrom that the probability of success of the POVM measurement of the particle 4 is
Figure BDA00024942751700000912
The probability of failure is
Figure BDA00024942751700000913
In step 3, the POVM measurement success probability analysis of the particles 6 is as follows:
POVM measurement result is E "0Probability of (c):
Figure BDA0002494275170000101
measurement result is E "1Probability of (c):
Figure BDA0002494275170000102
measurement result is E "2Probability of (c):
Figure BDA0002494275170000103
it follows that the probability of success of the POVM measurement of the particle 6 is
Figure BDA0002494275170000104
The probability of failure is
Figure BDA0002494275170000105
The invention has the beneficial effects that:
1. according to the invention, three-party communication can be smoothly carried out only under the condition that all the control parties agree, and compared with a single control party, the introduction of N control parties effectively improves the safety of the scheme.
2. According to the invention, the target state is prepared in a three-way pairwise combined cycle manner, so that the utilization rate of the channel and the preparation efficiency are effectively improved.
3. The invention breaks the limitation of the prior combined remote state preparation method under the maximum entangled channel and populates the combined remote state preparation to the scene of the non-maximum entangled channel.
Drawings
FIG. 1 is a flow chart of the method for preparing the combined cycle remote state based on the control of the non-maximum entangled channel N side.
FIG. 2 is a quantum channel schematic diagram of the combined cycle remote state preparation method based on non-maximum entanglement channel N-side control.
Detailed Description
The present invention is further described below in conjunction with the following figures and specific examples so that those skilled in the art may better understand the present invention and practice it, but the examples are not intended to limit the present invention.
The technical terms of the invention explain:
1. pauli array
Some unitary matrices, Pauli matrices, are used in the present invention. The specific form is as follows:
Figure BDA0002494275170000111
Figure BDA0002494275170000112
Figure BDA0002494275170000113
Figure BDA0002494275170000114
Figure BDA0002494275170000115
2. CNOT operations
The CNOT operation is a control not gate operation, and the not gate operation is a two-qubit logic gate, namely a control bit and a target bit. When the control bit is |0>, the target bit state is unchanged; when the control bit is |1>, the plaque bit state is inverted. The bit form that the CNOT operation contributes to the qubit is as follows:
Figure BDA0002494275170000116
referring to fig. 1 and 2, under the control of N, Alice, Bob, Charlie perform pairwise combined cycle preparation of the target state, including the following steps: the method comprises the following steps: combining Alice, Bob and Charlie in pairs to circularly prepare the required target states; the whole scheme consists of Alice, Bob, Charlie, N control parties and a non-maximum entanglement channel. Under the monitoring of N controllers, the non-maximum entangled state is used as a channel, Alice and Bob are in a Charlie joint preparation target state, Alice and Charlie are in a Bob joint preparation target state, and Bob and Charlie are in an Alice joint preparation target state. Therefore, the preparation efficiency can be effectively improved, the utilization rate of channels can be increased, and the complete process comprises the following steps:
step 1: the seven-bit quantum entanglement channel comprises the following specific steps:
Figure BDA0002494275170000121
the H transform is performed on particle 7 while introducing N-1 particles, and the channel becomes the following form:
Figure BDA0002494275170000122
to pair
Figure BDA0002494275170000123
Particles of (2)
Figure BDA0002494275170000124
Performing a CNOT operation, and combining the whole channels into:
Figure BDA0002494275170000125
wherein the real coefficient satisfies | α2+|β|2＝1(|α|≥|β|)，|λ|2+|δ|2＝1(|λ|≥|δ|)，|μ|2+|ρ|2＝1(|μ|≥|ρ|)。
Step 2: the target states are as follows:
alice and Charlie want to prepare an arbitrary quantum state for Bob, in the form:
Figure BDA0002494275170000126
wherein α and b are amplitude coefficients and satisfy a normalization condition | a2+|b|2＝1，0≤θ1＜2π。
Bob and Alice want to prepare an arbitrary quantum state for Charlie, in the form:
Figure BDA0002494275170000127
wherein c and d are amplitude coefficients and satisfy a normalization condition | c2+|d|2＝1，0≤θ2＜2π。
Charlie and Bob want to prepare an arbitrary quantum state for Alice, in the form:
Figure BDA0002494275170000128
wherein f and g are amplitude coefficients and satisfy a normalization condition | f2+|g|2＝1，0≤θ2＜2π。
The expression of the non-maximally entangled channel shared by Alice, Bob, Charlie, the controller is as follows:
Figure BDA0002494275170000129
wherein the real coefficient satisfies | α2+|β|2＝1(|α|≥|β|)，|λ|2+|δ|2＝1(|λ|≥|δ|)，|μ|2+|ρ|2＝1(|μ|≥|ρ|)。
Alice has particles 1,2, Bob has particles 3,4, Charlie has particles 5,6, D1,D2....DnAre control particles.
Respectively introducing auxiliary particles into Alice, Bob and Charlie
Figure BDA0002494275170000131
And to particle pair (1, omega)1),(3,ω2) And (5, ω)3) Implementing CNOT operations, the system expression is as follows:
Figure BDA0002494275170000132
and step 3: POVM measurement and phase measurement are specifically as follows:
alice, Bob, Charlie each separate the particle ω1，ω2，ω3Sending the measurement result to Charlie, Alice and Bob, and simultaneously Alice, Bob and Charlie select proper measurement bases to respectively carry out POVM measurement on the particles 2,4 and 6, and according to the measurement result of the amplitude, Charlie, Alice and Bob carry out the measurement on the particles omega1，ω2，ω3Performing the measurement;
and 4, step 4: control measurement and target state preparation
For the N controllers, the target states can only be jointly prepared if they all agree on the communication between Alice, Bob and Charlie. Therefore, if preparation is agreed, each control method needs to perform a single bit measurement, whose measurement basis is { | + >, | - >, expressed as follows:
Figure BDA0002494275170000133
Figure BDA0002494275170000134
the final system is written as:
Figure BDA0002494275170000141
alice, Bob, Charlie obtained 8 combinations of measurements, respectively. According to the measurement results of the N control parties, Charlie, Bob and Alice receive different measurement information, and execute a corresponding unitary operation i ═ 0 according to the respective measurement results><0|+|1><1|，σX＝|0><1|+|1><0|，σz＝|0><0|-|1><1|，σy＝i(|0><1|-|1><0|) to recover the target state.
In step 3, taking Alice as an example, the following details are provided:
alice and Charlie for particles 2 and ω, respectively3The measurement is performed.
Alice selects a set of measurement bases Mi>(ii) a POVM measurements were performed for particle 2 by i ∈ (0,1) }:
Figure BDA0002494275170000142
Figure BDA0002494275170000143
the collapse state of the measurement result is | M0>Or | M1>The following are:
get
Figure BDA0002494275170000144
The optimal POVM measurement basis (given by the matrix) is as follows:
Figure BDA0002494275170000145
where I is the identity matrix and the value of x ensures P2The non-negative operator is a non-negative operator,
Figure BDA0002494275170000146
Figure BDA0002494275170000147
when the measurement result of the particle 2 is E0When the state of the particle 2 is | M0>(ii) a Alice publishes the successful result of the measurement, and Charlie selects a proper measurement basis to measure the particle omega3Measurement, the measurement base is:
Figure BDA0002494275170000151
Figure BDA0002494275170000152
when the measurement result of the particle 2 is E1When the state of the particle 2 is | M1>(ii) a Alice publishes the successful result of the measurement, and Charlie selects a proper measurement basis to measure the particle omega3Measurement, the measurement base is:
Figure BDA0002494275170000153
Figure BDA0002494275170000154
when the measurement result of the particle 2 is E2When this happens, no inference can be made and the measurement fails.
Bob and Alice for particles 4 and ω, respectively1The measurement is performed.
Bob selects a set of measurement bases { | μi>(ii) a POVM measurement is performed for particle 4 by i ∈ (0,1) }:
Figure BDA0002494275170000155
Figure BDA0002494275170000156
the collapse state of the measurement result is | mu0>Or | μ1>The following are:
get
Figure BDA0002494275170000157
The optimal POVM measurement basis (given by the matrix) is as follows:
Figure BDA0002494275170000158
wherein I is a unit matrix, and the value of x is ensured to be P'2The non-negative operator is a non-negative operator,
Figure BDA0002494275170000159
Figure BDA00024942751700001510
when the measurement result of particle 4 is E'0When the state of the particle 4 is | μ0>. Bob publishes the successful result of the measurement, and Alice selects a proper measurement basis to measure the particle omega1Measurement, the measurement base is:
Figure BDA0002494275170000161
Figure BDA0002494275170000162
when the measurement result of particle 4 is E'1When the state of the particle 4 is | μ1>. Bob publishes the successful result of the measurement, and Alice selects a proper measurement basis to measure the particle omega1Measurement, the measurement base is:
Figure BDA0002494275170000163
Figure BDA0002494275170000164
when the measurement result of particle 4 is E'2When this happens, no inference can be made and the measurement fails.
Charlie and Bob for particle 6 and omega, respectively2The measurement is performed.
Charlie selects a set of measurement bases { | Tj>(j ═ 0,1) } POVM measurement was performed on the particles 6:
Figure BDA0002494275170000165
Figure BDA0002494275170000166
the collapse state of the measurement result is | T0>Or | T1>The following are:
get
Figure BDA0002494275170000167
The optimal POVM measurement basis (given by the matrix) is as follows:
Figure BDA0002494275170000168
wherein I is an identity matrix and the value of x is guaranteed to be P'2The non-negative operator is a non-negative operator,
Figure BDA0002494275170000169
Figure BDA00024942751700001610
when the measurement result of the particles 6 is E "0When the state of the particle 6 is | T0>(ii) a Charlie discloses the results of successful measurements, Bob selects the appropriate measurement basis for the particle omega2The measurement was performed on the basis of:
Figure BDA0002494275170000171
Figure BDA0002494275170000172
when the measurement result of the particles 6 is E "1When the state of the particle 6 is | T1>(ii) a Charlie discloses the results of successful measurements, Bob selects the appropriate measurement basis for the particle omega2The measurement was performed on the basis of:
Figure BDA0002494275170000173
Figure BDA0002494275170000174
when the measurement result of the particles 6 is E "2When the measurement fails, no inference can be made
In step 4, the controller measures the following specifically:
according to the measurement results of Alice, Bob and Charlie, if the controllers agree with the preparation among the controllers, each controller selects the measurement bases { | + >, | - >) to perform single particle measurement:
Figure BDA0002494275170000175
Figure BDA0002494275170000176
alice, Bob, Charlie respectively obtain 8 combinations of measurement results, wherein 4 combinations of results are invalid combinations, and the unitary operation corresponding to the specific measurement result is shown in table 1. According to the measurement results of the N control parties, Charlie, Bob and Alice receive different measurement information,executing corresponding unitary operation I ═ 0 according to the obtained measuring result><0|+|1><1|，σX＝|0><1|+|1><0|，σz＝|0><0|-|1><1|，σy＝i(|0><1|-|1><0|) to recover the target state. "is specifically shown in Table 1:
Figure BDA0002494275170000181
TABLE 1 measurement results and recovery of performed unitary operations for Bob, Charlie, Alice
As shown in Table 1, if Alice and Charlie have the measurement results of
Figure BDA0002494275170000182
Bob receives measurements that depend on N controlling parties. If Bob receives m ' 0's and (n-m) 1's, the particle measurements of the controlling party are analyzed as follows:
Figure BDA0002494275170000183
Figure BDA0002494275170000184
wherein, |0>nAlways has a positive expansion, |1>nThe expansion of (1) has positive and negative. To |1>nThe following discussion is expanded:
in the first case: if m is an even number, n is an odd number, or n is an even number, m is an odd number;
|1>nhas a negative expansion, Bob receives the measurement result of the control party as (| 0)>-|1>) I.e. | ->Bob obtains the information about the target state as
Figure BDA0002494275170000185
At this time, Bob only needs to execute on the obtained state
Figure BDA0002494275170000186
The target state can be obtained by operation.
In the second case: if m and n are both even numbers, or m and n are both odd numbers;
|1>nhas a positive expansion, Bob receives the measurement result of the control party as (| 0)>+|1>) I is | +>Bob obtains the information about the target state as
Figure BDA0002494275170000191
At this time, the information obtained by Bob is the target state.
In addition, the analysis process of Charlie and Alice for obtaining measurement information and recovering the target state is the same as Bob.
In step 3, the POVM measurement success probability analysis of the particle 2 is as follows:
POVM measurement result is E0Probability of (c):
Figure BDA0002494275170000192
measurement result is E1Probability of (c):
Figure BDA0002494275170000193
measurement result is E2Probability of (c):
Figure BDA0002494275170000194
it follows therefrom that the probability of success of the POVM measurement of the particle 2 is
Figure BDA0002494275170000195
The probability of failure is
Figure BDA0002494275170000196
In step 3, the probability analysis of successful POVM measurement of the particles 4 is as follows:
POVM measurement result is E'0Probability of (c):
Figure BDA0002494275170000197
measurement result is E1The probability of `:
Figure BDA0002494275170000198
measurement result was E'2Probability of (c):
Figure BDA0002494275170000199
it follows therefrom that the probability of success of the POVM measurement of the particle 4 is
Figure BDA00024942751700001910
The probability of failure is
Figure BDA00024942751700001911
In step 3, the POVM measurement success probability analysis of the particles 6 is as follows:
POVM measurement result is E "0Probability of (c):
Figure BDA00024942751700001912
measurement result is E "1Probability of (c):
Figure BDA00024942751700001913
measurement result is E "2Probability of (c):
Figure BDA00024942751700001914
it follows that the probability of success of the POVM measurement of the particle 6 is
Figure BDA00024942751700001915
The probability of failure is
Figure BDA00024942751700001916
A specific application scenario of the present invention is given below:
the first embodiment is as follows: under the monitoring of ten control parties, Alice and Bob jointly prepare for Charlie
Figure BDA00024942751700001917
Bob andalice federation for Charlie preparation
Figure BDA00024942751700001918
Charlie and Bob combine for Alice preparation
Figure BDA0002494275170000201
The complete process comprises the following steps:
step 1: the seven-bit quantum entanglement channel comprises the following specific steps:
Figure BDA0002494275170000202
the H transform is performed on particle 7 while introducing 9 particles, and the channel becomes the following form:
Figure BDA0002494275170000203
to pair
Figure BDA0002494275170000204
Particles of (2)
Figure BDA0002494275170000205
Performing a CNOT operation, and combining the whole channels into:
Figure BDA0002494275170000206
step 2: the target states are as follows:
alice and Charlie want to prepare an arbitrary quantum state for Bob, in the form:
Figure BDA0002494275170000207
bob and Alice want to prepare an arbitrary quantum state for Charlie, in the form:
Figure BDA0002494275170000208
charlie and Bob want to prepare an arbitrary quantum state for Alice, in the form:
Figure BDA0002494275170000209
the expression form of quantum channels shared by Alice, Bob, Charlie, the controller is as follows:
Figure BDA0002494275170000211
alice has particles 1,2, Bob has particles 3,4, Charlie has particles 5,6, D1,D2....D10Are control particles.
Respectively introducing auxiliary particles into Alice, Bob and Charlie
Figure BDA0002494275170000212
And to particle pair (1, omega)1),(3,ω2) And (5, ω)3) Implementing CNOT operations, the system expression is as follows:
Figure BDA0002494275170000213
and step 3: POVM measurement and phase measurement are specifically as follows:
alice, Bob, Charlie each separate the particle ω1，ω2，ω3Sending the measurement result to Charlie, Alice and Bob, and simultaneously Alice, Bob and Charlie select proper measurement bases to respectively carry out POVM measurement on the particles 2,4 and 6, and according to the measurement result of the amplitude, Charlie, Alice and Bob carry out the measurement on the particles omega1，ω2，ω3Performing the measurement;
alice and Charlie for particles 2 and ω, respectively3The measurement is performed.
Alice selects a set of measurement bases Mi>；i∈(0,1) } POVM measurement on particle 2:
Figure BDA0002494275170000214
Figure BDA0002494275170000215
the collapse state of the measurement result is | M0>Or | M1>The following are:
get
Figure BDA0002494275170000216
The optimal POVM measurement basis (given by the matrix) is as follows:
Figure BDA0002494275170000221
where I is the identity matrix and the value of x ensures P2The non-negative operator is a non-negative operator,
Figure BDA0002494275170000222
Figure BDA0002494275170000223
when the measurement result of the particle 2 is E0When the state of the particle 2 is | M0>(ii) a Alice publishes the successful result of the measurement, and Charlie selects a proper measurement basis to measure the particle omega3Measurement, the measurement base is:
Figure BDA0002494275170000224
Figure BDA0002494275170000225
when the measurement result of the particle 2 is E1When the state of the particle 2 is | M1>；AliceThe successful result of the measurement is published, and Charlie selects a proper measurement base for the particle omega3Measurement, the measurement base is:
Figure BDA0002494275170000226
Figure BDA0002494275170000227
when the measurement result of the particle 2 is E2When this happens, no inference can be made and the measurement fails.
Bob and Alice for particles 4 and ω, respectively1The measurement is performed.
Bob selects a set of measurement bases { | μi>(ii) a POVM measurement is performed for particle 4 by i ∈ (0,1) }:
Figure BDA0002494275170000228
Figure BDA0002494275170000229
the collapse state of the measurement result is | mu0>Or | μ1>The following are:
get
Figure BDA00024942751700002210
The optimal POVM measurement basis (given by the matrix) is as follows:
Figure BDA0002494275170000231
wherein I is a unit matrix, and the value of x is ensured to be P'2The non-negative operator is a non-negative operator,
Figure BDA0002494275170000232
Figure BDA0002494275170000233
when the measurement result of particle 4 is E'0When the state of the particle 4 is | μ0>(ii) a Bob publishes the successful result of the measurement, and Alice selects a proper measurement basis to measure the particle omega1Measurement, the measurement base is:
Figure BDA0002494275170000234
Figure BDA0002494275170000235
when the measurement result of particle 4 is E'1When the state of the particle 4 is | μ1>(ii) a Bob publishes the successful result of the measurement, and Alice selects a proper measurement basis to measure the particle omega1Measurement, the measurement base is:
Figure BDA0002494275170000236
Figure BDA0002494275170000237
when the measurement result of particle 4 is E'2When this happens, no inference can be made and the measurement fails.
Charlie and Bob for particle 6 and omega, respectively2The measurement is performed.
Charlie selects a set of measurement bases { | Tj>(j ═ 0,1) } POVM measurement was performed on the particles 6:
Figure BDA0002494275170000238
Figure BDA0002494275170000239
measuring the result ofThe collapsed state is | T0>Or | T1>The following are:
get
Figure BDA00024942751700002310
The optimal POVM measurement basis (given by the matrix) is as follows:
Figure BDA0002494275170000241
wherein I is an identity matrix and the value of x is guaranteed to be P'2The non-negative operator is a non-negative operator,
Figure BDA0002494275170000242
Figure BDA0002494275170000243
when the measurement result of the particles 6 is E "0When the state of the particle 6 is | T0>(ii) a Charlie discloses the results of successful measurements, Bob selects the appropriate measurement basis for the particle omega2The measurement was performed on the basis of:
Figure BDA0002494275170000244
Figure BDA0002494275170000245
when the measurement result of the particles 6 is E "1When the state of the particle 6 is | T1>(ii) a Charlie discloses the results of successful measurements, Bob selects the appropriate measurement basis for the particle omega2The measurement was performed on the basis of:
Figure BDA0002494275170000246
Figure BDA0002494275170000247
when the measurement result of the particles 6 is E "2When the measurement fails, no inference can be made
Control of the control measures
According to the measurement results of Alice, Bob and Charlie, if the controllers agree with the preparation among the controllers, the single particle measurement is carried out on the measurement basis { | + >, | - >) selected by each controller:
Figure BDA0002494275170000248
Figure BDA0002494275170000249
after measurement, the whole system form is as follows:
Figure BDA0002494275170000251
if Alice and Charlie have the measurement results as
Figure BDA0002494275170000252
Bob receives measurements that depend on 10 controlling parties, and if Bob receives particle measurements for m ' 0 ' and (10-m) 1 ' controlling parties, the following:
Figure BDA0002494275170000253
Figure BDA0002494275170000254
at this time, for |1>10The following discussion is expanded:
in the first case: bob receives an odd number of '0';
|1>nhas an expansion of negative, Bob receives controlThe measurement result of the square is (| 0)>-|1>) I.e. | ->Bob obtains the information about the target state as
Figure BDA0002494275170000255
At this time, Bob only needs to execute on the obtained state
Figure BDA0002494275170000257
The target state can be obtained by operation.
In the second case: bob receives an even number of '0';
|1>nhas a positive expansion, Bob receives the measurement result of the control party as (| 0)>+|1>) I is | +>Bob obtains the information about the target state as
Figure BDA0002494275170000256
At this time, the information obtained by Bob is the target state.
In addition, the analysis process of Charlie and Alice for obtaining measurement information and recovering the target state is the same as Bob.
The above-mentioned embodiments are merely preferred embodiments for fully illustrating the present invention, and the scope of the present invention is not limited thereto. The equivalent substitution or change made by the technical personnel in the technical field on the basis of the invention is all within the protection scope of the invention. The protection scope of the invention is subject to the claims.

Claims (4)

1. A combined cycle remote state preparation method based on non-maximum entanglement channel N-side control is characterized by comprising the following steps: combining Alice, Bob and Charlie in pairs to circularly prepare the required target states; the whole scheme consists of Alice, Bob, Charlie, N control parties and a non-maximum entanglement channel; under the monitoring of N controllers, a non-maximum entangled state is used as a channel, Alice and Bob are in a Charlie joint preparation target state, Alice and Charlie are in a Bob joint preparation target state, and Bob and Charlie are in an Alice joint preparation target state;
the method specifically comprises the following steps:
step 1: the seven-bit quantum channel specifically comprises the following components:
Figure FDA0003146069240000011
the H transform is performed on particle 7 while introducing N-1 particles, and the channel becomes the following form:
Figure FDA0003146069240000012
to pair
Figure FDA0003146069240000013
Particles of (2)
Figure FDA0003146069240000014
Performing a CNOT operation, and combining the whole channels into:
Figure FDA0003146069240000015
wherein the real coefficient satisfies | α2+|β|21, where | α | ≧ β |, | λ |,/y |, |2+|δ|21, where | λ | ≧ δ |, | μ |, | y |, y2+|ρ|21, where | μ | ≧ | ρ |;
step 2: the target states are as follows:
alice and Charlie want to prepare an arbitrary quantum state for Bob, in the form:
Figure FDA0003146069240000016
wherein α and b are amplitude coefficients and satisfy a normalization condition | a2+|b|2＝1，0≤θ1＜2π；
Bob and Alice want to prepare an arbitrary quantum state for Charlie, in the form:
Figure FDA0003146069240000021
wherein c and d are amplitude coefficients and satisfy a normalization condition | c2+|d|2＝1，0≤θ2＜2π；
Charlie and Bob want to prepare an arbitrary quantum state for Alice, in the form:
Figure FDA0003146069240000022
wherein f and g are amplitude coefficients and satisfy a normalization condition | f2+|g|2＝1，0≤θ2＜2π；
The expression of the non-maximally entangled channels shared by Alice, Bob, Charlie, the controller is as follows:
Figure FDA0003146069240000023
wherein the real coefficient satisfies | α2+|β|21, where | α | ≧ β |, | λ |,/y |, |2+|δ|21, where | λ | ≧ δ |, | μ |, | y |, y2+|ρ|21, where | μ | ≧ | ρ |;
alice has particles 1,2, Bob has particles 3,4, Charlie has particles 5,6, D1,D2....DnIs a control particle;
respectively introducing auxiliary particles into Alice, Bob and Charlie
Figure FDA0003146069240000024
And to particle pair (1, omega)1),(3,ω2) And (5, ω)3) The CNOT operation is implemented, and the system expression is as follows:
Figure FDA0003146069240000025
and step 3: POVM measurement and phase measurement are specifically as follows:
alice, Bob, Charlie each separate the particle ω1，ω2，ω3Sending the measurement result to Charlie, Alice and Bob, and simultaneously Alice, Bob and Charlie select proper measurement bases to respectively carry out POVM measurement on the particles 2,4 and 6, and according to the measurement result, Charlie, Alice and Bob carry out the POVM measurement on the particles omega1，ω2，ω3Performing the measurement;
and 4, step 4: control measurement and target state preparation
For N control parties, only if all the control parties agree to the communication among Alice, Bob and Charlie, the target states can be jointly prepared; thus, if preparation is agreed, each control party needs to perform a single bit measurement;
the final system is written as:
Figure FDA0003146069240000031
respectively obtaining 8 combinations of measurement results by Alice, Bob and Charlie; according to the measurement results of the N control parties, Charlie, Bob and Alice receive different measurement information, and execute a corresponding unitary operation i ═ 0 according to the respective measurement results><0|+|1><1|，σX＝|0><1|+|1><0|，σz＝|0><0|-|1><1|，σy＝i(|0><1|-|1><0|) to recover the target state;
in step 3, taking Alice as an example, the following details are provided:
alice and Charlie for particles 2 and ω, respectively3Performing the measurement;
alice selects a set of measurement bases Mi>(ii) a POVM measurements were performed for particle 2 by i ∈ (0,1) }:
Figure FDA0003146069240000032
Figure FDA0003146069240000033
the collapse state of the measurement result is | M0>Or | M1>The following are:
get
Figure FDA0003146069240000034
The optimal POVM measurement bases are as follows:
Figure FDA0003146069240000035
E2＝I-E0-E1
wherein I is an identity matrix, and the value of x is ensured to be E2The non-negative operator is a non-negative operator,
Figure FDA0003146069240000041
Figure FDA0003146069240000042
when the measurement result of the particle 2 is E0When the state of the particle 2 is | M0>(ii) a Alice publishes the successful result of the measurement, and Charlie selects a proper measurement basis to measure the particle omega3Measurement, the measurement base is:
Figure FDA0003146069240000043
Figure FDA0003146069240000044
when the measurement result of the particle 2 is E1When the state of the particle 2 is | M1>(ii) a Alice publishes the successful result of the measurement, and Charlie selects a proper measurement basis to measure the particle omega3Measurement, measurementThe amount base is as follows:
Figure FDA0003146069240000045
Figure FDA0003146069240000046
when the measurement result of the particle 2 is E2When, no inference can be made, the measurement fails;
bob and Alice for particles 4 and ω, respectively1Performing the measurement;
bob selects a set of measurement bases { | μi>(ii) a POVM measurement is performed for particle 4 by i ∈ (0,1) }:
Figure FDA0003146069240000047
Figure FDA0003146069240000048
the collapse state of the measurement result is | mu0>Or | μ1>The following are:
get
Figure FDA0003146069240000049
The optimal POVM measurement bases are as follows:
Figure FDA00031460692400000410
E′2＝I-E′0-E′1
wherein I is a unit matrix, and the value of x is ensured to be E'2The non-negative operator is a non-negative operator,
Figure FDA00031460692400000411
Figure FDA0003146069240000051
when the measurement result of particle 4 is E'0When the state of the particle 4 is | μ0>(ii) a Bob publishes the successful result of the measurement, and Alice selects a proper measurement basis to measure the particle omega1Measurement, the measurement base is:
Figure FDA0003146069240000052
Figure FDA0003146069240000053
when the measurement result of particle 4 is E'1When the state of the particle 4 is | μ1>(ii) a Bob publishes the successful result of the measurement, and Alice selects a proper measurement basis to measure the particle omega1Measurement, the measurement base is:
Figure FDA0003146069240000054
Figure FDA0003146069240000055
when the measurement result of particle 4 is E'2When, no inference can be made, the measurement fails;
charlie and Bob for particle 6 and omega, respectively2Performing the measurement;
charlie selects a set of measurement bases { | Tj>(j ═ 0,1) } POVM measurement was performed on the particles 6:
Figure FDA0003146069240000056
Figure FDA0003146069240000057
the collapse state of the measurement result is | T0>Or | T1>The following are:
get
Figure FDA0003146069240000058
The optimal POVM measurement bases are as follows:
Figure FDA0003146069240000059
E″2＝I-E″0-E″1
wherein, I is an identity matrix, and the value of x is ensured to be E ″)2The non-negative operator is a non-negative operator,
Figure FDA00031460692400000510
Figure FDA00031460692400000511
when the measurement result of the particles 6 is E ″)0When the state of the particle 6 is | T0>(ii) a Charlie discloses the results of successful measurements, Bob selects the appropriate measurement basis for the particle omega2The measurement was performed on the basis of:
Figure FDA0003146069240000061
Figure FDA0003146069240000062
when the measurement result of the particles 6 is E ″)1When the state of the particle 6 is | T1>(ii) a Charlie discloses the results of successful measurements, Bob selects the appropriate measurement basis for the particle omega2The measurement was performed on the basis of:
Figure FDA0003146069240000063
Figure FDA0003146069240000064
when the measurement result of the particles 6 is E ″)2When, no inference can be made, the measurement fails;
in step 4, the measurement basis for each control method to implement single-bit measurement is { + >, | - >, which is specifically expressed as follows:
Figure FDA0003146069240000065
Figure FDA0003146069240000066
alice, Bob and Charlie respectively obtain 8 combinations of measurement results, wherein 4 combinations of results are invalid combinations, and the unitary operation corresponding to the specific measurement results is shown in Table 1; according to the measurement results of the N control parties, Charlie, Bob and Alice receive different measurement information, and execute corresponding unitary operation I ═ 0 according to the respective measurement results><0|+|1><1|，σX＝|0><1|+|1><0|，σz＝|0><0|-|1><1|，σy＝i(|0><1|-|1><0|) to recover the target state; specifically, as shown in table 1:
Figure FDA0003146069240000071
TABLE 1 measurement results and recovery of performed unitary operations for Bob, Charlie, Alice
As shown in Table 1, if Alice and Charlie have the measurement results of
Figure FDA0003146069240000072
Bob receives measurements that depend on N controllers; if Bob receives m ' 0's and (n-m) 1's, the particle measurements of the controlling party are analyzed as follows:
Figure FDA0003146069240000073
Figure FDA0003146069240000074
wherein, |0>nAlways has a positive expansion, |1>nThe expansion of (A) has positive and negative;
to |1>nThe following discussion is expanded:
in the first case: if m is an even number, n is an odd number, or n is an even number, m is an odd number;
|1>nhas a negative expansion, Bob receives the measurement result of the control party as (| 0)>-|1>) I.e. | ->Bob obtains the information about the target state as
Figure FDA0003146069240000075
At this time, Bob only needs to execute on the obtained state
Figure FDA0003146069240000076
The target state can be obtained by operation;
in the second case: if m and n are both even numbers, or m and n are both odd numbers;
|1>nhas a positive expansion, Bob receives the measurement result of the control party as (| 0)>+|1>) I is | +>Bob obtains the information about the target state as
Figure FDA0003146069240000081
At this time, the information obtained by Bob is the target state.
2. The combined-cycle remote-state preparation method based on non-maximally entangled-channel N-party control according to claim 1, wherein the POVM measurement success probability analysis of particle 2 is as follows:
POVM measurement result is E0Probability of (c):
Figure FDA0003146069240000082
measurement result is E1Probability of (c):
Figure FDA0003146069240000083
measurement result is E2Probability of (c):
Figure FDA0003146069240000084
it follows therefrom that the probability of success of the POVM measurement of the particle 2 is
Figure FDA0003146069240000085
The probability of failure is
Figure FDA0003146069240000086
3. The combined-cycle remote-state preparation method based on non-maximally entangled-channel N-party control according to claim 1, wherein the POVM measurement success probability analysis of the particles 4 is as follows:
POVM measurement result is E'0Probability of (c):
Figure FDA0003146069240000087
measurement result is E1The probability of `:
Figure FDA0003146069240000088
measurement result was E'2Probability of (c):
Figure FDA0003146069240000089
from this, particles are obtainedProbability of success of POVM measurement of 1 is
Figure FDA00031460692400000810
The probability of failure is
Figure FDA00031460692400000811
4. The combined-cycle remote-state preparation method based on non-maximally entangled-channel N-party control according to claim 1, wherein the POVM measurement success probability analysis of the particles 6 is as follows:
the POVM measurement result is E ″)0Probability of (c):
Figure FDA00031460692400000812
the measurement result is E ″)1Probability of (c):
Figure FDA00031460692400000813
the measurement result is E ″)2Probability of (c):
Figure FDA00031460692400000814
it follows that the probability of success of the POVM measurement of the particle 6 is
Figure FDA00031460692400000815
The probability of failure is
Figure FDA00031460692400000816
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"Quantum Network Coding For Remote State Preparation of Multi-qudit States";Wang Xu et al;《2019 IEEE International Conference on Systems, Man and Cybernetics (SMC)》;20191009;全文 *
"Remote State Preparation for Multiple Parties";Wenhan Dai et al;《ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing(ICASSP)》;20190517;全文 *

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