CN116488807A - Method for remotely preparing quantum state based on bidirectional fault-tolerant multi-hop combination - Google Patents
Method for remotely preparing quantum state based on bidirectional fault-tolerant multi-hop combination Download PDFInfo
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Abstract
The invention relates to a method for remotely preparing a quantum state based on bidirectional fault-tolerant multi-hop combination, which comprises the steps of establishing a first sender Alice 1 Second sender Alice 2 Quantum entanglement channels of the receiver Bob and the plurality of intermediate nodes; the channel between Bob and the intermediate node has dislocations; the intermediate node performs Bell measurement on the particle pair carried by the intermediate node, and sends Bell measurement results and dislocation information to Bob; so that Bob constructs a dislocation matrix, performs dislocation correction, and determines a direct entanglement channel; bob performs a corresponding unitary operation on the particles carried by himself according to the format of the direct entangled channel; alice (Alice) 1 After CNOT operation and single particle measurement are carried out on the particle pair, POVM measurement is carried out, a single particle measurement result containing quantum state amplitude information to be prepared and a POVM measurement result are obtained and sent to Bob; alice (Alice) 2 Performing projection measurement on particles carried by the particles, obtaining a projection measurement result containing quantum state phase information to be prepared, and sending the projection measurement result to Bob; bob performs on the particles carried by Bob based on various measurement resultsAnd (5) carrying out corresponding unitary operation to recover the quantum state to be prepared.
Description
Technical Field
The invention relates to the technical field of communication networks and information propagation, in particular to a method for remotely preparing a quantum state based on bidirectional fault tolerance multi-hop combination.
Background
With the development of society and the progress of scientific technology, information technology has made an unprecedented great breakthrough, and information, substances and energy have become fundamental elements of the present and developing society. Information theory and quantum theory are two important findings in the twentieth century, and as research continues to go deep, they began to cross and fuse in the last twenty years of the last century, thereby generating quantum informatics based on quantum mechanics. The quantum information technology is an emerging front science for researching information processing based on the quantum mechanics as a state superposition principle, and comprises a plurality of aspects such as quantum cryptography, quantum communication, quantum algorithm, quantum computer and the like, and great breakthroughs are made in theory and experiment in recent years.
The quantum invisible transmission state restores the unknown state of the bit held by one party to the bit held by the other party without the need of a sender to directly transmit the entity bit carrying the unknown information to the receiver, thereby almost realizing the information transmission in hyperspace. In 1993 Bennett et al proposed for the first time that unknown quantum states were transmitted by classical channels and EPR channels. Since Bennett et al proposed the original protocol, there has been a great deal of attention due to its wonderful nature and positive progress has been made both theoretically and experimentally. For decades, in order to meet various different quantum communication scenarios, a series of quantum invisible transmission state protocols involving different quantum sub-channels, such as Bell state, GHZ state, W state, etc., have been proposed. At the same time, many protocols have been proposed successively, such as cavity quantum electrodynamics, ion traps, quantum dots, optics, nuclear magnetic resonance, etc.
But it is not practical to directly transfer the quantum state between two nodes that are further apart due to the unavoidable losses of the quantum channel. To overcome this limitation, quantum communication is also gradually moving toward the development of networking. Quantum networks are based on point-to-point quantum key distribution, enabling communication partners in the network to exchange theoretically secure keys, rather than the infrastructure for secure communication. The quantum network comprises a plurality of intermediate nodes, the adjacent nodes share a pair of entanglement sources, and communication between two nodes with a larger distance is finally realized through entanglement exchange. Quantum communication networks have several significant advantages over classical communication networks: the security and confidentiality of communication are higher; the stronger information transmission and processing capacity makes the communication efficiency higher; the communication complexity is low. At present, many research groups have proposed their own ideas for building quantum networks. In 2014 Wang et al, a quantum wireless multi-hop invisible state transmission method based on Bell pairs is proposed to construct a quantum communication network. In 2017, in order to improve transmission efficiency, zou et al proposed a multi-hop invisible state transmission protocol, and quantum invisible state transmission of unknown two particle states was realized through a composite GHZ-Bell channel. Recently, zhou et al proposed a multi-hop invisible transmission scheme, with the W state as the quantum channel for transmission. After Lo and Pati propose schemes of quantum state remote preparation (remote state preparation), as RSP has the advantage of being capable of reducing classical information and quantum resource consumption, along with bidirectional quantum state preparation, controlled quantum state preparation, multiparty quantum state remote preparation and the like, a plurality of quantum state remote preparation schemes are developed. In 2012, zhan et al propose an improvement scheme which skillfully utilizes a new unitary operation to realize the preparation of any two-bit and three-bit quantum states, and the success rate can reach 100%. In 2019, zhang et al proposed a JRSP scheme for mixed state through the three-qubit GHZ state. The probability of success of the scheme for the real coefficient mixture is determined, and probability for the complex coefficient.
In a practical quantum communication network, due to interaction between particles and environment, the quantum state of the quantum communication network is changed under the influence of quantum noise in the process of channel transmission, and the quantum dislocation possibly generated by the action of one quantum bit and the environment is as follows: i-no errors; x-bit anti-error; z-phase error; XZ-bit anti-error + bit error. Based on the method, the quantum communication network needs to adopt a quantum error correction technology to ensure the reliable transmission of the original unknown quantum state information, and the quantum communication network protects a quantum channel based on the quantum mechanics principle, so that the absolute reliability of communication between two places can be ensured.
Disclosure of Invention
Therefore, the invention aims to solve the technical problems that the information transmission efficiency is low and the bidirectional fault tolerance cannot be realized in the prior art.
In order to solve the technical problems, the invention provides a method for remotely preparing a quantum state based on bidirectional fault-tolerant multi-hop combination, which comprises the following steps:
establishment based on first sender Alice 1 Second sender Alice 2 Quantum entanglement channels of the receiver Bob and the plurality of intermediate nodes; the channel between the receiving party Bob and the intermediate node has dislocation;
the intermediate node performs Bell measurement on the particle pair carried by the intermediate node, and sends Bell measurement results and dislocation information to a receiver Bob;
the receiver Bob constructs a dislocation matrix according to dislocation information of each intermediate node, executes dislocation correction operation, and determines a direct entanglement channel according to the dislocation information and Bell measurement results;
the receiver Bob executes corresponding unitary operation on the particles carried by the receiver Bob according to the format of the direct entanglement channel;
first sender Alice 1 After CNOT operation is carried out on the particle pair, single particle measurement is carried out on the target quantum bit, POVM measurement is carried out on the control quantum bit, a single particle measurement result containing quantum state amplitude information to be prepared and a POVM measurement result are obtained, and the single particle measurement result and the POVM measurement result are sent to a receiver Bob;
second sender Alice 2 Performing projection measurement on particles carried by the particles to obtain projection measurement results containing quantum state phase information to be preparedTo the recipient Bob;
and the receiver Bob executes corresponding unitary operation on the particles carried by the receiver Bob according to the single particle measurement result, the POVM measurement result and the projection measurement result, and recovers the quantum state to be prepared.
In one embodiment of the invention, the establishment is based on a first sender Alice 1 Second sender Alice 2 A quantum entanglement channel for a recipient Bob with a plurality of intermediate nodes, comprising:
first sender Alice 1 Second sender Alice 2 The receiver Bob is connected with a plurality of intermediate nodes through non-maximum entanglement Bell channels, wherein the non-maximum entanglement Bell channels are expressed as:
wherein the coefficient a 0 ,a 1 Is real and satisfies the normalization condition:the first sender Alice 1 Holding amplitude information of quantum state to be prepared, possessing target quantum bit A 1 And control of qubit P 1 The method comprises the steps of carrying out a first treatment on the surface of the The second sender Alice 2 Holding the phase information of the quantum state to be prepared, having particles p 2 The method comprises the steps of carrying out a first treatment on the surface of the Recipient Bob owns particle B q+1 The method comprises the steps of carrying out a first treatment on the surface of the The kth intermediate node owns particle pair A k ,B k (k=1, 2, …, q+1); the quantum state can generate dislocation in the channel transmission process, channel bits are overturned by the interference of the environment, and the typical overturned states are in four forms of I, X, Z and XZ; suppose A in the kth node k ,B k The (k=1, 2, …, q+1) particle pair is dislocated, i.e. flip occurs at both ends of the Bell channel, the dislocation information is e fg ＝Z f X g Wherein f, g=0, 1;
the quantum entanglement channel, i.e. the system initial state, can be expressed as:
in one embodiment of the invention, the particle pair A of the intermediate node in the initial state of the system k ,B k (k=1, 2,., q+1) is expressed as:
wherein |phi uv >U, v of (1) is 0,1, expressed as:
m (k) ,n (k) representing the kth intermediate node particle pairBell measurement results;representing exclusive OR, representing AND operation; u (U) cor Dislocation matrix representing the structure, expressed as +.>
In one embodiment of the invention, the direct entanglement channel is represented as:
wherein the coefficient b 0 、b 1 Expressed as:
in one embodiment of the present invention, the joint system state of the intermediate node particle pair under the direct entanglement channel is expressed as:
in one embodiment of the present invention, the receiving party Bob performs a corresponding unitary operation on the particles carried by itself according to the format of the direct entangled channel, including:
when the direct entanglement channel isAt this time, particle B of recipient Bob q+1 Performing a unitary operation u=i= |0><0|+|1><1|；
When the direct entanglement channel isAt this time, particle B of recipient Bob q+1 Performing a unitary operation u=x= |0><1|+|1><0|；
When the direct entanglement channel isAt this time, particle B of recipient Bob q+1 Performing a unitary operation u=z= |0><0|-|1><1|；
When the direct entanglement channel isAt this time, particle B of recipient Bob q+1 Performing a unitary operation u=zx= (0><0|-|1><1|)(0><1|+|1><0|)。
In one embodiment of the present invention, the first sender Alice 1 After performing CNOT operation on the particle pair, performing single particle measurement on the target quantum bit, performing POVM measurement on the control quantum bit, obtaining a single particle measurement result containing quantum state amplitude information to be prepared and a POVM measurement result, and sending the single particle measurement result and the POVM measurement result to a receiver Bob, wherein the single particle measurement method comprises the following steps:
first sender Alice 1 For its particle pair (P 1 ,A 1 ) Performs CNOT operations and on particle A 1 The proceeding base is { |t>Single particle measurement of t=0, 1;
first sender Alice 1 For particle P 1 The POVM measurements are performed, expressed as:
wherein,,the coefficient x satisfies->
In one embodiment of the present invention, the first sender Alice 1 For its particle pair (P 1 ,A 1 ) After performing the CNOT operation, the joint system state of the intermediate node particle pair is expressed as:
wherein redefinition is performedThe joint system state is expressed as:
in one embodiment of the present invention, the second sender Alice 2 Performing projection measurement on particles carried by the particles, obtaining a projection measurement result containing quantum state phase information to be prepared, and after the projection measurement result is sent to a receiver Bob, the method comprises the following steps:
after the receiving party Bob obtains the projection measurement result, a preset phase measurement base |o is selected n >(n=0, 1) is:
in one embodiment of the present invention, the receiving party Bob performs a corresponding unitary operation on the particles carried by the receiving party Bob according to the single particle measurement result, the poim measurement result and the projection measurement result, and restores the quantum state to be prepared, including:
the receiving party Bob performs particle B according to the single particle measurement result q+1 The first unitary operation performed is denoted as:
b of the receiver Bob after the first unitary operation according to the POVM measurement result and the projection measurement result q+1 Performing a second unitary operation to recover the quantum state to be preparedComprising the following steps:
if the POVM measurement result is |P 0 >The projection measurement result is |O 0 >Then perform a second unitary operation
If the POVM measurement result is |P 0 >The projection measurement result is |O 1 >Then perform a second unitary operation
If the POVM measurement result is |P 1 >The projection measurement result is |O 0 >Then perform a second unitary operation
If the POVM measurement result is |P 1 >The projection measurement result is |O 1 >Then perform a second unitary operation
If the POVM measurement result is |P 2 >And returning to the preparation failure.
Compared with the prior art, the technical scheme of the invention has the following advantages:
the method for remotely preparing the quantum state based on the bidirectional fault-tolerant multi-hop combination provided by the invention is implemented by a first sender Alice 1 Second sender Alice 2 A multi-hop quantum network composed of a plurality of intermediate nodes is constructed between the multi-hop quantum network and the receiver Bob, the plurality of intermediate nodes can simultaneously carry out Bell measurement, and simultaneously send measurement results and dislocation information to the receiver Bob, so that the information transmission efficiency is improved; and the multi-hop quantum network solves the long-distance remote quantum state preparation problem, and can meet the requirement of constructing a complex quantum network. According to the invention, dislocation information generated by the quantum bit is directly transmitted to a receiver by utilizing particle pairs of a plurality of intermediate nodes, and the receiver Bob carries out corresponding unitary transformation on the particles carried by the dislocation information according to the acquired measurement result and the dislocation information, so that quantum errors are corrected, bidirectional fault tolerance is realized, and a quantum state to be prepared is correctly recovered; and the error correction operation and the to-be-prepared quantum state recovery operation are executed by the receiver Bob, so that the operation of the middle part is simplified, and the transmission efficiency is improved.
Drawings
In order that the invention may be more readily understood, a more particular description of the invention will be rendered by reference to specific embodiments thereof that are illustrated in the appended drawings, in which
FIG. 1 is a schematic diagram of the steps of a method for remotely preparing quantum states based on a bidirectional fault-tolerant multi-hop joint provided by the invention;
FIG. 2 is a schematic diagram of particle distribution of a method for remote preparation of quantum states based on bi-directional fault-tolerant multi-hop joint for a single intermediate node provided by an embodiment of the present invention;
fig. 3 is a schematic diagram of a quantum circuit of a method for remotely preparing a quantum state based on a bidirectional fault-tolerant multi-hop joint of a single intermediate node according to an embodiment of the present invention.
Detailed Description
The present invention will be further described with reference to the accompanying drawings and specific examples, which are not intended to be limiting, so that those skilled in the art will better understand the invention and practice it.
The invention provides a method for remotely preparing a quantum state based on bidirectional fault-tolerant multi-hop combination, which is characterized in that each communication node on a communication path directly transmits errors generated by quantum bits to an information receiver through a channel error correction technology, the information receiver executes proper matrix transformation, corrects quantum errors and correctly recovers transmitted unknown quantum state information.
Referring to fig. 1, the method for remotely preparing a quantum state based on bidirectional fault-tolerant multi-hop joint of the invention comprises the following steps:
s101: establishment based on first sender Alice 1 Second sender Alice 2 Quantum entanglement channels of the receiver Bob and the plurality of intermediate nodes; the channel between the receiving party Bob and the intermediate node has dislocation;
first sender Alice 1 Second sender Alice 2 And the receiver Bob are located in a multi-hop quantum network together, alice 1 And Alice 2 Respectively holding amplitude information and phase information of quantum states to be prepared, and remotely preparing any single-bit quantum state for recipient BobWherein alpha is 0 ,α 1 E R, and |alpha 0 | 2 +|α 1 | 2 ＝1，λ 1 E [0,2 pi); wherein Alice 1 Hold amplitude information, alice 2 Holding phase information;
first sender Alice 1 Second sender Alice 2 The receiver Bob is connected with a plurality of intermediate nodes through non-maximum entanglement Bell channels, wherein the non-maximum entanglement Bell channels are expressed as:
wherein the coefficient a 0 ,a 1 Is real and satisfies the normalization condition:the first sender Alice 1 Holding amplitude information of quantum state to be prepared, possessing target quantum bit A 1 And control of qubit P 1 The method comprises the steps of carrying out a first treatment on the surface of the The second hairSender Alice 2 Holding the phase information of the quantum state to be prepared, having particles p 2 The method comprises the steps of carrying out a first treatment on the surface of the Recipient Bob owns particle B q+1 The method comprises the steps of carrying out a first treatment on the surface of the The kth intermediate node owns particle pair A k ,B k (k=1, 2, …, q+1); the quantum state can generate dislocation in the channel transmission process, channel bits are overturned by the interference of the environment, and the typical overturned states are in four forms of I, X, Z and XZ; suppose A in the kth node k ,B k The (k=1, 2, …, q+1) particle pair is dislocated, i.e. flip occurs at both ends of the Bell channel, the dislocation information is e fg ＝Z f X g Wherein f, g=0, 1;
the quantum entanglement channel, i.e. the system initial state, can be expressed as:
particle pair A of intermediate node in initial state of system k ,B k The combined system states consisting of (k=1, 2, …, q+1) are expressed as:
wherein |phi uv >U, v of (1) is 0,1, expressed as:
m (k) ,n (k) bell measurements representing the kth intermediate node particle pair;representing exclusive OR, representing AND operation; u (U) cor Dislocation matrix representing the structure, expressed as +.>
S102: the intermediate node performs Bell measurement on the particle pair carried by the intermediate node, and sends Bell measurement results and dislocation information to a receiver Bob;
s103: the receiver Bob constructs a dislocation matrix according to dislocation information of each intermediate node, executes dislocation correction operation, and determines a direct entanglement channel according to the dislocation information and Bell measurement results;
the direct entanglement channel is expressed as:
wherein the coefficient b 0 、b 1 Expressed as:
the joint system state of the intermediate node particle pairs under the direct entanglement channel is expressed as:
s104: the receiver Bob executes corresponding unitary operation on the particles carried by the receiver Bob according to the format of the direct entanglement channel;
when the direct entanglement channel isAt this time, particle B of recipient Bob q+1 Performing a unitary operation u=i= |0><0|+|1><1|；
When the direct entanglement channel isAt this time, particle B of recipient Bob q+1 Performing a unitary operation u=x= |0><1|+|1><0|；
When the direct entanglement channel isAt this time, particle B of recipient Bob q+1 Performing a unitary operation u=z= |0><0|-|1><1|；
When the direct entanglement channel isAt this time, particle B of recipient Bob q+1 Performing a unitary operation u=zx= (|0)><0|-|1><1|)(0><1|+|1><0|)。
S105: first sender Alice 1 After CNOT operation is carried out on the particle pair, single particle measurement is carried out on the target quantum bit, POVM measurement is carried out on the control quantum bit, and the quantum state to be prepared is obtainedThe single particle measurement result and the POVM measurement result of the amplitude information are sent to a receiver Bob;
first sender Alice 1 For its particle pair (P 1 ,A 1 ) Performs CNOT operations and on particle A 1 The proceeding base is { |t>Single particle measurement of t=0, 1;
first sender Alice 1 For particle P 1 The POVM measurements are performed, expressed as:
wherein,,the coefficient x satisfies->
After performing the CNOT operation, the joint system state of the intermediate node particle pair is expressed as:
wherein redefinition is performedThe joint system state is then expressed as:
s106: second sender Alice 2 Performing projection measurement on particles carried by the particles, obtaining a projection measurement result containing quantum state phase information to be prepared, and sending the projection measurement result to a receiver Bob;
after the receiving party Bob obtains the projection measurement result, a preset phase measurement base |o is selected n >(n=0, 1) is:
s107: and the receiver Bob executes corresponding unitary operation on the particles carried by the receiver Bob according to the single particle measurement result, the POVM measurement result and the projection measurement result, and recovers the quantum state to be prepared.
The receiving party Bob performs particle B according to the single particle measurement result q+1 The first unitary operation performed is denoted as:
b of the receiver Bob after the first unitary operation according to the POVM measurement result and the projection measurement result q+1 Performing a second unitary operation to recover the quantum state to be preparedComprising the following steps:
if the POVM measurement result is |P 0 >The projection measurement result is |O 0 >Then perform a second unitary operation
If the POVM measurement result is |P 0 >The projection measurement result is |O 1 >Then perform a second unitary operation
If the POVM measurement result is |P 1 >The projection measurement result is |O 0 >Then perform a second unitary operation
If the POVM measurement result is |P 1 >The projection measurement result is |O 1 >Then perform a second unitary operation
If the POVM measurement result is |P 2 >And returning to the preparation failure.
Specifically, in the present invention, the quantum state of any single bit state to be prepared is expressed as:
bell's base is the largest entangled state composed of two particles, it forms a complete set of orthogonal bases for a four-dimensional Hilbert space, the specific form is as follows:
a control NOT gate (CNOT) having two input qubits, a control qubit and a target qubit, respectively; the function is as follows: when the control qubit state is |0>, the target qubit state is unchanged; when the control qubit state is |1>, the target qubit state is flipped from |0> to |1>, or from |1> to >0>. The corresponding matrix form is:
the unitary matrix, i.e., pauli matrix. The specific form is as follows:
the method for remotely preparing the quantum state based on the bidirectional fault-tolerant multi-hop combination provided by the invention is implemented by a first sender Alice 1 Second sender Alice 2 A multi-hop quantum network composed of a plurality of intermediate nodes is constructed between the multi-hop quantum network and the receiver Bob, the plurality of intermediate nodes can simultaneously carry out Bell measurement, and simultaneously send measurement results and dislocation information to the receiver Bob, so that the information transmission efficiency is improved; and the multi-hop quantum network solves the long-distance remote quantum state preparation problem, and can meet the requirement of constructing a complex quantum network. According to the invention, dislocation information generated by the quantum bit is directly transmitted to a receiver by utilizing particle pairs of a plurality of intermediate nodes, and the receiver Bob carries out corresponding unitary transformation on the particles carried by the dislocation information according to the acquired measurement result and the dislocation information, so that quantum errors are corrected, bidirectional fault tolerance is realized, and a quantum state to be prepared is correctly recovered; and the error correction operation and the to-be-prepared quantum state recovery operation are executed by the receiver Bob, so that the operation of the middle part is simplified, and the transmission efficiency is improved.
Based on the above embodiment, in this embodiment, based on a single intermediate node, the method for remotely preparing a quantum state based on bidirectional fault-tolerant multi-hop association is implemented, including:
s201: establishment based on first sender Alice 1 Second sender Alice 2 Quantum entanglement channels of the receiver Bob and the plurality of intermediate nodes; the channel between the receiving party Bob and the intermediate node has dislocation;
sender Alice 1 ，Alice 2 And the receiver Bob are located in a multi-hop quantum network together, alice 1 And Alice 2 Respectively holding amplitude and phase information of quantum state to be prepared, and remotely preparing any single bit state for receiving party BobWherein Alice 1 Hold amplitude information, alice 2 Holding phase information.
Referring to fig. 2, there are 1 intermediate nodes on the path between the sender and the receiver, and Alice is established 1 、Alice 2 Quantum entanglement channels of Bob and intermediate nodes; on the transmission path, alice 1 、Alice 2 And Bob and the intermediate node are connected in pairs through a non-maximum entangled Bell channel. The non-maximally entangled Bell channel format is as follows:
coefficient a 0 ,a 1 Is real and satisfies the normalization condition: a, a 1 2 +a 1 2 ＝1。Alice 1 Possessing particle A 1 ,p 1 ，Alice 2 Having particles p 2 Bob owns particle B 2 Intermediate node owns particle B 1 ,A 2 The channel generates dislocations during transmission, assuming particle a 1 ,A 2 ,B 1 ,B 2 The inversion occurs and the dislocation information is e fg ＝Z f X g Wherein f, g=0, 1; the system initial state can be written as:
in the initial state of the system, the particle A k ,B k The combined system state consisting of (k=1, 2, … q+1) can be written as:
the above formula can be further rewritten as:
where phi uv >Wherein u, v are 0,1 combinations, represented as follows:
wherein,,
L 00 ＝m·n；
in the above formula, m and n represent Bell measurement results of the intermediate node particle pairs; sign symbol"" and "-" represent exclusive or, and no operation, respectively; bob constructs a dislocation matrix U according to the dislocation information published by each intermediate node cor The method comprises the following steps:
s202: the intermediate node performs Bell measurement on the particle pair carried by the intermediate node, and sends Bell measurement results and dislocation information to a receiver Bob;
s203: the receiver Bob constructs a dislocation matrix according to the dislocation information of each intermediate node, executes dislocation correction operation, and determines a direct entanglement channel according to the dislocation information and Bell measurement results
For ease of analysis, redefining the representation of the coefficients is as follows:
thus, alice 1 The direct entanglement channel formed between Bob and Bob is:
the joint system state at this time is expressed as:
s204: the receiving party Bob carries the particle B on itself according to the format of the direct entanglement channel 2 Executing corresponding unitary operation;
the corresponding unitary operation is shown with reference to table 1:
table 1: direct entangled channel format corresponding unitary operation
S205: first sender Alice 1 For its particle pair (p 1 ,A 1 ) After CNOT operation is executed, single-particle measurement is executed on the target quantum bit, POVM measurement is executed on the control quantum bit, and a single-particle measurement result and P containing quantum state amplitude information to be prepared are obtainedThe OVM measurement result is sent to a receiving party Bob;
Alice 1 for its particles A 1 Is taken as { |t>Single particle measurement of t=0, 1} and sending single particle measurement result to Bob;
to prepare amplitude information, alice 1 For particle p 1 The POVM measurements are performed in the form of:
wherein,,to ensure P 2 The minimum value of the coefficient x needs to be satisfied by the non-negativity of all diagonal elements in +.>
At this time, the joint system state of the intermediate node particle pair is expressed as:
wherein redefinition is performedThe joint system state is then expressed as:
s206: second sender Alice 2 Performing projection measurement on particles carried by the particles, obtaining a projection measurement result containing quantum state phase information to be prepared, and sending the projection measurement result to a receiver Bob;
to prepare phase information, bob rootAccording to the information grasped by oneself, selecting correspondent phase measurement base I O n >(n=0, 1) is:
/>
s207: and the receiver Bob executes corresponding unitary operation on the particles carried by the receiver Bob according to the single particle measurement result, the POVM measurement result and the projection measurement result, and recovers the quantum state to be prepared.
S207-1: the receiving party Bob performs particle B according to the single particle measurement result 2 The first unitary operation performed is denoted as:
s207-2: b of the receiver Bob after the first unitary operation according to the POVM measurement result and the projection measurement result q+1 Performing a second unitary operation; referring to table 2, for unitary operation corresponding to the measurement result:
table 2: unitary operation corresponding to measurement result
As can be seen from the above table, when Alice 1 The POVM measurement result of (2) is |P 0 >Or |P 1 >At the time, user Bob is accepted for B 2 Particle execution corresponding U l Operation can recover the target stateWhen Alice 2 The POVM measurement result of (2) is |P 2 >In this case, it is difficult for the recipient Bob to determine particle B 2 What quantum state is in, which will lead to failure of the remote state preparation scheme;
neglecting global phase, when Alice 1 For particle p 1 The POVM measurement result of (2) is |P 0 >，|P 1 >When determining that the receiving user needs to hold the particle B 2 Execution U l In operation, bob can recover the state to be prepared
Specifically, referring to fig. 3, in an embodiment of the present invention, for a second receiver Alice 2 Particles P of (2) 2 Performing a projective measurement PM for a first recipient Alice 1 Particle pair (P) 1 ,A 1 ) Performs CNOT operations and on particle A 1 Performing a single particle measurement SM on particle P 1 Performing a POVN measurement; pair of intermediate node particles (B) 1 ,A 2 ) Performing Bell measurement BM, passing the Bell measurement result U and projection measurement result U 1 Single particle measurement result U' and POVM measurement result U 1 At the receiving side Bob, for particle B 2 And performing error correction operation and recovering the quantum state to be prepared, thereby realizing remote preparation of the quantum state.
In the method for remotely preparing the quantum state based on the bidirectional fault-tolerant multi-hop combination, the error correction operation and the original unknown quantum state recovery operation are uniformly executed by the receiver Bob, so that the operation of an intermediate party is simplified; according to the invention, each node on the communication path can simultaneously carry out Bell measurement, and dislocation information and Bell measurement results are simultaneously sent to the receiver Bob, so that the efficiency of information transmission is improved; the invention solves the problem of long-distance remote quantum state preparation through the multi-hop network, can meet the requirement of constructing a complex quantum network, and directly transmits the errors generated by the quantum bits to the information receiver Bob by utilizing each communication node on a communication path, so that the information receiver Bob executes proper matrix transformation to correct the quantum errors, thereby correctly recovering the transmitted unknown quantum state information and realizing the preparation of the quantum state.
It will be appreciated by those skilled in the art that embodiments of the present application may be provided as a method, system, or computer program product. Accordingly, the present application may take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present application may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, and the like) having computer-usable program code embodied therein.
The present application is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems) and computer program products according to embodiments of the application. It will be understood that each flow and/or block of the flowchart illustrations and/or block diagrams, and combinations of flows and/or blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
It is apparent that the above examples are given by way of illustration only and are not limiting of the embodiments. Other variations and modifications of the present invention will be apparent to those of ordinary skill in the art in light of the foregoing description. It is not necessary here nor is it exhaustive of all embodiments. While still being apparent from variations or modifications that may be made by those skilled in the art are within the scope of the invention.
Claims (10)
1. The method for remotely preparing the quantum state based on the bidirectional fault-tolerant multi-hop combination is characterized by comprising the following steps of:
establishment based on first sender Alice 1 Second sender Alice 2 Quantum entanglement channels of the receiver Bob and the plurality of intermediate nodes; the channel between the receiving party Bob and the intermediate node has dislocation;
the intermediate node performs Bell measurement on the particle pair carried by the intermediate node, and sends Bell measurement results and dislocation information to a receiver Bob;
the receiver Bob constructs a dislocation matrix according to dislocation information of each intermediate node, executes dislocation correction operation, and determines a direct entanglement channel according to the dislocation information and Bell measurement results;
the receiver Bob executes corresponding unitary operation on the particles carried by the receiver Bob according to the format of the direct entanglement channel;
first sender Alice 1 After CNOT operation is carried out on the particle pair, single particle measurement is carried out on the target quantum bit, POVM measurement is carried out on the control quantum bit, a single particle measurement result containing quantum state amplitude information to be prepared and a POVM measurement result are obtained, and the single particle measurement result and the POVM measurement result are sent to a receiver Bob;
second sender Alice 2 Performing projection measurement on particles carried by the particles, obtaining a projection measurement result containing quantum state phase information to be prepared, and sending the projection measurement result to a receiver Bob;
and the receiver Bob executes corresponding unitary operation on the particles carried by the receiver Bob according to the single particle measurement result, the POVM measurement result and the projection measurement result, and recovers the quantum state to be prepared.
2. The method for remotely preparing quantum state based on bidirectional fault-tolerant multi-hop joint as claimed in claim 1, wherein the method comprises the following steps ofIn that the establishment is based on the first sender Alice 1 Second sender Alice 2 A quantum entanglement channel for a recipient Bob with a plurality of intermediate nodes, comprising:
first sender Alice 1 Second sender Alice 2 The receiver Bob is connected with a plurality of intermediate nodes through non-maximum entanglement Bell channels, wherein the non-maximum entanglement Bell channels are expressed as:
wherein the coefficient a 0 ,a 1 Is real and satisfies the normalization condition:the first sender Alice 1 Holding amplitude information of quantum state to be prepared, possessing target quantum bit A 1 And control of qubit P 1 The method comprises the steps of carrying out a first treatment on the surface of the The second sender Alice 2 Holding the phase information of the quantum state to be prepared, having particles p 2 The method comprises the steps of carrying out a first treatment on the surface of the Recipient Bob owns particle B q+1 The method comprises the steps of carrying out a first treatment on the surface of the The kth intermediate node owns particle pair A k ,B k (k=1, 2, …, q+1); the quantum state can generate dislocation in the channel transmission process, channel bits are overturned by the interference of the environment, and the typical overturned states are in four forms of I, X, Z and XZ; suppose A in the kth node k ,B k The (k=1, 2, …, q+1) particle pair is dislocated, i.e. flip occurs at both ends of the Bell channel, the dislocation information is e fg ＝Z f X g Wherein f, g=0, 1;
the quantum entanglement channel, i.e. the system initial state, can be expressed as:
3. the bi-directional fault-tolerant multi-hop based joint remote of claim 2A method for preparing quantum state is characterized in that the particle pair A of intermediate node in initial state of the system k ,B k The combined system states consisting of (k=1, 2, …, q+1) are expressed as:
wherein |phi uv >U, v of (1) is 0,1, expressed as:
m (k) ,n (k) bell representing the kth intermediate node particle pairMeasuring results;representing exclusive OR, representing AND operation; u (U) cor Dislocation matrix representing the structure, expressed as +.>
4. The method for remote preparation of quantum states based on bi-directional fault-tolerant multi-hop joint of claim 2, wherein the direct entanglement channel is expressed as:
wherein the coefficient b 0 、b 1 Expressed as:
5. the method for remotely preparing a quantum state based on bidirectional fault-tolerant multi-hop joint as claimed in claim 4, wherein the joint system state of the intermediate node particle pair under the direct entanglement channel is expressed as:
6. the method for remote preparation of quantum states based on bi-directional fault-tolerant multi-hop association of claim 4, wherein the receiving party Bob performs a corresponding unitary operation on the particles carried by itself according to the format of the direct entangled channel, comprising:
when the direct entanglement channel isAt this time, particle B of recipient Bob q+1 Performing a unitary operation u=i= |0><0|+|1><1|；
When the direct entanglement channel isAt this time, particle B of recipient Bob q+1 Performing a unitary operation u=x= |0><1|+|1><0|；
When the direct entanglement channel isAt this time, particle B of recipient Bob q+1 Performing a unitary operation u=z= |0><0|-|1><1|；
When the direct entanglement channel isAt this time, particle B of recipient Bob q+1 Performing a unitary operation u=zx= (|0)><0|-|1><1|)(|0><1|+|1><0|)。
7. The method for remote preparation of quantum states based on bi-directional fault-tolerant multi-hop joint of claim 2, wherein the first sender Alice 1 After performing CNOT operation on the particle pair, performing single particle measurement on the target quantum bit, performing POVM measurement on the control quantum bit, obtaining a single particle measurement result containing quantum state amplitude information to be prepared and a POVM measurement result, and sending the single particle measurement result and the POVM measurement result to a receiver Bob, wherein the single particle measurement method comprises the following steps:
first sender Alice 1 For its particle pair (P 1 ,A 1 ) Performs CNOT operations and on particle A 1 The proceeding base is { |t>Single particle measurement of t=0, 1;
first sender Alice 1 For particle P 1 The POVM measurements are performed, expressed as:
wherein,, the coefficient x satisfies->
8. The method for remote preparation of quantum states based on bi-directional fault-tolerant multi-hop joint of claim 7, wherein the first sender Alice 1 For its particle pair (P 1 ,A 1 ) After performing the CNOT operation, the joint system state of the intermediate node particle pair is expressed as:
wherein redefinition is performedThe joint system state is then expressed as:
9. the method for remote preparation of quantum states based on bi-directional fault-tolerant multi-hop joint of claim 2, wherein the second sender Alice 2 Performing projection measurement on particles carried by the particles, obtaining a projection measurement result containing quantum state phase information to be prepared, and after the projection measurement result is sent to a receiver Bob, the method comprises the following steps:
after the receiving party Bob obtains the projection measurement result, a preset phase measurement base |o is selected n >(n=0, 1) is:
10. the method for remotely preparing a quantum state based on bidirectional fault-tolerant multi-hop joint according to claim 2, wherein the receiving party Bob performs corresponding unitary operation on particles carried by the receiving party Bob according to the single particle measurement result, the POVM measurement result and the projection measurement result, and recovers the quantum state to be prepared, including:
the receiving party Bob performs particle B according to the single particle measurement result q+1 The first unitary operation performed is denoted as:
b of the receiver Bob after the first unitary operation according to the POVM measurement result and the projection measurement result q+1 Performing a second unitary operation to recover the quantum state to be preparedComprising the following steps:
if the POVM measurement result is |P 0 >The projection measurement result is |O 0 >Then perform a second unitary operation
If the POVM measurement result is |P 0 >The projection measurement result is |O 1 >Then perform a second unitary operation
If the POVM measurement result is |P 1 >The projection measurement result is |O 0 >Then perform a second unitary operation
If the POVM measurement result is |P 1 >The projection measurement result is |O 1 >Then perform a second unitary operation
If the POVM measurement result is |P 2 >And returning to the preparation failure.
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