A method for sharing high-dimensional quantum states in an amplitude-damping channel

Abstract

The present application relates to a method for sharing high-dimensional quantum states in an amplitude damping channel, which includes two stages: the first stage is a process of sharing an entangled pure state among multiple parties in an amplitude damping channel based on coherence compensation. In the amplitude damping channel, it includes a distributor sharing an entangled pure state of a d-level quantum system with (n - 1) participants. The second stage is a process of probabilistically sharing quantum states based on the shared entangled pure state. If the distributor wants to share a single-particle unknown quantum state to be shared with (n - 1) participants, then the n-particle entangled pure state shared in the first stage is used as a quantum channel to obtain the tensor product state of the quantum state to be shared and the quantum channel. The present invention can perform coherence compensation on a high-energy-level quantum system, achieve a fidelity of 1 for reconstructing quantum states, and overcome the defect that the noise of the amplitude damping channel will cause the quantum system to decohere and reduce the fidelity of the reconstructed quantum state in the quantum state sharing scheme.

Description

Technical Field

[0001] This application relates to the technical field of multi-party quantum state sharing in quantum communication, and specifically relates to a method for sharing high-dimensional quantum states in an amplitude damping channel. Background Art

[0002] Quantum state sharing belongs to one of the basic researches in quantum cryptography and is crucial for constructing large-scale quantum communication networks. Based on the non-local property of quantum entanglement, quantum state sharing technology can achieve the long-distance transmission of unknown quantum states without transmitting the quantum states themselves, greatly improving the security of information transmission. On the other hand, multi-party sharing indicates that the secret quantum state can only be reconstructed when multiple participants cooperate with each other, further enhancing the security of quantum information.

[0003] Compared with two-level quantum systems, quantum communication based on high-level quantum systems (d-level quantum systems, qudits) has a larger information encoding space and can accommodate more channel noise. As a typical noise model in quantum channels, amplitude damping noise will cause the quantum system to decohere, so that the distribution of the maximally entangled state in the amplitude damping channel will degenerate into an entangled mixed state, and even in severe cases, the phenomenon of sudden death of entanglement will occur. Therefore, the amplitude damping noise in the channel will affect the performance of the quantum state sharing scheme, that is, reduce the fidelity of the reconstructed quantum state and cause the loss of quantum information. Summary of the Invention

[0004] The purpose of the present invention is to provide a method for sharing high-dimensional quantum states in an amplitude damping channel, which can compensate for the coherence of high-level quantum systems and achieve a fidelity of 1 for the reconstructed quantum state, thereby overcoming the defect that the amplitude damping channel noise will cause the quantum system to decohere and reduce the fidelity of the reconstructed quantum state in the quantum state sharing scheme.

[0005] The technical solution adopted by the present invention is: A method for sharing high-dimensional quantum states in an amplitude damping channel includes two stages:

[0006] The first stage is a process of sharing an entangled pure state among multiple parties in an amplitude damping channel based on a coherence compensation method. In the amplitude damping channel, the distributor Alice who includes the secret quantum state wants to share the entangled pure state of the d-level quantum system with (n - 1) participants Bob1, Bob2,..., Bob n-1 The specific steps of coherence compensation are as follows:

[0007] S101: The distributor Alice prepares the maximally entangled Bell state |ψ of two particles according to the single particle A and single particle T she has 0,0 > A,T, and send the single particle T to the participant Bob1 through the amplitude damping channel;

[0008] S102: When the participant Bob1 receives the single particle T, the participant Bob1 performs a joint unitary operation RS on the single particle B1 he prepares and the received single particle T C , the initial state of the single particle B1 is prepared in the computational basis state |0>, where |·> is a ket vector indicating that the object is a vector, and the single particle T and the single particle B1 are the control qubit and the target qubit respectively;

[0009] Subsequently, the participant Bob1 sends the single particle T to the participant Bob2 through the amplitude damping channel;

[0010] S103: The participant Bob2 and the remaining (n - 3) participants repeat the operation in step S102, and the last participant Bob n-1 sends the single particle T to the distributor Alice through the amplitude damping channel;

[0011] S104: After the distributor Alice receives the single particle T, she performs a joint unitary operation LS on the single particle A and the single particle T C , where the single particle A and the single particle T are the control qubit and the target qubit respectively;

[0012] Subsequently, the distributor Alice performs a single-particle measurement SPM operation on the single particle T in the Z basis {|Z j >, j = 0, 1,..., d - 1};

[0013] If the measurement result of the distributor Alice is the computational basis state |Z0>, it indicates that the coherence compensation is successful this time, and the distributor Alice and the (n - 1) participants have successfully realized the sharing of the GHZ entangled pure state. At this time, the (n - 1) participants and the distributor Alice share an n-particle entangled pure state If the measurement result of the distributor Alice is not the computational basis state |Z0>, it indicates that the coherence compensation is not successful this time. The distributor Alice and the (n - 1) participants continue to repeat the above process until the coherence compensation is successful and the sharing of the entangled pure state is realized;

[0014] The second stage is a probabilistic quantum state sharing process based on the shared entangled pure state. Under the amplitude damping channel, if the distributor Alice wants to share a single-particle unknown quantum state to be shared n-1 among the (n - 1) participants Bob1, Bob2,..., Bob then use the n-particle entangled pure state shared in the first stage as the quantum channel to obtain the quantum state |φ> to be shared S and the quantum channel tensor product state; where, β j is a complex number, |β j | 2 represents the probability that the quantum state |φ> is in the computational basis state |j>;

[0015] For the m-th participant Bob m , 1 ≤ m ≤ n - 1, when the participant Bob m is the reconstructing party, the remaining (n - 2) participants Bob1, …, Bob m-1 , Bob m+1 , …, Bob n-1 act as assisting parties to help the reconstructing party reconstruct the secret quantum state shared by the distributor Alice. The multi-party quantum state sharing based on entangled pure states in a high-dimensional quantum system includes the following steps:

[0016] S201: The distributor Alice performs a Bell state basis measurement (BSBM) operation on her 2-qudit (S, A); when the measurement result of the distributor Alice is |ψ t,h >, t, h ∈ {0, 1, …, d - 1}, calculate the collapsed state of the (n - 1)-qudit system owned by the (n - 1) participants where, |ψ t,h >, represents the maximum entangled Bell state of 2-qudit;

[0017] S202: The (n - 2) assisting parties perform single-particle measurements (SPM) on their respective single particles on the X basis {|X j >, j = 0, 1, …, d - 1}, where, the reconstructing party calculates the quantum state m collapsed by the local particle B

[0018] S203: The distributor Alice and the (n - 2) assisting parties transmit their respective measurement results to the participant Bob through an authenticated classical channel m , the participant Bob m after obtaining the measurement results, performs a unitary operation U m F on his local particle B m to achieve the quantum state evolution of the local particle B m

[0019] S204: The participant Bob m introduces an auxiliary particle Aux, and the initial state of the auxiliary particle Aux is prepared in the computational basis state |0>; based on the basis τ, the participant Bob m on his local particle B mPerform the joint unitary operation U on the local particle B and the auxiliary particle Aux Aux , and transform the local particle B m and the auxiliary particle Aux into a quantum state

[0020] S205: Participant Bob m Perform a single-particle measurement SPM operation in the Z basis on the auxiliary particle Aux. If the measurement result |Z0> is obtained, it indicates that Participant Bob m successfully restores the quantum state |φ shared by the distributor Alice on the local particle B m ; otherwise, the quantum state sharing fails and the quantum state |φ shared by Alice cannot be reconstructed S . S .

[0021] Furthermore, the calculation formula for the maximum entangled Bell state |ψ 0,0 > of the single particle A and the single particle T is: A,T

[0022]

[0023] The calculation formula for the maximum entangled Bell state |ψ t,h > of the 2 - qudit is:

[0024]

[0025] where the symbol represents the modulo operation of the modulo d operation, that is

[0026] Furthermore, the expression of the unitary operation RS C is:

[0027]

[0028] where j,k ∈ {0,1,…,d - 1},

[0029] The expression of the unitary operation LS C is:

[0030]

[0031] where the symbol represents the modulo operation of the modulo d operation, that is

[0032] Furthermore, the expression of the entangled pure state of the n - particle is:

[0033] ​

[0034] Among them, p represents the intensity of the noise in the amplitude damping channel.

[0035] Furthermore, the expression for the distributor Alice to perform the Bell state basis measurement (BSBM) operation on the 2 - qudit (S, A) she owns is:

[0036]

[0037] Among them, <·| is the bra vector, representing the dual vector of |·>, that is, <·| is equal to the conjugate transpose of the vector |·>;

[0038] The collapsed state of the (n - 1)-qudit system owned by the (n - 1) participants has the expression:

[0039]

[0040] Furthermore, the specific calculation process of the SPM operation on the X - basis in the step S202 is:

[0041]

[0042] Among them, represents the tensor product operation, and I d represents the d - dimensional identity matrix,

[0043] Furthermore, the collapsed quantum state has the expression:

[0044]

[0045] Furthermore, the expression of the unitary operation U F is:

[0046]

[0047] In the step S203, the quantum state of the particle B m is transformed into

[0048] Furthermore, the expression of the basis τ is:

[0049]

[0050] The expression of the joint unitary operation U Aux is:

[0051]

[0052] The quantum state has the following expression:

[0053]

[0054] where E1 and E2 are d×d diagonal matrices, 0 d represents a d-dimensional all-zero matrix,

[0055] The beneficial effects of the present invention are as follows:

[0056] (1) By cyclically transmitting single particles in the maximally entangled Bell state among all participants, during the particle transmission process, each participant can perform a joint controlled summation operation, the distributor performs a reverse controlled summation operation, and finally a Z-basis measurement on a single particle, thereby realizing the coherence compensation process of the entire quantum system, and thus realizing the sharing of entangled pure states of high-dimensional quantum systems in an amplitude damping channel;

[0057] (2) Based on the shared entangled pure state to construct a quantum channel, the distributor performs a Bell state basis measurement, the assisting party performs a single-particle X-basis measurement, classical channel communication, and the reconstructing party can realize the probabilistic quantum state sharing with a fidelity of 1 based on the entangled state under a high-dimensional quantum system by introducing auxiliary particles, performing a joint unitary operation, and performing a single-particle Z-basis measurement. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following will briefly introduce the drawings required to be used in the embodiments. Obviously, the drawings in the following description are only some embodiments of the present application. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0059] Figure 1 is the flowchart of the first stage of the embodiment of the present invention;

[0060] Figure 2 is the flowchart of the second stage of the embodiment of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0061] To better understand the above objects, features, and advantages of the present invention, the present invention will be further described in detail below in conjunction with the accompanying drawings and specific embodiments. In the following description, many specific details are set forth to fully understand the present invention. However, the present invention can also be implemented in other ways different from those described herein. Therefore, the present invention is not limited by the specific embodiments disclosed below.

[0062] A method for sharing high-dimensional quantum states in an amplitude-damping channel includes two stages:

[0063] As Figure 1 shown, the first stage is a process of sharing an entangled pure state among multiple parties in an amplitude-damping channel based on coherence compensation. In the amplitude-damping channel, the distributor Alice of the secret quantum state wants to share an entangled pure state of a d-level quantum system (qudit) with (n - 1) participants Bob1, Bob2,..., Bob n-1 where d is a positive integer greater than or equal to 3. The specific steps of coherence compensation are as follows:

[0064] S101: The distributor Alice prepares a two-particle maximally entangled Bell state |ψ 0,0 > A,T using the single particle A and single particle T she possesses, and sends the single particle T through the amplitude-damping channel to the participant Bob1. The formula for the two-particle maximally entangled Bell state |ψ 0,0 > A,T is:

[0065]

[0066] S102: When the participant Bob1 receives the single particle T, the participant Bob1 performs a joint unitary operation RS C on the single particle B1 he prepares and the received single particle T. The initial state of the single particle B1 is prepared in the computational basis state |0>, where |·> is a ket vector indicating that the object is a vector. The single particle T and the single particle B1 are the control qubit and the target qubit respectively. The expression of the unitary operation RS C is:

[0067]

[0068] where j, k ∈ {0, 1,..., d - 1}, and the symbol represents the modulo operation of modulo d arithmetic, that is

[0069] Subsequently, the participant Bob1 sends the single particle T through the amplitude-damping channel to the participant Bob2.

[0070] S103: Participant Bob2 and the remaining (n - 3) participants repeat the operation of step S102, and the last participant Bob n-1 sends the single - particle T to the distributor Alice through an amplitude - damping channel.

[0071] S104: After receiving the single - particle T, the distributor Alice performs a joint unitary operation LS on the single - particle A and the single - particle T C , where the single - particle A and the single - particle T are the control qubit and the target qubit respectively. The unitary operation LS C has the following expression:

[0072]

[0073] where the symbol represents the modulo operation of the modulo - d operation, that is

[0074] Subsequently, the distributor Alice performs a single - particle measurement SPM operation on the single - particle T in the Z - basis {|Zj>, j = 0, 1,..., d - 1}. j >, j = 0, 1,..., d - 1}.

[0075] If the measurement result of the distributor Alice is the computational basis state |Z0>, it indicates that the coherence compensation is successful this time, and the distributor Alice and the (n - 1) participants have successfully achieved the sharing of the GHZ entangled pure state. At this time, the (n - 1) - particle entangled pure state shared between the distributor Alice and the (n - 1) participants is If the measurement result of the distributor Alice is not the computational basis state |Z0>, it indicates that the coherence compensation is not successful this time. The distributor Alice and the (n - 1) participants continue to repeat the above process until the coherence compensation is successful and the sharing of the entangled pure state is achieved. The n - particle entangled pure state has the following expression:

[0076]

[0077] where p represents the intensity of the noise in the amplitude - damping channel.

[0078] As Figure 2 shown, the second stage is a process of probabilistically sharing a quantum state based on the shared entangled pure state. Under the amplitude - damping channel, if the distributor Alice wants to share a single - particle unknown quantum state to be shared n-1 with the (n - 1) participants Bob1, Bob2,..., Bob then uses the n - particle entangled pure state shared in the first stage as a quantum channel to obtain the quantum state to be shared |φ>S and the quantum channel in a tensor product state; where β j is a complex number, |β j | 2 represents the probability that the quantum state |φ> is in the computational basis state |j>. The quantum state |φ> to be shared S and the quantum channel in a tensor product state is expressed as:

[0079]

[0080] For the m-th participant Bob m , 1 ≤ m ≤ n - 1, when the participant Bob m is the reconstructing party, the remaining (n - 2) participants Bob1,..., Bob m-1 , Bob m+1 ,..., Bob n-1 act as assisting parties to help the reconstructing party reconstruct the secret quantum state shared by the distributor Alice. The multi-party quantum state sharing based on entangled pure states in a high-dimensional quantum system includes the following steps:

[0081] S201: The distributor Alice performs a Bell state basis measurement BSBM operation on her 2 - qudit (S, A), and the specific expression is:

[0082]

[0083] where <·| is the bra, representing the dual vector of |·>, that is, <·| is equal to the conjugate transpose of the vector |·>.

[0084] When the measurement result of the distributor Alice is |ψ t,h >, t, h ∈ {0, 1,..., d - 1}, calculate the collapsed state of the (n - 1)-qudit system owned by the (n - 1) participants where |ψ t,h > represents the maximally entangled Bell state of 2 - qudit:

[0085]

[0086] where the symbol represents the modulo operation of the modulo d operation, that is

[0087] The collapsed state of the (n - 1)-qudit system owned by the (n - 1) participants is expressed as:

[0088]

[0089] S202: (n - 2) assisting parties perform single - particle measurement SPM operations on the single particles they each own in the X - basis {|X j >, j = 0, 1, …, d - 1}, where the reconstructing party calculates the quantum state of the collapsed local particle B according to the X - basis measurement results of the (n - 2) assisting parties m the collapsed quantum state

[0090] The specific calculation process of the single - particle measurement SPM operation on the X - basis is as follows:

[0091]

[0092] where denotes the tensor - product operation, and I d denotes the d - dimensional identity matrix

[0093] The collapsed quantum state has the following expression:

[0094]

[0095] S203: The distributor Alice and the (n - 2) assisting parties transmit their respective measurement results to the participant Bob through an authenticated classical channel m , and the participant Bob m , after obtaining the measurement results, performs a unitary operation U m on the local particle B he owns F , and the unitary operation U F has the following expression:

[0096]

[0097] By performing the unitary operation U F on the quantum state of the local particle B m , the quantum state of the local particle B m is transformed into

[0098]

[0099] S204: The participant Bob m introduces an auxiliary particle Aux, and the initial state of the auxiliary particle Aux is prepared in the computational - basis state |0>; based on the basis τ, the participant Bob m performs a joint unitary operation U m on the local particle B Aux and the auxiliary particle Aux, and transforms the local particle B m and the auxiliary particle Aux into a quantum state

[0100] The expression of the base τ is as follows:

[0101]

[0102] The joint unitary operation U Aux has the following expression:

[0103]

[0104] The quantum state has the following expression:

[0105]

[0106] where E1 and E2 are d×d diagonal matrices, and 0 d represents a d-dimensional all-zero matrix,

[0107] S205: Participant Bob m performs a single-particle measurement SPM operation in the Z basis on the auxiliary particle Aux. If the measurement result |Z0> is obtained, it indicates that participant Bob m has successfully restored the quantum state |φ> shared by the distributor Alice on the local particle B m ; otherwise, the sharing of the quantum state fails, and the quantum state |φ> shared by Alice cannot be reconstructed S . S .

[0108] In the embodiments of the present invention, by cyclically transmitting single particles in the maximally entangled Bell state among all participants, during the particle transmission process, each participant can achieve the coherence compensation process of the entire quantum system by performing a joint controlled summation operation, the distributor finally performs a reverse controlled summation operation, and a single-particle Z-basis measurement, thereby realizing the sharing of entangled pure states of high-dimensional quantum systems in an amplitude damping channel. Based on the shared entangled pure state, a quantum channel is constructed. The distributor performs a Bell state basis measurement, the assistant performs a single-particle X-basis measurement, classical channel communication, and the reconstructor realizes the precise restoration of the quantum state shared by the distributor Alice by introducing auxiliary particles, performing a joint unitary operation, and performing a single-particle Z-basis measurement, that is, it can realize the probabilistic quantum state sharing with a fidelity of 1 based on entangled states in a high-dimensional quantum system.

[0109] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. For those skilled in the art, the present invention may have various modifications and changes. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A method for sharing high-dimensional quantum states in an amplitude-damping channel, characterized in that, It includes two stages: The first stage is a process of sharing an entangled pure state among multiple parties in an amplitude damping channel based on a coherence compensation method. In the amplitude damping channel, the distributor Alice, who holds the secret quantum state, wants to share the entangled pure state of a d-level quantum system with (n - 1) participants Bob1, Bob2, …, Bob n-1 The specific steps of coherence compensation are as follows: S101: The distributor Alice prepares the maximally entangled Bell state |ψ of two particles according to the single particle A and single particle T she owns 0,0 > A,T , and sends the single particle T to the participant Bob1 through an amplitude damping channel; S102: When participant Bob1 receives the single particle T, participant Bob1 performs a joint unitary operation RS on the single particle B1 he prepared and the received single particle T C , the initial state of the single particle B1 is prepared in the computational basis state |0>, where |·> is a ket vector indicating that the object is a vector, and the single particle T and the single particle B1 are the control qubit and the target qubit respectively; Subsequently, the participant Bob1 sends a single particle T to the participant Bob2 through an amplitude damping channel; S103: Participant Bob2 and the remaining (n - 3) participants repeat the operation in step S102, and the last participant Bob n-1 sends the single particle T to the distributor Alice through an amplitude damping channel; S104: After the distributor Alice receives the single particle T, perform a joint unitary operation LS on the single particle A and the single particle T C , where the single particle A and the single particle T are the control qubit and the target qubit respectively; Subsequently, the distributor Alice performs a single-particle measurement SPM operation on the single particle T in the Z basis {|Z j >, j = 0, 1, ..., d - 1}; If the measurement result of the distributor Alice is the computational ground state |Z0>, it indicates that the coherence compensation is successful this time, and the distributor Alice and (n - 1) participants have successfully achieved the sharing of the GHZ entangled pure state. At this time, what is shared between the distributor Alice and (n - 1) participants is an n-particle entangled pure state If the measurement result of the distributor Alice is not the computational ground state |Z0>, it indicates that the coherence compensation is not successful this time. The distributor Alice and (n - 1) participants continue to repeat the above process until the coherence compensation is successful and the sharing of the entangled pure state is achieved; The second stage is a probabilistic quantum state sharing process based on the shared entangled pure state. Under the amplitude damping channel, if the distributor Alice wants to share an unknown single-particle quantum state to be shared with (n - 1) participants Bob1, Bob2, …, Bob n-1 share the unknown single-particle quantum state to be shared among them then use the n-particle entangled pure state shared in the first stage as the quantum channel to obtain the quantum state |φ> to be shared S and the tensor product state of the quantum channel ; where β j is a complex number, and |β j | 2 represents the probability that the quantum state |φ> is in the computational basis state |j>; For the m-th participant Bob m , where 1 ≤ m ≤ n - 1, when the participant Bob m is the reconstructing party, the remaining (n - 2) participants Bob1, …, Bob m-1 , Bob m+1 , …, Bob n-1 act as assisting parties to help the reconstructing party reconstruct the secret quantum state shared by the distributor Alice. The multi-party quantum state sharing based on entangled pure states under a high-dimensional quantum system includes the following steps: S201: The distributor Alice performs the Bell state basis measurement (BSBM) operation on the 2 - qudit (S, A) she owns; when the measurement result of the distributor Alice is |ψ t,h >, where t, h ∈ {0, 1, …, d - 1}, calculate the collapsed state of the (n - 1) - qudit system owned by the (n - 1) participants where, |ψ t,h > represents the maximally entangled Bell state of 2 - qudits; S202: (n - 2) assisting parties perform single - particle measurement SPM operations on the single particles they each own in the X - basis {|X j >, j = 0, 1, …, d - 1}, where The reconstructing party calculates the quantum state after the collapse of the local particle B according to the X - basis measurement results of the (n - 2) assisting parties m Collapsed quantum state S203: The distributor Alice and (n - 2) assisting parties transmit their respective measurement results to the participant Bob through an authenticated classical channel m , the participant Bob m After obtaining the measurement results, performs the unitary operation U on the local particle B m owned by him F to realize the quantum state evolution of the local particle B m ; S204: Participant Bob m introduces an auxiliary particle Aux, and the initial state of the auxiliary particle Aux is prepared in the computational ground state |0>; based on the basis τ, Participant Bob m performs a joint unitary operation U m on the local particle B Aux and the auxiliary particle Aux, and transforms the local particle B m and the auxiliary particle Aux into a quantum state S205: Participant Bob m Perform a single-particle measurement (SPM) operation in the Z basis on the auxiliary particle Aux. If the measurement result |Z0> is obtained, it indicates that Participant Bob m has successfully restored the quantum state |φ> shared by the distributor Alice on the local particle B m ; otherwise, the sharing of the quantum state fails and the quantum state |φ> shared by Alice cannot be reconstructed S . S .

2. A method for sharing high-dimensional quantum states in an amplitude-damping channel according to claim 1, characterized in that The maximum entangled Bell state |ψ of single particle A and single particle T 0,0 > A,T The calculation formula is as follows: The calculation formula for the maximum entangled Bell state |ψ t,h > of 2 - qudits is as follows: Among them, the defined symbol represents the modulo operation of modulo d, that is 3. A method for sharing high-dimensional quantum states in an amplitude-damping channel according to claim 2, characterized in that, The unitary operation RS C has the following expression: where \(j,k\in\{0,1,\ldots,d - 1\}\), The unitary operation LS C has the following expression: Among them, the defined symbol represents the modulo operation of modulo d, that is 4. A method for sharing high-dimensional quantum states in an amplitude-damping channel according to claim 3, characterized in that, The entangled pure state of the n particles has the following expression: where p represents the intensity of the noise in the amplitude damping channel.

5. A method for sharing high-dimensional quantum states in an amplitude-damping channel according to claim 4, wherein The expression for the distributor Alice to perform the Bell state basis measurement BSBM operation on the 2 - qudit (S,A) she owns is: where <·| is the bra, representing the dual vector of |·>, that is, <·| is equal to the conjugate transpose of the vector |·>; The collapsed state of the (n - 1)-qudit system owned by the said (n - 1) participants has the following expression:

6. A method for sharing high-dimensional quantum states in an amplitude-damped channel according to claim 5, characterized in that, The specific calculation process of the SPM operation on the X - basis in the step S202 is: Among them, represents the tensor product operation, and I d represents the d-dimensional identity matrix, 7. A method for sharing high-dimensional quantum states in an amplitude-damped channel according to claim 6, characterized in that The collapsed quantum state has the following expression:

8. A method for sharing high-dimensional quantum states in an amplitude-damping channel, according to claim 7, wherein The unitary operation U F has the following expression: The particle B in the step S203 m has its quantum state transformed into 9. A method for sharing high-dimensional quantum states in an amplitude-damping channel according to claim 8, characterized in that The expression for the basis τ is: The combined unitary operation U Aux has the following expression: The quantum state has the following expression: where E1 and E2 are d×d diagonal matrices, and 0 d represents a d-dimensional all-zero matrix,

Images