CN103368658A - Four-photon entangled W state-based super-dense coding method for quantum signaling - Google Patents

Abstract

The invention discloses a four-photon entangled W state-based super-dense coding method for quantum signaling. The problems of low signaling coding capacity, low efficiency and poor security in a quantum communication process are mainly solved. The method is implemented by a process comprising the following steps of preparing four-photon (A, B, C and D) entangled W state as the quantum signaling, and transmitting the photons C and D to a receiver by using a transmitter; performing unitary transformation on the photons A and B to obtain 16 new quantum states; forming an orthogonal basis by using the 16 new quantum states; coding the new quantum states respectively, and sequentially transmitting photons A' and B' obtained by the unitary transformation to the receiver; and after the receiver receives the photons A' and B', arbitrarily selecting a measurement basis from the orthogonal basis, and performing joint measurement on the photons A', B', C and D to measure a new quantum state to finish coding. The method has the advantages of high coding efficiency, high capacity and high security, and can be used for a signaling system for quantum mobile communication.

Description

Quantum signaling ultra-dense coding method based on four-photon entangled W state

Technical Field

The invention belongs to the technical field of coding, in particular to a quantum signaling ultra-dense coding method which can be used for a quantum communication network.

Background

Signaling is an essential and important component of any communication system, and quantum communication is no exception. In a multi-user quantum communication network, the encoding mode of quantum signaling directly affects the transmission efficiency and security of a signaling system.

The existing quantum coding scheme mainly comprises amplitude coding, frequency coding, phase coding, differential phase coding, two-photon entanglement super-dense coding, three-photon entanglement controlled super-dense coding and four-photon entanglement controlled super-dense coding. Wherein:

the coding efficiency of amplitude coding, frequency coding, phase coding and differential phase coding is low, and only 1 bit of information can be transmitted by one-time coding.

Two photons are entangled and encoded in an ultra-dense way, 2 bits of classical information can be transmitted by one encoding, but the capacity of a signaling system is large in the process of communication among multiple users, and the encoding mode needs to be encoded for multiple times to meet the communication requirement and is complicated;

three-photon entanglement controlled ultra-dense coding needs a control party besides a receiving party and a sending party, the coding efficiency is mainly related to the measurement angle when the control party carries out von Neumann measurement on own photons, the classical information transmitted by one-time coding is less than 3 bits due to the limitation of the angle, the coding process is particularly complicated, and the coding efficiency is low;

the four-photon entanglement controlled super-dense coding has two control parties besides a receiving party and a sending party, and as with three photons, the coding efficiency is mainly related to the measurement angle of the control party when the control party carries out von Neumann measurement on own photons, because of the limitation of the angle, the coding efficiency of the four-photon entanglement controlled super-dense coding is less than 4 bits, compared with three photons, the coding capacity is improved, but the coding efficiency is basically unchanged.

Disclosure of Invention

The invention aims to provide a quantum signaling ultra-dense coding method based on a four-photon entanglement W state aiming at the defects of the four-photon entanglement controlled ultra-dense coding method so as to improve the capacity and efficiency of primary coding.

The technical idea for realizing the invention is that quantum signaling is expressed by using the four-photon entangled W state, an entangled channel is directly established, the four-photon entangled state is subjected to corresponding unitary transformation according to the classical information to be transmitted to obtain a new quantum state, and then the new quantum state is coded. The method comprises the following specific steps:

(1) sender Alice uses ultraviolet pulsed lightTwo barium metaborate crystals are continuously passed to generate a W state phi entangled by four photons, namely a first photon A, a second photon B, a third photon C and a fourth photon D>_ABCDAs quantum signaling, and distributing the third photon C and the fourth photon D to the receiver Bob through the channel, so that the sender Alice and the receiver Bob share the W state | phi |>_ABCD；

(2) The sender Alice performs unitary transformation on the first photon A and the second photon B owned by the sender Alice to complete coding, and obtains a new quantum state | phi_m>_A′B′CDWherein, a 'and B' are photons obtained by performing unitary transformation on the photons a and B, and m = {1,2,3,4,5,6,7,8, 9,10,11,12,13,14,15,16 };

(3) after the sender Alice finishes coding, the photons A 'and B' obtained through the unitary-to-unitary conversion are sequentially rearranged and sent to a receiver Bob through a channel according to the sequence of A 'B' or B 'A';

(4) after receiving the photons A 'and B', the receiver Bob selects a measuring base, and performs combined measurement on the four photons A ', B' and C, D to obtain a new quantum state | phi_m>_A′B′CDTo complete decoding and obtain the classical signaling information transmitted by Alice.

The invention has the following advantages:

1. according to the invention, the quantum signaling is represented by adopting the four-photon entanglement W state, so that an entanglement channel can be directly established, channel resources are saved, and communication efficiency is improved;

2. the invention improves the coding capacity by performing unitary conversion on the quantum state of the signaling, so that 4 bits of classical signaling information can be transmitted by one-time coding;

3. in the encoding process, the sender Alice can transmit 4 bits of classical signaling information only by transmitting two photons, so the encoding efficiency is high;

4. after the encoding is finished, the sending party Alice rearranges the sending sequence of each photon when the photon obtained after the unitary conversion is sent to the receiving party Bob, thereby improving the encoding safety and ensuring the smooth proceeding of the quantum communication.

Drawings

FIG. 1 is a flow chart of quantum signaling encoding of the present invention

Fig. 2 is a transmission diagram of quantum signaling according to the present invention.

FIG. 3 is a diagram of a quantum signaling encoding format in the present invention;

Detailed Description

Referring to fig. 1, the specific implementation steps of the present invention are as follows:

step 1, preparing and distributing four-photon entanglement pairs;

1.1) Alice is used as a signaling sender, and ultraviolet pulsed light is utilized to continuously pass through two barium metaborate crystals to prepare n ordered four-photon A_k,B_k,C_k,D_kEntangled W state of

$|\mathrm{\Φ}{>}_{A_k{B}_k{C}_k{D}_k}=\frac{1}{2}|0001>+\frac{1}{2}|0010>+\frac{1}{2}|0100>+\frac{1}{2}|1000>,$

Wherein k =1,2,3,4, …, n, n ≧ 1, 0 denotes that the photon spin direction is horizontal state, 1 denotes that the photon spin direction is vertical state;

1.2) Alice will emit a photon C_kAnd D_kForm a sequence Q_B={C_k,D_kSending the photons to a receiver Bob, and storing the photons A by the receiver Bob_kAnd B_kComposed of sequence Q_A={A_k，B_k}，

As shown in FIG. 2, Alice converts the photon sequence Q_B={C_k,D_kThe quantum exchange A transmits the photon sequence Q to a quantum exchange A through a quantum channel, and the quantum exchange A transmits the photon sequence Q according to the address of a switching center where a receiver Bob is positioned_B={C_k,D_kThe quantum exchange B selects a proper route according to the destination address of Bob and transmits the photon sequence Q to a quantum exchange B of the exchange center through a quantum channel_B={C_k,D_kIs transmitted to Bob;

1.3) when Bob receives the sequence Q_BThen, the sender Alice and the receiver Bob establish an entangled channel, and the two parties share the four-photon W state

Step 2, Alice entangles the four photons into W state

As quantum signaling, and for photon A_kAnd B_kAnd performing unitary transformation to complete the coding of the quantum signaling.

As shown in fig. 3, the format of quantum signaling coding is composed of signaling units, the signaling unit is the minimum unit of signaling messages, the length is 8 bits, and the coding can be implemented by coding the quantum signaling twice.

2.1) the sender Alice selects a corresponding unitary operator from the unitary operator set according to the classic signaling message to be transmitted;

the sum of the unitary computations is: { U₀,U₁,U₂,U₃,U₄,U₅,U₆,U₇,U₈,U₉,U₁₀,U₁₁,U₁₂,U₁₃,U₁₄,U₁₅And each element in the operator set is a 16-dimensional matrix, and each matrix is represented as:

$U_0={\mathrm{\δ}}_0\⊗{\mathrm{\δ}}_0\⊗{\mathrm{\δ}}_0\⊗{\mathrm{\δ}}_0,U_1={\mathrm{\δ}}_3\⊗{\mathrm{\δ}}_3\⊗{\mathrm{\δ}}_0\⊗{\mathrm{\δ}}_0,U_2={\mathrm{\δ}}_1\⊗{\mathrm{\δ}}_0\⊗{\mathrm{\δ}}_0\⊗{\mathrm{\δ}}_0,$

$U_3={-\mathrm{i\δ}}_2\⊗{\mathrm{\δ}}_3\⊗{\mathrm{\δ}}_0\⊗{\mathrm{\δ}}_0,U_4={\mathrm{\δ}}_1\⊗{\mathrm{\δ}}_1\⊗{\mathrm{\δ}}_1\⊗{\mathrm{\δ}}_1,U_5={-\mathrm{\δ}}_2\⊗{\mathrm{\δ}}_2\⊗{\mathrm{\δ}}_1\⊗{\mathrm{\δ}}_1,$

$U_6={\mathrm{\δ}}_0\⊗{\mathrm{\δ}}_1\⊗{\mathrm{\δ}}_1\⊗{\mathrm{\δ}}_1,U_7={\mathrm{\δ}}_0\⊗{\mathrm{\δ}}_1\⊗{\mathrm{\δ}}_2\⊗{\mathrm{\δ}}_2,U_8={\mathrm{\δ}}_0\⊗{\mathrm{\δ}}_3\⊗{\mathrm{\δ}}_1\⊗{\mathrm{\δ}}_2,$

$U_{12}={-\mathrm{\δ}}_1\⊗{\mathrm{i\δ}}_2\⊗{\mathrm{\δ}}_0\⊗{\mathrm{\δ}}_3,U_{13}={-\mathrm{\δ}}_1\⊗{\mathrm{i\δ}}_2\⊗{\mathrm{\δ}}_3\⊗{\mathrm{\δ}}_0,U_{14}={\mathrm{\δ}}_0\⊗{\mathrm{i\δ}}_2\⊗{\mathrm{\δ}}_0\⊗{\mathrm{\δ}}_3,$

$U_{15}={\mathrm{\δ}}_0\⊗{-\mathrm{i\δ}}_2\⊗{\mathrm{\δ}}_3\⊗{\mathrm{\δ}}_0;$

wherein, i represents an imaginary number,

representing the tensor product, δ₀,δ₁,δ₂,δ₃In order to be a Pauli matrix, ${\mathrm{\δ}}_0=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],$ ${\mathrm{\δ}}_1=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],{\mathrm{\δ}}_2=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right],{\mathrm{\δ}}_3=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right];$

2.2) after Alice selects an operator, it will select photon A_kAnd B_kPerforming unitary conversion to obtain new quantum state

Wherein, A'_kAnd B_′k is a photon A_kAnd photon B_kThe photon obtained by unitary transformation, m = {1,2,3,4,5,6,7,8, 9,10,11,12,13,14,15,16}, each new quantum state

The quantum representation of (a) is:

$|{\mathrm{\Φ}}_1{>}_{A_k^{\′}{B}_k^{\′}{C}_k{D}_k}=U_0|\mathrm{\Φ}{>}_{A_k{B}_k{C}_k{D}_k}=\frac{1}{2}|0001>+\frac{1}{2}|0010>+\frac{1}{2}|0100>+\frac{1}{2}|1000>,$

$|{\mathrm{\&Phi;}}_2{>}_{A_k^{\&prime;}{B}_k^{\&prime;}{C}_k{D}_k}={\mathrm{<figure-callout\; id="1"\; label="U"\; filenames="HDA00003465918000022.png"\; state="\{\{state\}\}">U</figure-callout>}}_1|\mathrm{\&Phi;}{>}_{A_k{B}_k{C}_k{D}_k}=\frac{1}{2}|0001>+\frac{1}{2}|0010>-\frac{1}{2}|0100>-\frac{1}{2}|1000>,$

$|{\mathrm{\&Phi;}}_3{>}_{A_k^{\&prime;}{B}_k^{\&prime;}{C}_k{D}_k}=U_2|\mathrm{\&Phi;}{>}_{A_k{B}_k{C}_k{D}_k}=\frac{1}{2}|1001>+\frac{1}{2}|1010>+\frac{1}{2}|1100>+\frac{1}{2}|0000>,$

$|{\mathrm{\&Phi;}}_4{>}_{A_k^{\&prime;}{B}_k^{\&prime;}{C}_k{D}_k}=U_3|\mathrm{\&Phi;}{>}_{A_k{B}_k{C}_k{D}_k}=\frac{1}{2}|1001>+\frac{1}{2}|1010>-\frac{1}{2}|1100>-\frac{1}{2}|0000>,$

$|{\mathrm{\&Phi;}}_5{>}_{A_k^{\&prime;}{B}_k^{\&prime;}{C}_k{D}_k}=U_4|\mathrm{\&Phi;}{>}_{A_k{B}_k{C}_k{D}_k}=\frac{1}{2}|1110>+\frac{1}{2}|1101>+\frac{1}{2}|1011>+\frac{1}{2}|0111>,$

$|{\mathrm{\&Phi;}}_6{>}_{A_k^{\&prime;}{B}_k^{\&prime;}{C}_k{D}_k}=U_5|\mathrm{\&Phi;}{>}_{A_k{B}_k{C}_k{D}_k}=\frac{1}{2}|1110>+\frac{1}{2}|1101>-\frac{1}{2}|1011>-\frac{1}{2}|0111>,$

$|{\mathrm{\&Phi;}}_7{>}_{A_k^{\&prime;}{B}_k^{\&prime;}{C}_k{D}_k}=U_6|\mathrm{\&Phi;}{>}_{A_k{B}_k{C}_k{D}_k}=\frac{1}{2}|0110>+\frac{1}{2}|0101>+\frac{1}{2}|0011>+\frac{1}{2}|1111>,$

$|{\mathrm{\&Phi;}}_8{>}_{A_k^{\&prime;}{B}_k^{\&prime;}{C}_k{D}_k}=U_7|\mathrm{\&Phi;}{>}_{A_k{B}_k{C}_k{D}_k}=\frac{1}{2}|0110>+\frac{1}{2}|0101>-\frac{1}{2}|0011>-\frac{1}{2}|1111>,$

$|{\mathrm{\&Phi;}}_9{>}_{A_k^{\&prime;}{B}_k^{\&prime;}{C}_k{D}_k}=U_8|\mathrm{\&Phi;}{>}_{A_k{B}_k{C}_k{D}_k}=\frac{1}{2}|0010>-\frac{1}{2}|0001>+\frac{1}{2}|0111>-\frac{1}{2}|1011>,$

$|{\mathrm{\&Phi;}}_{10}{>}_{A_k^{\&prime;}{B}_k^{\&prime;}{C}_k{D}_k}=U_9|\mathrm{\&Phi;}{>}_{A_k{B}_k{C}_k{D}_k}=\frac{1}{2}|0010>-\frac{1}{2}|0001>-\frac{1}{2}|0111>+\frac{1}{2}|1011>,$

$|{\mathrm{\&Phi;}}_{11}{>}_{A_k^{\&prime;}{B}_k^{\&prime;}{C}_k{D}_k}=U_{10}|\mathrm{\&Phi;}{>}_{A_k{B}_k{C}_k{D}_k}=\frac{1}{2}|1001>+\frac{1}{2}|0011>-\frac{1}{2}|1010>-\frac{1}{2}|1111>,$

$|{\mathrm{\&Phi;}}_{12}{>}_{A_k^{\&prime;}{B}_k^{\&prime;}{C}_k{D}_k}=U_{11}|\mathrm{\&Phi;}{>}_{A_k{B}_k{C}_k{D}_k}=\frac{1}{2}|1001>-\frac{1}{2}|0011>-\frac{1}{2}|1010>+\frac{1}{2}|1111>,$

$|{\mathrm{\&Phi;}}_{13}{>}_{A_k^{\&prime;}{B}_k^{\&prime;}{C}_k{D}_k}=U_{12}|\mathrm{\&Phi;}{>}_{A_k{B}_k{C}_k{D}_k}=\frac{1}{2}|0100>-\frac{1}{2}|1000>-\frac{1}{2}|1101>+\frac{1}{2}|1110>,$

$|{\mathrm{\&Phi;}}_{14}{>}_{A_k^{\&prime;}{B}_k^{\&prime;}{C}_k{D}_k}=U_{13}|\mathrm{\&Phi;}{>}_{A_k{B}_k{C}_k{D}_k}=\frac{1}{2}|0100>-\frac{1}{2}|1000>+\frac{1}{2}|1101>-\frac{1}{2}|1110>,$

$|{\mathrm{\&Phi;}}_{15}{>}_{A_k^{\&prime;}{B}_k^{\&prime;}{C}_k{D}_k}=U_{14}|\mathrm{\&Phi;}{>}_{A_k{B}_k{C}_k{D}_k}=\frac{1}{2}|0101>+\frac{1}{2}|0000>-\frac{1}{2}|0110>-\frac{1}{2}|1100>,$

$|{\mathrm{\&Phi;}}_{16}{>}_{A_k^{\&prime;}{B}_k^{\&prime;}{C}_k{D}_k}=U_{15}|\mathrm{\&Phi;}{>}_{A_k{B}_k{C}_k{D}_k}=\frac{1}{2}|0101>-\frac{1}{2}|0000>-\frac{1}{2}|0110>+\frac{1}{2}|1100>,$

these 16 new quantum states constitute a complete orthonormal basis

New quantum states

Satisfy the relation:

$\left\{\begin{array}{c}<{\mathrm{\&Phi;}}_m|{\mathrm{\&Phi;}}_j>={\mathrm{\&delta;}}_{\mathrm{mj}}\\ \underset{m}{\mathrm{\&Sigma;}}\underset{j}{\mathrm{\&Sigma;}}<{\mathrm{\&Phi;}}_m|{\mathrm{\&Phi;}}_j>=I\end{array}\right.$

wherein,<Φ_m|Φ_j>representing quantum states

And

the inner product of (a) is, ${\mathrm{\&delta;}}_{\mathrm{ij}}=\left\{\begin{array}{c}1,i=j\\ 0,i\&NotEqual;j\end{array},\right.$ $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],$ m,j∈[1,16]；

2.3) Alice pairs each new quantum state obtained

Are coded separately, i.e. about

The code is a code of 0000 and the code is a code of 0000,

the code is a code of 0001 and is a code of 0001,

the code is 0010 and the code is,

the code is 0101 and the code is,

the code is 0100, and the code is,

the code is 0101 and the code is,

the code is 0110 and the code is,

the code is 0111 and the code is,

the code is a code of 1000 and the code is a code,

the code is a code of 1001 and the code is,

the code is a code of 1010, and the code is,

the code is 1011 of the number of codes,

the code is coded into a code of 1100,

the code is 1101 and the code is provided with,

the code is a code of 1110 (one-time code),

the code is 1111.

And 3, the sender Alice sends the photon sequence to the receiver Bob.

3.1) after the coding is finished, Alice converts the unitary to obtain the photon A_k' and photon B_k' component photon sequence Q_A′={A₁′，B₁′，A₂′，B₂′…A_k′，B_k′…A_n′，B_n′}；

3.2) Alice to photon sequence Q_A′={A₁′，B₁′，A₂′，B₂′…A_k′，B_k′…A_n′，B_n'all photons in the' are arranged according to the first order B_k', rear array A_k' the order is rearranged to obtain a new photon sequence Q_A′′；

3.3) Alice converts the new photon sequence Q_A'' is sent to the recipient Bob over a quantum channel and informs Bob of the photon A over a classical channel_k' and photon B_k' in a novel photon sequence Q_A'' in the specification.

And 4, decoding the received quantum signaling by the receiver Bob.

4.1) the receiver Bob receives the new photon sequence Q_A'' thereafter, the photon sequence Q_A'' recovery to photon sequence Q_A′={A₁′，B₁′，A₂′，B₂′…A_k′，B_k′…A_n′，B_n′}；

4.2) Bob from complete orthogonal baseSelecting any one of them as a measuring base;

4.3) Bob uses the measurement basis to said four photons A_k′，B_k′，C_k，D_kPerforming combined measurement to obtain new quantum state

And finishing decoding to obtain the classical signaling information sent by Alice.

Concluding sentence

The foregoing description is only an example of the present invention and should not be construed as limiting the invention in any way, and it will be apparent to those skilled in the art that various changes and modifications in form and detail may be made therein without departing from the principles and arrangements of the invention, but such changes and modifications are within the scope of the invention as defined by the appended claims.

Claims (4)

1. A quantum signaling ultra-dense coding method based on four-photon entangled W state comprises the following steps:

(1) a sender Alice uses ultraviolet pulsed light to continuously pass through two barium metaborate crystals to generate a W state | phi which is entangled by four photons, namely a first photon A, a second photon B, a third photon C and a fourth photon D>_ABCDAs quantum signaling, and distributing the third photon C and the fourth photon D to the receiver Bob through the channel, so that the sender Alice and the receiver Bob share the W state | phi |>_ABCD；

(2) SendingThe party Alice performs unitary transformation on the first photon A and the second photon B owned by the party Alice to complete coding, and obtains a new quantum state | phi_m>_A′B′CDWherein, a 'and B' are photons obtained by performing unitary transformation on the photons a and B, and m = {1,2,3,4,5,6,7,8, 9,10,11,12,13,14,15,16 };

(3) after the sender Alice finishes coding, the photons A 'and B' obtained through the unitary-to-unitary conversion are sequentially rearranged and sent to a receiver Bob through a channel according to the sequence of A 'B' or B 'A';

(4) after receiving the photons A 'and B', the receiver Bob selects a measuring base, and performs combined measurement on the four photons A ', B' and C, D to obtain a new quantum state | phi_m>_A′B′CDTo complete decoding and obtain the classical signaling information transmitted by Alice.

2. The quantum signaling ultra-dense coding method based on four-photon entangled W state according to claim 1, wherein the four-photon entangled W state | Φ in the step (1)>_ABCDThe quantum state expression is as follows:

$|\mathrm{\&Phi;}{>}_{ABCD}=\frac{1}{2}|0001>+\frac{1}{2}|0010>+\frac{1}{2}|0100>+\frac{1}{2}|1000>$

where 0 indicates that the spin direction of the photon is the horizontal state and 1 indicates that the spin direction of the photon is the vertical state.

3. The quantum signaling super-dense coding method based on the four-photon entangled W state according to claim 1, wherein in the step (2), the sender Alice performs unitary transformation on the first photon A and the second photon B owned by itself, and the method comprises the following steps:

(2a) from unitary operator set { U₀,U₁,U₂,U₃,U₄,U₅,U₆,U₇,U₈,U₉,U₁₀,U₁₁,U₁₂,U₁₃,U₁₄,U₁₅Selecting a unitary operator, wherein each element in the operator set is a 16-dimensional matrix, and each matrix is represented as:

$U_0={\mathrm{\&delta;}}_0\&CircleTimes;{\mathrm{\&delta;}}_0\&CircleTimes;{\mathrm{\&delta;}}_0\&CircleTimes;{\mathrm{\&delta;}}_0,U_1={\mathrm{\&delta;}}_3\&CircleTimes;{\mathrm{\&delta;}}_3\&CircleTimes;{\mathrm{\&delta;}}_0\&CircleTimes;{\mathrm{\&delta;}}_0,U_2={\mathrm{\&delta;}}_1\&CircleTimes;{\mathrm{\&delta;}}_0\&CircleTimes;{\mathrm{\&delta;}}_0\&CircleTimes;{\mathrm{\&delta;}}_0,$

$U_3={-\mathrm{i\&delta;}}_2\&CircleTimes;{\mathrm{\&delta;}}_3\&CircleTimes;{\mathrm{\&delta;}}_0\&CircleTimes;{\mathrm{\&delta;}}_0,U_4={\mathrm{\&delta;}}_1\&CircleTimes;{\mathrm{\&delta;}}_1\&CircleTimes;{\mathrm{\&delta;}}_1\&CircleTimes;{\mathrm{\&delta;}}_1,U_5={-\mathrm{\&delta;}}_2\&CircleTimes;{\mathrm{\&delta;}}_2\&CircleTimes;{\mathrm{\&delta;}}_1\&CircleTimes;{\mathrm{\&delta;}}_1,$

$U_6={\mathrm{\&delta;}}_0\&CircleTimes;{\mathrm{\&delta;}}_1\&CircleTimes;{\mathrm{\&delta;}}_1\&CircleTimes;{\mathrm{\&delta;}}_1,U_7={\mathrm{\&delta;}}_0\&CircleTimes;{\mathrm{\&delta;}}_1\&CircleTimes;{\mathrm{\&delta;}}_2\&CircleTimes;{\mathrm{\&delta;}}_2,U_8={\mathrm{\&delta;}}_0\&CircleTimes;{\mathrm{\&delta;}}_3\&CircleTimes;{\mathrm{\&delta;}}_1\&CircleTimes;{\mathrm{\&delta;}}_2,$

$U_{12}={-\mathrm{\&delta;}}_1\&CircleTimes;{\mathrm{i\&delta;}}_2\&CircleTimes;{\mathrm{\&delta;}}_0\&CircleTimes;{\mathrm{\&delta;}}_3,U_{13}={-\mathrm{\&delta;}}_1\&CircleTimes;{\mathrm{i\&delta;}}_2\&CircleTimes;{\mathrm{\&delta;}}_3\&CircleTimes;{\mathrm{\&delta;}}_0,U_{14}={\mathrm{\&delta;}}_0\&CircleTimes;{\mathrm{i\&delta;}}_2\&CircleTimes;{\mathrm{\&delta;}}_0\&CircleTimes;{\mathrm{\&delta;}}_3,$

$U_{15}={\mathrm{\&delta;}}_0\&CircleTimes;{-\mathrm{i\&delta;}}_2\&CircleTimes;{\mathrm{\&delta;}}_3\&CircleTimes;{\mathrm{\&delta;}}_0;$

wherein, i represents an imaginary number,

representing the tensor product, δ₀,δ₁,δ₂,δ₃In order to be a Pauli matrix, ${\mathrm{\&delta;}}_0=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],$ ${\mathrm{\&delta;}}_1=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],{\mathrm{\&delta;}}_2=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right],{\mathrm{\&delta;}}_3=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right];$

(2b) after Alice selects a unitary operator, the quantum state | phi is subjected to>_ABCDPerforming unitary conversion to obtainTo a new quantum state | Φ_m>_A′B′CDEach new quantum state | Φ_m>_A′B′CDThe quantum representation of (a) is:

$|{\mathrm{\&Phi;}}_1{>}_{A^{\&prime;}{B}^{\&prime;}\mathrm{CD}}=U_0|\mathrm{\&Phi;}{>}_{\mathrm{ABCD}}=\frac{1}{2}|0001>+\frac{1}{2}|0010>+\frac{1}{2}|0100>+\frac{1}{2}|1000>,$

$|{\mathrm{\&Phi;}}_2{>}_{A^{\&prime;}{B}^{\&prime;}\mathrm{CD}}=U_1|\mathrm{\&Phi;}{>}_{\mathrm{ABCD}}=\frac{1}{2}|0001>+\frac{1}{2}|0010>-\frac{1}{2}|0100>-\frac{1}{2}|1000>,$

$|{\mathrm{\&Phi;}}_3{>}_{A^{\&prime;}{B}^{\&prime;}\mathrm{CD}}=U_2|\mathrm{\&Phi;}{>}_{\mathrm{ABCD}}=\frac{1}{2}|1001>+\frac{1}{2}|1010>+\frac{1}{2}|1100>+\frac{1}{2}|0000>,$

$|{\mathrm{\&Phi;}}_4{>}_{A^{\&prime;}{B}^{\&prime;}\mathrm{CD}}=U_3|\mathrm{\&Phi;}{>}_{\mathrm{ABCD}}=\frac{1}{2}|1001>+\frac{1}{2}|1010>-\frac{1}{2}|1100>-\frac{1}{2}|0000>,$

$|{\mathrm{\&Phi;}}_5{>}_{A^{\&prime;}{B}^{\&prime;}\mathrm{CD}}=U_4|\mathrm{\&Phi;}{>}_{\mathrm{ABCD}}=\frac{1}{2}|1110>+\frac{1}{2}|1101>+\frac{1}{2}|1011>+\frac{1}{2}|0111>,$

$|{\mathrm{\&Phi;}}_6{>}_{A^{\&prime;}{B}^{\&prime;}\mathrm{CD}}=U_5|\mathrm{\&Phi;}{>}_{\mathrm{ABCD}}=\frac{1}{2}|1110>+\frac{1}{2}|1101>-\frac{1}{2}|1011>-\frac{1}{2}|0111>,$

$|{\mathrm{\&Phi;}}_7{>}_{A^{\&prime;}{B}^{\&prime;}\mathrm{CD}}=U_6|\mathrm{\&Phi;}{>}_{\mathrm{ABCD}}=\frac{1}{2}|0110>+\frac{1}{2}|0101>+\frac{1}{2}|0011>+\frac{1}{2}|1111>,$

$|{\mathrm{\&Phi;}}_8{>}_{A^{\&prime;}{B}^{\&prime;}\mathrm{CD}}=U_7|\mathrm{\&Phi;}{>}_{\mathrm{ABCD}}=\frac{1}{2}|0110>+\frac{1}{2}|0101>-\frac{1}{2}|0011>-\frac{1}{2}|1111>,$

$|{\mathrm{\&Phi;}}_9{>}_{A^{\&prime;}{B}^{\&prime;}\mathrm{CD}}=U_8|\mathrm{\&Phi;}{>}_{\mathrm{ABCD}}=\frac{1}{2}|0010>-\frac{1}{2}|0001>+\frac{1}{2}|0111>-\frac{1}{2}|1011>,$

$|{\mathrm{\&Phi;}}_{10}{>}_{A^{\&prime;}{B}^{\&prime;}\mathrm{CD}}=U_9|\mathrm{\&Phi;}{>}_{\mathrm{ABCD}}=\frac{1}{2}|0010>-\frac{1}{2}|0001>-\frac{1}{2}|0111>+\frac{1}{2}|1011>,$

$|{\mathrm{\&Phi;}}_{11}{>}_{A^{\&prime;}{B}^{\&prime;}\mathrm{CD}}=U_{10}|\mathrm{\&Phi;}{>}_{\mathrm{ABCD}}=\frac{1}{2}|1001>+\frac{1}{2}|0011>-\frac{1}{2}|1010>-\frac{1}{2}|1111>,$

$|{\mathrm{\&Phi;}}_{12}{>}_{A^{\&prime;}{B}^{\&prime;}\mathrm{CD}}=U_{11}|\mathrm{\&Phi;}{>}_{\mathrm{ABCD}}=\frac{1}{2}|1001>-\frac{1}{2}|0011>-\frac{1}{2}|1010>+\frac{1}{2}|1111>,$

$|{\mathrm{\&Phi;}}_{13}{>}_{A^{\&prime;}{B}^{\&prime;}\mathrm{CD}}=U_{12}|\mathrm{\&Phi;}{>}_{\mathrm{ABCD}}=\frac{1}{2}|0100>-\frac{1}{2}|1000>-\frac{1}{2}|1101>+\frac{1}{2}|1110>,$

$|{\mathrm{\&Phi;}}_{14}{>}_{A^{\&prime;}{B}^{\&prime;}\mathrm{CD}}=U_{13}|\mathrm{\&Phi;}{>}_{\mathrm{ABCD}}=\frac{1}{2}|0100>-\frac{1}{2}|1000>+\frac{1}{2}|1101>-\frac{1}{2}|1110>,$

$|{\mathrm{\&Phi;}}_{15}{>}_{A^{\&prime;}{B}^{\&prime;}\mathrm{CD}}=U_{14}|\mathrm{\&Phi;}{>}_{\mathrm{ABCD}}=\frac{1}{2}|0100>+\frac{1}{2}|0000>-\frac{1}{2}|0110>-\frac{1}{2}|1100>,$

$|{\mathrm{\&Phi;}}_{16}{>}_{A^{\&prime;}{B}^{\&prime;}\mathrm{CD}}=U_{15}|\mathrm{\&Phi;}{>}_{\mathrm{ABCD}}=\frac{1}{2}|0101>-\frac{1}{2}|0000>-\frac{1}{2}|0110>+\frac{1}{2}|1100>,$

the 16 quantum states form a complete orthogonal basis phi_m>_A′B′CD}，|Φ_m>_A′B′CDSatisfy the relation:

$\left\{\begin{array}{c}<{\mathrm{\&Phi;}}_m|{\mathrm{\&Phi;}}_j>={\mathrm{\&delta;}}_{\mathrm{mj}}\\ \underset{m}{\mathrm{\&Sigma;}}\underset{j}{\mathrm{\&Sigma;}}<{\mathrm{\&Phi;}}_m|{\mathrm{\&Phi;}}_j>=I\end{array}\right.$

wherein,<Φ_m|Φ_j>representing a quantum state | Φ_m>_A′B′CDAnd | Φ_j>_A′B′CDThe inner product of (a) is, ${\mathrm{\&delta;}}_{\mathrm{ij}}=\left\{\begin{array}{c}1,i=j\\ 0,i\&NotEqual;j\end{array},\right.$ $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],$ m,j∈[1,16]；

(2c) alice pairs each new quantum state | Φ_m>_A′B′CDAre coded separately, i.e. about

|Φ₁>_A′B′CDThe code is a code of 0000 and the code is a code of 0000,

|Φ₂>_A′B′CDthe code is a code of 0001 and is a code of 0001,

|Φ₃>_A′B′CDthe code is 0010 and the code is,

|Φ₄>_A′B′CDthe code is 0101 and the code is,

|Φ₅>_A′B′CDthe code is 0100, and the code is,

|Φ₆>_A′B′CDthe code is 0101 and the code is,

|Φ₇>_A′B′CDthe code is 0110 and the code is,

|Φ₈>_A′B′CDthe code is 0111 and the code is,

|Φ₉>_A′B′CDthe code is a code of 1000 and the code is a code,

|Φ₁₀>_A′B′CDthe code is a code of 1001 and the code is,

|Φ₁₁>_A′B′CDthe code is a code of 1010, and the code is,

|Φ₁₂>_A′B′CDthe code is 1011 of the number of codes,

|Φ₁₃>_A′B′CDthe code is coded into a code of 1100,

|Φ₁₄>_A′B′CDthe code is 1101 and the code is provided with,

|Φ₁₅>_A′B′CDthe code is a code of 1110 (one-time code),

|Φ₁₆>_A′B′CDthe code is 1111.

4. The quantum signaling ultra-dense coding method based on the four-photon entangled W state according to claim 1, wherein the measurement basis in the step (4) is a complete orthogonal basis { | Φ { |)_m>_A′B′CDAny one of them.

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