CN103368658B - Based on the ultra dense coding method of quantum signaling of four photon entanglement W states - Google Patents

Abstract

The invention discloses a kind of ultra dense coding method of quantum signaling based on four photon entanglement W states, mainly solve the problem that in quantum communications process, signalling coding capacity is little, efficiency is low, fail safe is not high.Its implementation procedure is, transmit leg prepares four photons (A, B, C, D) entangled W state | Φ & gt; _aBCDrepresent quantum signaling, and photon C, D are sent to recipient; Then by carrying out unitary transformation to photon A and B, 16 kinds of new quantum states are obtained | Φ _maMP.AMp.Amp gt; _{a ' B ' CD}; With this new quantum state in 16 | Φ _maMP.AMp.Amp gt; _{a ' B ' CD}formation orthogonal basis | Φ _maMP.AMp.Amp gt; _{a ' B ' CD}; To each new quantum state | Φ _maMP.AMp.Amp gt; _{a ' B ' CD}encode respectively, and the photon A ' that unitary transformation is obtained and B ', send to recipient in order; After recipient receives photon A ' and B ', from orthogonal basis | Φ _maMP.AMp.Amp gt; _{a ' B ' CD}in select arbitrarily a kind ofly to measure base, combined measurement is carried out to photon A ', B ', C, D, records new quantum state, complete decoding.It is high that the present invention has code efficiency, and capacity is large, and the advantage that fail safe is good, can be used for the signaling system of quantum mobile communication.

Description

Based on the ultra dense coding method of quantum signaling of four photon entanglement W states

Technical field

The invention belongs to coding techniques field, particularly the method for the ultra dense coding of a kind of quantum signaling, can be used for quantum communication network.

Background technology

Signaling is the requisite important component part of any communication system, and quantum communications are no exception.In multi-user quantum communication network, the coded system of quantum signaling directly affects efficiency of transmission and the fail safe of signaling system.

Existing quantum coding scheme mainly contains optical amplitude encoding, frequency coding, phase code, differential phase coding, and the ultra dense coding of two photon entanglement, three-photon tangle controlled ultra dense coding and the controlled ultra dense coding of four photon entanglement.Wherein:

The code efficiency of optical amplitude encoding, frequency coding, phase code and differential phase coding is lower, and first encoding can only transmit the information of 1 bit.

The ultra dense coding of two photon entanglement, first encoding just can transmit the classical information of 2 bits, but, carry out in the process communicated between multi-user, the Capacity Ratio of signaling system is comparatively large, and this coded system needs repeatedly to encode, can communicating requirement be met, more loaded down with trivial details;

Three-photon tangles controlled ultra dense coding, except recipient and transmit leg, also need a controlling party, and taking measurement of an angle when code efficiency is main and controlling party carries out von Neumann measurement to the photon of oneself is relevant, because the restriction of angle, so the classical information of first encoding transmission is less than 3 bits, and cataloged procedure is loaded down with trivial details especially, and code efficiency is lower;

The controlled ultra dense coding of four photon entanglement, except recipient and transmit leg, also there are two controlling parties, the same with three-photon, taking measurement of an angle when the main and controlling party of code efficiency carries out von Neumann measurement to the photon of oneself is relevant, because the restriction of angle, the code efficiency of the controlled ultra dense coding of four photon entanglement is less than 4 bits, and compared to three-photon, code capacity increases, but code efficiency is substantially constant.

Summary of the invention

The object of the invention is to the deficiency for the controlled ultra dense coding method of above-mentioned four photon entanglement, propose a kind of ultra dense coding method of quantum signaling based on four photon entanglement W states, to improve capacity and the efficiency of first encoding.

Realizing technical thought of the present invention is, utilize four photon entanglement W states to represent quantum signaling, directly establish and tangle channel, and according to the classical information that will transmit, corresponding unitary transformation is carried out to four photon entanglement states, obtain new quantum state, newer quantum state is encoded.Concrete steps are as follows:

(1) transmit leg Alice utilizes ultraviolet pulse light to continue through two BBO Crystals, produces by the W state of the first photon A, the second photon B, three-photon C, the 4th these four photon entanglement of photon D | Φ > _aBCD, as quantum signaling, and by three-photon C and the 4th photon D by channel distribution to recipient Bob, such transmit leg Alice and recipient Bob just have shared W state | Φ > _aBCD;

(2) transmit leg Alice carries out unitary transformation to one's own first photon A and the second photon B, to complete coding, obtains new quantum state | Φ _m> _{a ' B ' CD}, wherein A ' and B ' is the photon that photon A and photon B obtain through unitary transformation, m={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16};

(3), after transmit leg Alice completes coding, the photon A ' obtained and B ' is carried out order rearrangement, with the order of A ' B ' or B ' A ', send to recipient Bob by channel through unitary transformation;

(4) recipient Bob receives photon A ' and B ' and selects measurement base afterwards, carries out combined measurement, record new quantum state to described four photon A ', B ', C, D | Φ _m> _{a ' B ' CD}, to complete decoding, obtain the classical signaling information of Alice transmission.

Tool of the present invention has the following advantages:

1. the present invention represents quantum signaling owing to adopting four photon entanglement W states, directly can set up and tangle channel, save channel resource, improve communication efficiency;

2. the present invention is due to by carrying out unitary transformation to the quantum state of signaling, improves code capacity, makes first encoding can transmit the classical signaling information of 4 bits;

3. the present invention in an encoding process, and transmit leg Alice only transmits two photons, just can realize the classical signaling information of transmission 4 bit, so code efficiency is high;

4. transmit leg Alice is after coding completes, and when the photon obtained is sent to recipient Bob, rearranges the sending order of each photon, which enhance the fail safe of coding after unitary transformation, ensure that carrying out smoothly of quantum communications.

Accompanying drawing explanation

Fig. 1 quantum signalling coding of the present invention flow chart;

Fig. 2 is the transmission diagram that the present invention carries out quantum signaling.

Quantum signalling coding format chart in Fig. 3 the present invention;

Embodiment

With reference to Fig. 1, specific implementation step of the present invention is as follows:

Step 1, the preparation that four photon entanglement are right and distribution;

1.1) Alice is as signaling transmit leg, utilizes ultraviolet pulse light to continue through two BBO Crystals, prepares orderly n four photon A _k, B _k, C _k, D _kentangled W state

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

Wherein, k=1,2,3,4 ..., n, n >=1,0 represents that spin of photon direction is level, and 1 represents that spin of photon direction is plumbness;

1.2) Alice is by photon C _kand D _kcomposition sequence Q _b={ C _k, D _k, issue recipient Bob, oneself preserves photon A _kand B _kthe sequence Q of composition _a={ A _k, B _k,

As shown in Figure 2, Alice is by photon sequence Q _b={ C _k, D _kthroughput subchannel is sent to quantum communicativity machine A, quantum communicativity machine A according to the switching center address at recipient Bob place, then photon sequence Q _b={ C _k, D _kthroughput subchannel is sent to the quantum communicativity machine B of this switching center, quantum communicativity machine B, according to the destination address of Bob, selects suitable route, then photon sequence Q _b={ C _k, D _ksend Bob to;

1.3) when Bob receives sequence Q _bafter, transmit leg Alice and recipient Bob sets up and tangles channel, and both sides share four photon W states

Step 2, Alice is by four photon entanglement W states as quantum signaling, and to photon A _kand B _kcarry out unitary transformation, to complete the coding to quantum signaling.

As shown in Figure 3, the form of quantum signalling coding is made up of signaling unit, and signaling unit is the minimum unit of signaling message, and length is 8 bits, carries out twice coding just can realize quantum signaling.

2.1) transmit leg Alice is according to the classical signaling message that will transmit, and chooses a corresponding unitary operator from unitary operator set;

Unitary operator set is: { U ₀, U ₁, U ₂, U ₃, U ₄, U ₅, U ₆, U ₇, U ₈, U ₉, U ₁₀, U ₁₁, U ₁₂, U ₁₃, U ₁₄, U ₁₅, each element in operator set is the matrix of 16 dimensions, and each matrix is expressed as:

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

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

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

$U_9={\mathrm{\δ}}_3\⊗{\mathrm{\δ}}_0\⊗{\mathrm{\δ}}_1\⊗{\mathrm{i\δ}}_2,U_{10}=-{\mathrm{i\δ}}_2\⊗{\mathrm{\δ}}_0\⊗{\mathrm{i\δ}}_2\⊗{\mathrm{\δ}}_1,U_{11}=-{\mathrm{\δ}}_2\⊗{\mathrm{\δ}}_0\⊗{\mathrm{\δ}}_1\⊗{\mathrm{\δ}}_2,$

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

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

Wherein, i represents imaginary number, represent tensor product, δ ₀, δ ₁, δ ₂, δ ₃for 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 chooses operator, to photon A _kand B _kcarry out unitary transformation, obtain new quantum state wherein, A ' _kwith B ' _kphoton A _kwith photon B _kthrough the photon that unitary transformation obtains, m={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}, each new quantum state quantum expression formula be:

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

$|{\mathrm{\Φ}}_2{>}_{A_k^{\′}{B}_k^{\′}{C}_k{D}_k}=U_1|\mathrm{\Φ}{>}_{A_k{B}_k{C}_k{D}_k}=\frac{1}{2}|0001>+\frac{1}{2}|0010>-\frac{1}{2}|0100>-\frac{1}{2}|1000>,$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Above-mentioned these 16 kinds new quantum states form a Complete Orthogonal base new quantum state meet relational expression:

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

Wherein, < Φ _m| Φ _j> represents quantum state with inner product, ${\mathrm{\&delta;}}_{ij}=\left\{\begin{array}{c}1,i=j\\ 0,i\&NotEqual;j\end{array}\right.,$ $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],$ m,j∈[1,16]；

2.3) Alice is to each new quantum state obtained encode respectively, be about to compile

Code is 0000,

be encoded to 0001,

be encoded to 0010,

be encoded to 0011,

be encoded to 0100,

be encoded to 0101,

be encoded to 0110,

be encoded to 0111,

be encoded to 1000,

be encoded to 1001,

be encoded to 1010,

be encoded to 1011,

be encoded to 1100,

be encoded to 1101,

be encoded to 1110,

be encoded to 1111.

Step 3, photon sequence is sent to recipient Bob by transmit leg Alice.

3.1), after having encoded, Alice is the photon A obtained after unitary transformation _k' and photon B _k' composition photon sequence Q _a'={ A ₁', B ₁', A ₂', B ₂' ... A _k', B _k' ... A _n', B _n';

3.2) Alice is to photon sequence Q _a'={ A ₁', B ₁', A ₂', B ₂' ... A _k', B _k' ... A _n', B _n' the inside all photons, by first arranging B _k', rear arrangement A _k' order rearrange, obtain new photon sequence Q _a";

3.3) Alice is new photon sequence Q _a" throughput subchannel sends to recipient Bob, and informs Bob photon A by classical channel _k' and photon B _k' at new photon sequence Q _a" in position.

Step 4, recipient Bob decodes to the quantum signaling received.

4.1) recipient Bob receives new photon sequence Q _a" after, photon sequence Q _a" return to photon sequence Q _a'={ A ₁', B ₁', A ₂', B ₂' ... A _k', B _k' ... A _n', B _n';

4.2) Bob is from Complete Orthogonal base in, select wherein any one as measurement base;

4.3) Bob utilizes and measures base to described four photon A _k', B _k', C _k, D _kcarry out combined measurement, obtain new quantum state, complete decoding, obtain the classical signaling information that Alice sends.

Conclusion

More than describing is only example of the present invention; do not form any limitation of the invention; obviously for those skilled in the art; after having understood content of the present invention and principle; all may when not deviating from the principle of the invention, structure; carry out the various amendment in form and details and change, but these corrections based on inventive concept and change are still within claims of the present invention.

Claims (4)

1., based on the ultra dense coding method of quantum signaling of four photon entanglement W states, comprise the following steps:

(1) transmit leg Alice utilizes ultraviolet pulse light to continue through two BBO Crystals, produces by the W state of the first photon A, the second photon B, three-photon C, the 4th these four photon entanglement of photon D | Φ > _aBCD, as quantum signaling, and by three-photon C and the 4th photon D by channel distribution to recipient Bob, such transmit leg Alice and recipient Bob just have shared W state | Φ > _aBCD;

(2) transmit leg Alice carries out unitary transformation to one's own first photon A and the second photon B, to complete coding, obtains new quantum state | Φ _m> _{a ' B ' CD}, wherein A ' and B ' is the photon that photon A and photon B obtain through unitary transformation, m={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16};

(3), after transmit leg Alice completes coding, the photon A ' obtained and B ' is carried out order rearrangement, with the sequencing of A ' B ' or B ' A ', send to recipient Bob by channel through unitary transformation;

(4) recipient Bob receives photon A ' and B ' and selects measurement base afterwards, carries out combined measurement, record new quantum state to described four photon A ', B ', C, D | Φ _m> _{a ' B ' CD}, to complete decoding, obtain the classical signaling information of Alice transmission.

2. the ultra dense coding method of quantum signaling based on four photon entanglement W states according to right 1, the W state of four photon entanglement in wherein said step (1) | Φ > _aBCD, its quantum state expression is:

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

Wherein, the spin direction of 0 expression photon is level, and the spin direction of 1 expression photon is plumbness.

3. the ultra dense coding method of quantum signaling based on four photon entanglement W states according to right 1, wherein in step (2), transmit leg Alice carries out unitary transformation to one's own first photon A and the second photon B, carries out as follows:

(2a) from unitary operator set { U ₀, U ₁, U ₂, U ₃, U ₄, U ₅, U ₆, U ₇, U ₈, U ₉, U ₁₀, U ₁₁, U ₁₂, U ₁₃, U ₁₄, U ₁₅a middle selection unitary operator, each element in operator set is the matrix of 16 dimensions, and each matrix is expressed as:

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

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

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

$U_9={\mathrm{\&delta;}}_3\&CircleTimes;{\mathrm{\&delta;}}_0\&CircleTimes;{\mathrm{\&delta;}}_1\&CircleTimes;{\mathrm{i\&delta;}}_2,$ $U_{10}=-{\mathrm{i\&delta;}}_2\&CircleTimes;{\mathrm{\&delta;}}_0\&CircleTimes;{\mathrm{i\&delta;}}_2\&CircleTimes;{\mathrm{\&delta;}}_1,$ $U_{11}=-{\mathrm{\&delta;}}_2\&CircleTimes;{\mathrm{\&delta;}}_0\&CircleTimes;{\mathrm{\&delta;}}_1\&CircleTimes;{\mathrm{\&delta;}}_2,$

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

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

Wherein, i represents imaginary number, represent tensor product, δ ₀, δ ₁, δ ₂, δ ₃for 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 chooses a unitary operator, then carry out unitary transformation to the first photon A and the second photon B, obtain new quantum state | Φ _m> _{a ' B ' CD}, each new quantum state | Φ _m> _{a ' B ' CD}quantum expression formula be:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

${|{\mathrm{\&Phi;}}_{15}\&rang;}_{A^{\&prime;}{B}^{\&prime;}CD}=U_{14}{|\mathrm{\&Phi;}\&rang;}_{ABCD}=\frac{1}{2}|0101\&rang;+\frac{1}{2}|0000\&rang;-\frac{1}{2}|0110\&rang;-\frac{1}{2}|1100\&rang;,$

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

These 16 kinds of quantum states above-mentioned form a Complete Orthogonal base | Φ _m> _{a ' B ' CD}, | Φ _m> _{a ' B ' CD}meet relational expression:

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

Wherein, < Φ _m| Φ _j> represents quantum state | Φ _m> _{a ' B ' CD}with | Φ _j> _{a ' B ' CD}inner product, ${\mathrm{\&delta;}}_{ij}=\left\{\begin{array}{c}1,i=j\\ 0,i\&NotEqual;j\end{array}\right.,$ $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],$ m,j∈[1,16]；

(2c) Alice is to each new quantum state obtained | Φ _m> _{a ' B ' CD}encode respectively, be about to

| Φ ₁> _{a ' B ' CD}be encoded to 0000,

| Φ ₂> _{a ' B ' CD}be encoded to 0001,

| Φ ₃> _{a ' B ' CD}be encoded to 0010,

| Φ ₄> _{a ' B ' CD}be encoded to 0011,

| Φ ₅> _{a ' B ' CD}be encoded to 0100,

| Φ ₆> _{a ' B ' CD}be encoded to 0101,

| Φ ₇> _{a ' B ' CD}be encoded to 0110,

| Φ ₈> _{a ' B ' CD}be encoded to 0111,

| Φ ₉> _{a ' B ' CD}be encoded to 1000,

| Φ ₁₀> _{a ' B ' CD}be encoded to 1001,

| Φ ₁₁> _{a ' B ' CD}be encoded to 1010,

| Φ ₁₂> _{a ' B ' CD}be encoded to 1011,

| Φ ₁₃> _{a ' B ' CD}be encoded to 1100,

| Φ ₁₄> _{a ' B ' CD}be encoded to 1101,

| Φ ₁₅> _{a ' B ' CD}be encoded to 1110,

| Φ ₁₆> _{a ' B ' CD}be encoded to 1111.

4. the ultra dense coding method of quantum signaling based on four photon entanglement W states according to right 1, the measurement base in wherein said step (4), for Complete Orthogonal base | Φ _m> _{a ' B ' CD}in any one.