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

The ultra dense coding method of quantum signaling based on four photon entangled W states

Technical field

The invention belongs to the 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 the 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, and phase code, differential phase coding, and two photons tangle ultra dense coding, three-photon tangles controlled ultra dense coding and four photons tangle controlled ultra dense coding.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.

Two photons tangle ultra dense coding, and first encoding just can transmit the classical information of 2 bits, still, in the process that communicates between the multi-user, the Capacity Ratio of signaling system is larger, and this coded system need to repeatedly be encoded, can satisfy communicating requirement, more loaded down with trivial details;

Three-photon tangles controlled ultra dense coding, except recipient and transmit leg, also need a controlling party, and it is relevant with controlling party taking measurement of an angle when oneself photon is carried out the von Neumann measurement that code efficiency is main, 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, code efficiency is lower;

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

Summary of the invention

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

Realize that technical thought of the present invention is, utilize four photon entangled W states to represent the quantum signaling, directly set up and tangled channel, and according to the classical information that will transmit four photon Entangled States are carried out corresponding unitary transformation, obtain new quantum state, newer quantum state is encoded.Concrete steps are as follows:

(1) transmit leg Alice utilizes ultraviolet pulse light continuously by two BBO Crystals, produces the W attitude of being tangled by the first photon A, the second photon B, three-photon C, these four photons of the 4th photon D | Φ 〉 _ABCD, as the quantum signaling, and with three-photon C and the 4th photon D by channel distribution to recipient Bob, transmit leg Alice and recipient Bob have just shared the W attitude like this | Φ _ABCD

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

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

(4) recipient Bob receives photon A ' and the rear State selective measurements base of B ', and described four photon A ', B ', C, D are carried out combined measurement, records new quantum state | Φ _{m A ' B ' CD}, to finish decoding, obtain the classical signaling information of Alice transmission.

The present invention has following advantage:

1. the present invention can directly set up and tangle channel owing to adopt four photon entangled W states to represent the quantum signaling, has saved channel resource, has improved communication efficiency;

2. the present invention has improved code capacity owing to carry out unitary transformation by the quantum state to signaling, makes first encoding can transmit the classical signaling information of 4 bits;

3. the present invention is in cataloged procedure, and transmit leg Alice only transmits two photons, just can realize transmitting the classical signaling information of 4 bits, so code efficiency is high;

4. transmit leg Alice is after coding is finished, and when the photon that obtains behind the unitary transformation is sent to recipient Bob, the sending order of each photon rearranged, and improved like this fail safe of coding, guaranteed carrying out smoothly of quantum communications.

Description of drawings

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

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

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

Embodiment

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

Step 1, four photons tangle right preparation and distribution;

1.1) Alice is as the signaling transmit leg, utilizes ultraviolet pulse light continuously by two BBO Crystals, prepare the individual four photon A of orderly n _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 expression spin of photon direction is level, 1 expression spin of photon direction is plumbness;

1.2) Alice is photon C _kAnd D _kComposition sequence Q _B={ C _k, D _k, issue recipient Bob, oneself preserves photon A _kAnd B _kThe sequence Q that forms _A={ A _k, B _k,

As shown in Figure 2, Alice is with photon sequence Q _B={ C _k, D _kThe throughput subchannel is sent to the quantum switch A, the quantum switch A is according to the switching center address at recipient Bob place, again photon sequence Q _B={ C _k, D _kThe throughput subchannel is sent to the quantum switch b of this switching center, the quantum switch b is selected suitable route according to the destination address of Bob, again photon sequence Q _B={ C _k, D _kSend Bob to;

1.3) receive sequence Q as Bob _BAfter, transmit leg Alice and recipient Bob set up and tangle channel, and both sides share four photon W attitudes

Step 2, Alice is with four photon entangled W states

As the quantum signaling, and to photon A _kAnd B _kCarry out unitary transformation, to finish the coding to the quantum signaling.

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

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

The 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 the operator set is the matrix of 16 dimensions, each matrix is expressed as respectively:

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

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

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

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

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

Wherein, i represents imaginary number,

Representation ' Tensor PFoduct, δ ₀, δ ₁, δ ₂, δ ₃Be the 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 ' _kAnd B _'K is photon 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

The quantum expression formula be:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Above-mentioned these 16 kinds new quantum states consist of a Complete Orthogonal base

New quantum state

Satisfy relational expression:

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

Wherein,＜Φ _m| Φ _jThe expression quantum state

With

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

2.3) Alice each new quantum state to obtaining

Encode respectively, be about to

Be encoded to 0000,

Be encoded to 0001,

Be encoded to 0010,

Be encoded to 0101,

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, transmit leg Alice sends to recipient Bob with photon sequence.

3.1) after coding finishes, the photon A that Alice obtains 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 arranging first 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 the position.

Step 4, recipient Bob decodes to the quantum signaling that receives.

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 the Complete Orthogonal base In, select wherein any one as measuring base;

4.3) Bob utilize to measure base to described four photon A _k', B _k', C _k, D _kCarry out combined measurement, obtain new quantum state

Finish decoding, obtain the classical signaling information that Alice sends.

Conclusion

More than describing only is example of the present invention; do not consist of any limitation of the invention; obviously for those skilled in the art; after having understood content of the present invention and principle; all may be in the situation that does not deviate from the principle of the invention, structure; carry out various modifications and change on form and the details, but these are based on the correction of inventive concept with change still within claim protection range of the present invention.

Claims (4)

1. ultra dense coding method of quantum signaling based on four photon entangled W states may further comprise the steps:

(1) transmit leg Alice utilizes ultraviolet pulse light continuously by two BBO Crystals, produces the W attitude of being tangled by the first photon A, the second photon B, three-photon C, these four photons of the 4th photon D | Φ 〉 _ABCD, as the quantum signaling, and with three-photon C and the 4th photon D by channel distribution to recipient Bob, transmit leg Alice and recipient Bob have just shared the W attitude like this | Φ _ABCD

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

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

(4) recipient Bob receives photon A ' and the rear State selective measurements base of B ', and described four photon A ', B ', C, D are carried out combined measurement, records new quantum state | Φ _{m A ' B ' CD}, to finish decoding, obtain the classical signaling information of Alice transmission.

2. according to the right 1 described ultra dense coding method of quantum signaling based on four photon entangled W states, the W attitude that four photons in the wherein said step (1) tangle | Φ 〉 _ABCD, its quantum state expression is:

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

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

3. according to the right 1 described ultra dense coding method of quantum signaling based on four photon entangled W states, wherein transmit leg Alice carries out unitary transformation to one's own the first photon A and the second photon B in the step (2), 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 ₁₅Unitary operator of middle selection, and each element in the operator set is the matrixes of 16 dimensions, each matrix is expressed as respectively:

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

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

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

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

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

Wherein, i represents imaginary number,

Representation ' Tensor PFoduct, δ ₀, δ ₁, δ ₂, δ ₃Be the Pauli matrix, ${\mathrm{\&delta;}}_0=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],$ ${\mathrm{\&delta;}}_1=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],{\mathrm{\&delta;}}_2=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right],{\mathrm{\&delta;}}_3=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right];$

After (2b) Alice chooses a unitary operator, again to quantum state | Φ 〉 _ABCDCarry out unitary transformation, obtain new quantum state | Φ _{m A ' B ' CD}, each new quantum state | Φ _{m A ' B ' CD}The quantum expression formula be:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Complete Orthogonal base of above-mentioned this 16 kinds of quantum states formations | Φ _{m A ' B ' CD}, | Φ _{m A ' B ' CD}Satisfy relational expression:

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

Wherein,＜Φ _m| Φ _jThe expression quantum state | Φ _{m A ' B ' CD}With | Φ _{j A ' B ' CD}Inner product, ${\mathrm{\&delta;}}_{\mathrm{ij}}=\left\{\begin{array}{c}1,i=j\\ 0,i\&NotEqual;j\end{array},\right.$ $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],$ m,j∈[1,16]；

(2c) Alice each new quantum state to obtaining | 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 0101,

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

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