CN103916240A - Method for implementing quantum dense coding of three-particle one-class W states in 3*2 nonsymmetrical channel - Google Patents

Abstract

The invention discloses a method for implementing quantum dense coding of three-particle one-class W states in a 3*2 nonsymmetrical channel. The method includes the first step of enabling a quantum signal source to prepare the three-particle W-class state, distributing a particle A and a particle B to Alice and distributing a particle C to Bob, the second step of allowing Alice to select one code to carry out unitary transformation on the particle A or the particle B correspondingly, or carry out unitary transformation on the particle A and the particle B simultaneously according to classic information to be transmitted, wherein it is assumed that the initial state of the three particles is , the third step of allowing Alice to send the particle A and the particle B to Bob after unitary transformation, and the fourth step of carrying out joint measurement on the particle A, the particle B and the particle C by selecting a measurement basis corresponding to the unitary transformation based on a complete orthogonal basis after Bob receives the particle A and the particle B, and then obtaining the classic information sent by Alice. Through the method, the dense coding of the three-particle one-class W states in the 3*2 nonsymmetrical channel is realized, and the information transmission efficiency is improved.

Description

The quantum dense coding method of three particle one class W states in 3 × 2 nonsymmetric channels

Technical field

The invention belongs to quantum secret communication field, be specifically related to the quantum dense coding method of a kind of three particle one class W states in 3 × 2 nonsymmetric channels.

Background technology

Document 1 " PENG Zhao-hui; JIA Chun-xia.Scheme for implementing perfect quantum dense coding with three-atom W-class state in cavity QED[J] .Optics Communications; 2008; 281 (6): 1745-1750. " a kind of quantum dense coding scheme based on three particle W states disclosed, the channel of its foundation is symmetric channel, and information transfer efficiency is lower.Document 2 " YAN Feng-li; WANG Mei-yu; A Scheme for Dense Coding in the Non-Symmetric Quantum Channel; Chin.Phys.Lett.Volume21, Number7, July2004; pp.1195-1197. " discloses a kind of scheme that realizes dense coding based on GHZ state, GHZ state is maximal entangled state, and more difficult preparation is unfavorable for the practical of scheme.

Summary of the invention

In order to overcome the deficiency of the difficult preparation of GHZ, improve information transfer efficiency, the object of the invention is to, the quantum dense coding method of a kind of three particle one class W states in 3 × 2 nonsymmetric channels is provided, the method has realized the dense coding of three particle one class W states in nonsymmetric channel, and has improved information transfer efficiency.

To achieve these goals, the present invention adopts following technical scheme to be solved:

The quantum dense coding method of a kind of three particle one class W states in 3 × 2 nonsymmetric channels, specifically comprises the steps:

(1) make the preparation of quantum signal source meet three particle one class W states shown in formula 1; Then particle A and particle B are distributed to Alice, particle C is distributed to Bob; Particle A and particle B are all in three-dimensional Hilbert space, and particle C is in two-dimentional Hilbert space, form like this 3 × 2 nonsymmetric channel between Alice and Bob:

${|W>}_{\mathrm{ABC}}=a|001>+b|010>+\frac{\sqrt{2}}{2}|100>$ (formula 1)

Wherein, ${\left|a\right|}^2+{\left|b\right|}^2=\frac{1}{2}$

(2) Alice, according to the classical information that will transmit, selects a coding accordingly particle A or particle B to be carried out to unitary transformation, or particle A and particle B is carried out to unitary transformation simultaneously; The initial state of supposing 3 particles is

(3), after unitary transformation, particle A and particle B are sent to Bob by Alice; , suppose that channel is desirable;

(4) Bob receives particle A and particle B, at Complete Orthogonal base under, by selecting the measurement base corresponding with unitary transformation to carry out combined measurement to particle A, particle B and particle C, can obtain the classical information that Alice sends.

Further, select a coding accordingly particle A or particle B are carried out unitary transformation or particle A and particle B carried out to unitary transformation simultaneously described in described step (2), concrete execution is as follows:

If select the one in following 6 kinds of codings, only particle A is carried out to unitary transformation, in corresponding each coding, include unitary transformation and with this unitary transformation after the new quantum state that produces, this new quantum state will use as measurement base corresponding to Bob:

Coding 1: $U=U_{00}\⊗I_3\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_1^{+}=\left(a\right|001>+b|010>+\sqrt{2}/2|100>);$

Coding 2: $U=U_{01}\⊗I_3\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_1^{-}=\left(a\right|001>+b|010>-\sqrt{2}/2|100>);$

Coding 3: $U=U_{10}\⊗I_3\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_2^{+}=\left(a\right|101>+b|110>+\sqrt{2}/2|200>);$

Coding 4: $U=U_{11}\⊗I_3\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_2^{-}=\left(a\right|101>+b|110>-\sqrt{2}/2|200>);$

Coding 5: $U=U_{20}\⊗I_3\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_3^{+}=\left(a\right|201>+b|210>+\sqrt{2}/2|000>);$

Coding 6: $U=U_{21}\⊗I_3\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_3^{-}=\left(a\right|201>+b|210>-\sqrt{2}/2|000>);$

If select the one in following 2 kinds of codings, only particle B is carried out to unitary transformation, in corresponding each coding, include unitary transformation and with this unitary transformation after the new quantum state that produces, this new quantum state will use as measurement base corresponding to Bob:

Coding 7: $U=I_3\⊗U_{10}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_4^{+}=\left(a\right|011>+b|020>+\sqrt{2}/2|110>);$

Coding 8: $U=I_3\⊗U_{20}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_5^{+}=\left(a\right|021>+b|000>+\sqrt{2}/2|120>);$

If select the one in following 10 kinds of codings, particle A and particle B are combined to unitary transformation, in corresponding each coding, include unitary transformation and with this unitary transformation after the new quantum state that produces, this new quantum state will use as measurement base corresponding to Bob:

Coding 9: $U=U_{10}\⊗U_{10}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_6^{+}=\left(a\right|111>+b|120>+\sqrt{2}/2|210>);$

Coding 10: $U=U_{10}\⊗U_{20}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_7^{+}=\left(a\right|121>+b|100>+\sqrt{2}/2|220>);$

Coding 11: $U=U_{20}\⊗U_{10}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_8^{+}=\left(a\right|211>+b|220>+\sqrt{2}/2|010>);$

Coding 12: $U=U_{20}\⊗U_{20}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_9^{+}=\left(a\right|221>+b|200>+\sqrt{2}/2|020>);$

Coding 13: $U=U_{01}\⊗U_{10}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_4^{-}=\left(a\right|011>+b|020>-\sqrt{2}/2|110>);$

Coding 14: $U=U_{01}\⊗U_{20}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_5^{-}=\left(a\right|021>+b|000>-\sqrt{2}/2|120>);$

Coding 15: $U=U_{11}\⊗U_{10}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_6^{-}=\left(a\right|111>+b|120>-\sqrt{2}/2|210>);$

Coding 16: $U=U_{11}\⊗U_{20}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_7^{-}=\left(a\right|121>+b|100>-\sqrt{2}/2|220>);$

Coding 17: $U=U_{21}\⊗U_{10}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_8^{-}=\left(a\right|211>+b|220>-\sqrt{2}/2|010>);$

Coding 18: $U=U_{21}\⊗U_{20}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_9^{-}=\left(a\right|221>+b|200>-\sqrt{2}/2|020>);$

Wherein:

$\begin{array}{cc}{U}_{00}=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),& U_{01}=\left(\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right);\end{array}$

$\begin{array}{cc}{U}_{10}=\left(\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right),& U_{11}=\left(\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& -1& 0\end{array}\right);\end{array}$

$\begin{array}{cc}{U}_{20}=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right),& U_{21}=\left(\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right);\end{array}$

$I_3=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],I_2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right];$

Principle of the present invention and advantage are as follows:

Because W state is non-maximal entangled state, it is easily prepared than GHZ state, and therefore the present invention proposes available three particle one class W states; By making the particle A of Alice and particle B in three-dimensional Hilbert space, make the particle C of Bob in two-dimentional Hilbert space, construct 3 × 2 nonsymmetric channel; Alice, by A particle, beta particle are carried out to unitary transformation, completes the coding to classical information, and Bob utilizes corresponding orthogonal basis to carry out combined measurement to A, B and C tri-particles, obtains log ₂the classical information of 18 bits, and corresponding symmetric channel can only obtain log ₂the information of 8 bits, thus realize the dense coding of three particle one class W states in nonsymmetric channel, and improved information transfer efficiency.

Accompanying drawing explanation

Fig. 1 is the schematic diagram of the quantum dense coding method of three particle one class W states of the present invention in 3 × 2 nonsymmetric channels.

Below in conjunction with the drawings and specific embodiments, the present invention is further explained.

Embodiment

Main thought of the present invention is, by making the particle A of Alice and B in three-dimensional Hilbert space, the particle C of Bob, in two-dimentional Hilbert space, constructs 3 × 2 nonsymmetric channel;

And A particle, beta particle are carried out to unitary transformation, Bob utilizes corresponding orthogonal basis to carry out combined measurement to A, B and C tri-particles, realizes the dense coding of three particle one class W states in nonsymmetric channel.

As shown in Figure 1, suppose that transmit leg and the recipient of communication is respectively Alice and Bob, U conversion represents unitary transformation, and M measurement represents combined measurement.

The concrete steps that the present invention is based on the quantum dense coding method of three particle one class W states in 3 × 2 nonsymmetric channels are as follows:

(1) make the preparation of quantum signal source meet three particle one class W states shown in formula (1); Then particle A and particle B are distributed to Alice, particle C is distributed to Bob; Particle A and particle B are all in three-dimensional Hilbert space, and particle C is in two-dimentional Hilbert space, form like this 3 × 2 nonsymmetric channel between Alice and Bob:

${|W>}_{\mathrm{ABC}}=a|001>+b|010>+\frac{\sqrt{2}}{2}|100>$ (formula 1)

Wherein, ${\left|a\right|}^2+{\left|b\right|}^2=\frac{1}{2}$

(2) Alice, according to the classical information that will transmit, selects a coding accordingly particle A or particle B to be carried out to unitary transformation, or particle A and particle B is carried out to unitary transformation simultaneously; The initial state of supposing 3 particles is

If select the one in following 6 kinds of codings, only particle A is carried out to unitary transformation, in each coding, include unitary transformation and with this unitary transformation after the new quantum state that produces, this new quantum state will use as measurement base corresponding to Bob:

Coding 1: $U=U_{00}\⊗I_3\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_1^{+}=\left(a\right|001>+b|010>+\sqrt{2}/2|100>);$

Coding 2: $U=U_{01}\⊗I_3\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_1^{-}=\left(a\right|001>+b|010>-\sqrt{2}/2|100>);$

Coding 3: $U=U_{10}\⊗I_3\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_2^{+}=\left(a\right|101>+b|110>+\sqrt{2}/2|200>);$

Coding 4: $U=U_{11}\⊗I_3\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_2^{-}=\left(a\right|101>+b|110>-\sqrt{2}/2|200>);$

Coding 5: $U=U_{20}\⊗I_3\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_3^{+}=\left(a\right|201>+b|210>+\sqrt{2}/2|000>);$

Coding 6: $U=U_{21}\⊗I_3\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_3^{-}=\left(a\right|201>+b|210>-\sqrt{2}/2|000>);$

If select the one in following 2 kinds of codings, only particle B is carried out to unitary transformation, in each coding, include unitary transformation and with this unitary transformation after the new quantum state that produces, this new quantum state will use as measurement base corresponding to Bob:

Coding 7: $U=I_3\⊗U_{10}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_4^{+}=\left(a\right|011>+b|020>+\sqrt{2}/2|110>);$

Coding 8: $U=I_3\⊗U_{20}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_5^{+}=\left(a\right|021>+b|000>+\sqrt{2}/2|120>);$

If select the one in following 10 kinds of codings, particle A and particle B are carried out to unitary transformation simultaneously, in each coding, include unitary transformation and with this unitary transformation after the new quantum state that produces, this new quantum state will use as measurement base corresponding to Bob:

Coding 9: $U=U_{10}\⊗U_{10}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_6^{+}=\left(a\right|111>+b|120>+\sqrt{2}/2|210>);$

Coding 10: $U=U_{10}\⊗U_{20}\⊗I_2,$ Measure base ${\mathrm{\Φ}}_7^{+}=\left(a\right|121>+b|100>+\sqrt{2}/2|220>);$

Coding 11: $U=U_{20}\⊗U_{10}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_8^{+}=\left(a\right|211>+b|220>+\sqrt{2}/2|010>);$

Coding 12: $U=U_{20}\⊗U_{20}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_9^{+}=\left(a\right|221>+b|200>+\sqrt{2}/2|020>);$

Coding 13: $U=U_{01}\⊗U_{10}\⊗I_2,$ Measure base ${\mathrm{\Φ}}_4^{-}=\left(a\right|011>+b|020>-\sqrt{2}/2|110>);$

Coding 14: $U=U_{01}\⊗U_{20}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_5^{-}=\left(a\right|021>+b|000>-\sqrt{2}/2|120>);$

Coding 15: $U=U_{11}\⊗U_{10}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_6^{-}=\left(a\right|111>+b|120>-\sqrt{2}/2|210>);$

Coding 16: $U=U_{11}\⊗U_{20}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_7^{-}=\left(a\right|121>+b|100>-\sqrt{2}/2|220>);$

Coding 17: $U=U_{21}\⊗U_{10}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_8^{-}=\left(a\right|211>+b|220>-\sqrt{2}/2|010>);$

Coding 18: $U=U_{21}\⊗U_{20}\⊗I_2,$ Measure base: ${\mathrm{\Φ}}_9^{-}=\left(a\right|221>+b|200>-\sqrt{2}/2|020>);$

Wherein:

$\begin{array}{cc}{U}_{00}=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),& U_{01}=\left(\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right);\end{array}$

$\begin{array}{cc}{U}_{10}=\left(\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right),& U_{11}=\left(\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& -1& 0\end{array}\right);\end{array}$

$\begin{array}{cc}{U}_{20}=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right),& U_{21}=\left(\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right);\end{array}$

$I_3=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],I_2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right];$

From encoded content above, Alice, by particle A and B are carried out to different unitary transformations, can obtain 18 kinds of different quantum states and corresponding one by one with 18 codings; By formula (3), these 18 kinds of quantum states are done to inner product between two mutually, result is equal to 0, illustrates that 18 kinds of quantum states are mutually orthogonal between two.

$\left[\begin{array}{ccc}<{\mathrm{\&Phi;}}_i^j|{\mathrm{\&Phi;}}_k^m>={\mathrm{\&delta;}}_{\mathrm{ik}}{\mathrm{\&delta;}}_{\mathrm{jm}},& i,k\&Element;\left[\mathrm{1,9}\right],& j,m\end{array},\&Element;[+,-]\right]$ (formula 3)

In Hilbert space, corresponding 18 the mutually orthogonal vectors between two of these 18 quantum states, and the mould of vector is 1.Therefore, in the Hilbert space of 18 dimensions, it meets formula (4), and these 18 quantum states have just in time formed one group of Complete Orthogonal base

$\begin{array}{ccc}\underset{i}{\mathrm{\&Sigma;}}\underset{k}{\mathrm{\&Sigma;}}|{\mathrm{\&Phi;}}_i^j><{\mathrm{\&Phi;}}_k^m|=I,& i,k\&Element;\left[\mathrm{1,9}\right],& j,m\&Element;[+,-]\end{array}$ (formula 4)

For example: if Alice has selected coding 5 according to the classical information that will transmit, need only particle A to be carried out $U=U_{20}\&CircleTimes;I_3\&CircleTimes;I_2,$ Unitary transformation, suc as formula (2), be the measurement base that will select when Bob will carry out combined measurement in step (4), this measurement base is corresponding with this unitary transformation:

${U|W>}_{\mathrm{ABC}}=\left(a\right|201>+b|210>+\sqrt{2}/2|000>)={\mathrm{\&Phi;}}_3^{+}$ (formula 2)

(3), after unitary transformation, particle A and particle B are sent to Bob by Alice; , suppose that channel is desirable;

(4) Bob receives particle A and particle B, at Complete Orthogonal base under, by selecting the measurement base corresponding with unitary transformation to carry out combined measurement to particle A, particle B and particle C, can obtain the classical information that Alice sends.

For example: Alice has selected coding 5, should only do particle A unitary transformation, and Bob should select corresponding measurement base ${\mathrm{\&Phi;}}_3^{+}=\left(a\right|201>+b|210>+\sqrt{2}/2|000>),$ Particle A, particle B and particle C are carried out to combined measurement, the classical information that reduction Alice sends.

Claims (2)

1. the quantum dense coding method of three particle one class W states in 3 × 2 nonsymmetric channels, is characterized in that, specifically comprises the steps:

(1) make the preparation of quantum signal source meet three particle one class W states shown in formula 1; Then particle A and particle B are distributed to Alice, particle C is distributed to Bob; Particle A and particle B are all in three-dimensional Hilbert space, and particle C is in two-dimentional Hilbert space, form like this 3 × 2 nonsymmetric channel between Alice and Bob:

(formula 1)

Wherein,

(2) Alice, according to the classical information that will transmit, selects a coding accordingly particle A or particle B to be carried out to unitary transformation, or particle A and particle B is carried out to unitary transformation simultaneously; The initial state of supposing 3 particles is

(3), after unitary transformation, particle A and particle B are sent to Bob by Alice; , suppose that channel is desirable;

(4) Bob receives particle A and particle B, at Complete Orthogonal base under, by selecting the measurement base corresponding with unitary transformation to carry out combined measurement to particle A, particle B and particle C, can obtain the classical information that Alice sends.

2. the quantum dense coding method of three particle one class W states as claimed in claim 1 in 3 × 2 nonsymmetric channels, it is characterized in that, described in described step (2), select a coding accordingly particle A or particle B are carried out unitary transformation or particle A and particle B carried out to unitary transformation simultaneously, concrete execution is as follows:

If select the one in following 6 kinds of codings, only particle A is carried out to unitary transformation, in corresponding each coding, include unitary transformation and with this unitary transformation after the new quantum state that produces, this new quantum state will use as measurement base corresponding to Bob:

Coding 1: measure base:

Coding 2: measure base:

Coding 3: measure base:

Coding 4: measure base:

Coding 5: measure base:

Coding 6: measure base:

If select the one in following 2 kinds of codings, only particle B is carried out to unitary transformation, in each coding, include unitary transformation and with this unitary transformation after the new quantum state that produces, this new quantum state will use as measurement base corresponding to Bob:

Coding 7: measure base:

Coding 8: measure base:

If select the one in following 10 kinds of codings, particle A and particle B are carried out to unitary transformation simultaneously, in each coding, include unitary transformation and with this unitary transformation after the new quantum state that produces, this new quantum state will use as measurement base corresponding to Bob:

Coding 9: measure base:

Coding 10: measure base:

Coding 11: measure base:

Coding 12: measure base:

Coding 13: measure base:

Coding 14: measure base:

Coding 15: measure base:

Coding 16: measure base:

Coding 17: measure base:

Coding 18: measure base: wherein:

。