CN104297938B - Single complete mutual unbiased base of quantum bit realize device - Google Patents

Abstract

The invention provides the complete mutual unbiased base of a kind of single quantum bit realizes device.This realizes device and is made up of a wave plate and polaroid: the phase place of wave plate is δ, incident illumination normal incidence wave plate；Polaroid is positioned at the light path rear end of wave plate, for being separated with the composition vertically polarized by the composition polarized in the horizontal direction in wave plate incident illumination；Wherein, the optical axis of definition wave plate is θ relative to the angle of horizontal direction, when this θ value is respectively θ₁, θ₂, θ₃Time, it is achieved three measurement bases of single quantum bit, these three measurement bases constitute complete mutual unbiased base.The present invention realizes the complete mutual unbiased base of quantum bit only with a wave plate, compared with two traditional wave plate implementation methods, not only saves a wave plate, and mean error is reduced to original half.

Description

Single complete mutual unbiased base of quantum bit realize device

Technical field

The invention belongs to the distribution of quantum key, quantum state chromatographs, and quantum communications etc. relate to optical system and place an order the field of quantum bit mutual unbiased base, and particularly relate to the complete mutual unbiased base of a kind of single quantum bit realizes device.

Background technology

Quantum communications are one of core contents of quantum information, and it is that the safe transmission of information provides new method.After BB84 quantum-key distribution scheme is suggested, quantum communications achieve substantial amounts of achievement in research in theory and experimentally.Mutually unbiased base is as the carrier of quantum key in quantum key is distributed, and is to ensure that one of factor that quantum communications are perfectly safe.

Additionally, quantum state is the underlying carrier of quantum information task.The identification of quantum state needs to use quantum state chromatographic technique, and complete mutual unbiased base is one of fixation measuring base optimum in quantum state chromatography.The error of the quantum state that the experimental data using complete mutual unbiased base measurement to obtain reconstructs out is minimum on average.

In current quantum communications research work, use a polaroid and a half-wave plate or a quarter-wave plate can realize two mutual unbiased bases, as shown in Figure 1 (see reference document 1).But for three mutual unbiased bases (constituting complete mutual unbiased base), two wave plate combined of quarter-wave plate and half-wave plate and polaroid currently can only be used could to realize (see reference document 2).But, owing to wave plate number is the most, the experimental error of introducing is the biggest.Additionally, two wave plate combined and polaroid implementation cost are the highest, regulation inconvenience.

List of references 1:Gisin N, Ribordy G G, Tittel W and Zbinden H 2002 Rev.Mod.Phys.74145-195

List of references 2:Damask J N 2005 Polarization optics in telecommunications (New York:Springer)

Summary of the invention

(1) to solve the technical problem that

The technical problem to be solved in the present invention is how to realize the complete mutual unbiased base of single polarization quantum bit with a wave plate and a polaroid.

(2) technical scheme

According to an aspect of the invention, it is provided the complete mutual unbiased base of a kind of single quantum bit realize device.This realizes device and is made up of a wave plate and polaroid: the phase place of wave plate is δ, incident illumination normal incidence wave plate；Polaroid is positioned at the light path rear end of wave plate, for being separated with the composition vertically polarized by the composition polarized in the horizontal direction in wave plate incident illumination；Wherein, the optical axis of definition wave plate is θ relative to the angle of horizontal direction, when this θ value is respectively θ₁, θ₂, θ₃Time, it is achieved three measurement bases of single quantum bit, these three measurement bases constitute complete mutual unbiased base.

(3) beneficial effect

From technique scheme it can be seen that the unicast sheet of the complete mutual unbiased base of list quantum bit of the present invention realizes device and has the advantages that

(1) can mostly come from faulty experimental provision with intuitivism apprehension for systematic error, and instrument is the fewest, the error of introducing is the fewest.Realizing the complete mutual unbiased base of quantum bit only with a wave plate in the present invention, compared with traditional implementation method, mean error is reduced to original half；

(2) single wave plate realizes device and saves a wave plate, reduces cost, and has only to, to a wave plate rotational angle, greatly facilitate experimental implementation every time.

Accompanying drawing explanation

Fig. 1 is that prior art uses quarter-wave plate 1 and the wave plate combined of half-wave plate 2 composition and polaroid 3 to realize the schematic diagram of complete mutual unbiased base；

Fig. 2 is to use the complete mutual unbiased base of single quantum bit of single wave plate and polaroid to realize the structural representation of device according to the embodiment of the present invention；

Fig. 3 is that shown in Fig. 2, single complete mutual unbiased base of quantum bit realizes framework gesture Φ (δ) of wave plate in device and the relation of phase place δ；

Fig. 4 a and Fig. 4 b is that single complete mutual unbiased base of quantum bit shown in Fig. 2 realizes wave plate in device and realizes the relation of angle corresponding during complete mutual unbiased base and wave plate phase place, and wherein, Fig. 4 a and Fig. 4 b respectively represents a class solution.

[main element symbol description]

1-quarter-wave plate；2-half-wave plate；

3-polaroid；4-wave plate.

Detailed description of the invention

For making the object, technical solutions and advantages of the present invention clearer, below in conjunction with specific embodiment, and referring to the drawings, the present invention is described in more detail.It should be noted that in accompanying drawing or description describe, similar or identical part all uses identical figure number.The implementation not illustrated in accompanying drawing or describe, for form known to a person of ordinary skill in the art in art.Although it addition, the demonstration of the parameter comprising particular value can be provided herein, it is to be understood that parameter is worth equal to corresponding without definite, but can be similar to be worth accordingly in acceptable error margin or design constraint.The direction term mentioned in embodiment, such as " on ", D score, "front", "rear", "left", "right" etc., be only the direction with reference to accompanying drawing.Therefore, the direction term of use is used to illustrate not for limiting the scope of the invention.

The present invention is by rationally selecting phase place and the anglec of rotation of wave plate, it is achieved that the complete mutual unbiased base of quantum bit.

In one exemplary embodiment of the present invention, it is provided that the unicast sheet of the complete mutual unbiased base of a kind of single quantum bit realizes device.Fig. 2 is the schematic diagram realizing device according to the complete mutual unbiased base of embodiment of the present invention list quantum bit.As in figure 2 it is shown, the device that realizes of the complete mutual unbiased base of the present embodiment list quantum bit is made up of a wave plate 4 and polaroid 3.

In the present embodiment, incident illumination normal incidence wave plate 4.The phase place of wave plate 4 is δ, and optical axis is θ relative to the angle of horizontal direction.In the range of phase place δ is in k × 360 ° ± [111.5 °, 141.76 °], three θ (θ can be found₁, θ₂, θ₃), these three θ (θ₁, θ₂, θ₃) three measurement bases of respectively corresponding single quantum bit, these three measurement bases constitute complete mutual unbiased bases.Wherein k is integer.These θ (θ₁, θ₂, θ₃) value depend on phase place δ.

For phase place δ, only in δ=k × 360 ° ± 120 °, at a few class specific phase such as k × 360 ° ± 126.32 ° and k × 360 ° ± 141.76 °, analytic solutions, one of which analytic solutions are just had to be shown in Table 1.And for being in the wave plate of other interval phase places of k × 360 ° ± [111.5 °, 141.76 °], only numerical solution, the detailed demonstration part that the method for solving of numerical solution sees below, the result of numerical solution is shown in Fig. 4 a and Fig. 4 b.

Three the angle solutions meeting the complete mutual unbiased base of composition are not unique, have symmetry, even θ_i(i=1,2,3) constitutes complete mutual unbiased base, then 90 ° × k ± θ_iConstitute additionally other complete mutual unbiased bases.As shown in figures 4 a and 4b, one group of angle of a kind of given shape realizes one group of complete mutual unbiased base.

Table 1

δ	120°	126.32°	141.76°
				θ1	0°	15.66°	31.90°
θ2	27.37°	45°	58.10°
				θ3	117.37°	74.34°	135°

In the present embodiment, polaroid 3 is only used for separating the horizontal polarization composition in incident illumination with vertical polarized component, not other particular/special requirements.Those skilled in the art can the most rationally select polaroid to use, and the most no longer repeats.

The present invention uses single wave plate and a polaroid to realize complete mutual unbiased base, and its error is about the half of the systematic error realized with half-wave plate and quarter-wave plate combination.Additionally, also there is great advantage for simplifying experimental provision and saving composition.

Mutual unbiased base complete to above-mentioned single quantum bit realizes the principle of device and each ingredient is described in detail below:

1. the Bloch vector measuring base that unicast sheet realizes

Single wave plate measuring apparatus is as in figure 2 it is shown, be made up of wave plate 4 and polaroid 3.Wave plate phase place is δ, and when the angle of optical axis relative level is θ, the one corresponding battle array is changed to:

$U(\mathrm{\θ},\mathrm{\δ})=\left(\begin{array}{cc}{\cos }^2\mathrm{\θ}+e^{\mathrm{i\δ}}{\sin }^2\mathrm{\θ}& \frac{1-e^{\mathrm{i\δ}}}{2}\sin 2\mathrm{\θ}\\ \frac{1-e^{\mathrm{i\δ}}}{2}\sin 2\mathrm{\θ}& {\sin }^2\mathrm{\θ}+e^{\mathrm{i\δ}}{\cos }^2\mathrm{\θ}\end{array}\right)---\left(1\right)$

Definition horizontal polarization and vertical polarization direction are basic vector, and wave plate realizes measuring the Bloch vector form of base and is after being applied to polaroid: $\stackrel{\→}{r}(\mathrm{\δ},\mathrm{\θ})=\left(\begin{array}{ccc}{\sin }^2\frac{\mathrm{\δ}}{2}\sin 4\mathrm{\θ}& -\sin \mathrm{\δ}\sin 2\mathrm{\θ}& 1-2{\sin }^2\frac{\mathrm{\δ}}{2}{\sin }^{2}2\mathrm{\θ}\end{array}\right).$

The most complete mutual unbiased base

Under single quantum bit system, between the eigenvector of two mechanical quantities, the condition of unbiased is that their Bloch vector of correspondence is orthogonal each other.The most only three mutual unbiased bases under single quantum bit system, they constitute complete mutual unbiased base.The framework gesture of definition wave plate 4 is:

$\mathrm{\Φ}\left(\mathrm{\δ}\right)=\underset{{\mathrm{\θ}}_1,{\mathrm{\θ}}_2{,\mathrm{\θ}}_3}{\min }\{{[\stackrel{\→}{r}(\mathrm{\δ},{\mathrm{\θ}}_1)\·\stackrel{\→}{r}(\mathrm{\δ},{\mathrm{\θ}}_2)]}^2+{[\stackrel{\→}{r}(\mathrm{\δ},{\mathrm{\θ}}_1)\·\stackrel{\→}{r}(\mathrm{\δ},{\mathrm{\θ}}_3)]}^2+{[\stackrel{\→}{r}(\mathrm{\δ},{\mathrm{\θ}}_2)\·\stackrel{\→}{r}(\mathrm{\δ},{\mathrm{\θ}}_3)]}^2\}---\left(2\right)$

Then wave plate 4 can realize framework gesture Φ (δ)=0 that the condition of complete mutual unbiased base is it.Framework gesture is that the single wave plate of the phase place of zero correspondence can realize complete mutual unbiased base.

From figure 3, it can be seen that framework gesture is zero corresponding phase scope is [111.5 °, 141.76 °].Those solutions making framework gesture be zero are i.e. three angles that wave plate is corresponding, and the numerical solution of these angles is shown in Fig. 4 a and Fig. 4 b.Method of value solving is to use the algorithm lsqnonlin after optimizing in matlab for problems, and in order to avoid local optimum, the random initial point that have employed 100 groups of angles solves.

3. phase place and the symmetry of angle

Due to r (θ, δ)=r (θ, δ+360 °) andSo Φ (δ)=Φ (k × 360 ° ± δ).Additionally, due toIf therefore θ_i(i=1,2,3) constitutes complete mutual unbiased base, then 90 ° × k ± θ_i(i=1,2,3) also constitutes complete mutual unbiased base.Due to r (θ, δ)=r (θ+180 °, δ), so as shown in figures 4 a and 4b, angle, θ uses the angle in the range of [0 °, 180 °].Meet the one of complete mutual unbiased base at [0 °, 180 °] and have four kinds of symmetric case, namely θ_i, 90 ° of-θ_i, 90 ° of+θ_i, 180 ° of-θ_i(i=1,2,3) mould with 180 °.In figs. 4 a and 4b, these four groups complete mutual unbiased bases respectively with symbol "+", " ☆ ", ". " and " ◇ " represents.The solution being derived by symmetry is referred to as a class, Fig. 4 a and Fig. 4 b and respectively represents a class solution.The solution of Fig. 4 a and Fig. 4 b does not has symmetry each other.It is recognised that have two class solutions in the range of [0 °, 126.32 °] from Fig. 4 a and Fig. 4 b, (126.32 °, 141.76 °] in the range of an only class solution.

It should be noted that in figs. 4 a and 4b, (such as 120 °) at some phase place, the solution represented with distinct symbols overlaps.Overlapping position represents that the symbol shape (prismatic as solid) of solution is formed by above-mentioned Overlapping Symbol, is not new symbol.

4. unicast sheet realizes the systematic error of complete unbiased base

Unicast sheet realizes the systematic error of complete unbiased base and is calculated as follows:

Due to $\frac{\∂\stackrel{\→}{r}}{\∂\mathrm{\θ}}=(4{\sin }^2\frac{\mathrm{\δ}}{2}\cos 4\mathrm{\θ},-2\sin \mathrm{\δ}\cos 2\mathrm{\θ},-4{\sin }^2\frac{\mathrm{\δ}}{2}\sin 4\mathrm{\θ}),$ Then its square is ${\left|\frac{\∂\stackrel{\→}{r}}{\∂\mathrm{\θ}}\right|}^2={16\sin }^2\frac{\mathrm{\δ}}{2}-4{\sin }^2\mathrm{\δ}{\sin }^{2}2\mathrm{\θ}.$ $\frac{\∂\stackrel{\→}{r}}{\∂\mathrm{\δ}}=(\frac{1}{2}\sin \mathrm{\δ}\sin 4\mathrm{\θ},-\cos \mathrm{\δ}\sin 2\mathrm{\θ},-\sin \mathrm{\δ}{\sin }^{2}2\mathrm{\θ}),$ Then its square isDue to the orthogonal normalizing of Bloch vector of one group of complete mutual unbiased base, thenThe systematic error realizing complete unbiased base is:

$\begin{array}{c}\underset{i=1}{\overset{3}{\mathrm{\Σ}}}{\left|\frac{\∂\stackrel{\→}{r}\left({\mathrm{\θ}}_i\right)}{{\∂\mathrm{\θ}}_i}\right|}^2{\left(\mathrm{\Δ\θ}\right)}^2+{\left|\frac{\∂\stackrel{\→}{r}\left({\mathrm{\θ}}_i\right)}{\∂\mathrm{\δ}}\right|}^2{\left(\mathrm{\Δ\δ}\right)}^2\\ =\underset{i-1}{\overset{3}{\mathrm{\Σ}}}(16{\sin }^2\frac{\mathrm{\δ}}{2}-4{\sin }^2\mathrm{\δ}{\sin }^2{2\mathrm{\θ}}_i){\left(\mathrm{\Δ\θ}\right)}^2+{\sin }^2{2\mathrm{\θ}}_i{\left(\mathrm{\Δ\δ}\right)}^2\\ =(48{\sin }^2\frac{\mathrm{\δ}}{2}-4){\left(\mathrm{\Δ\δ}\right)}^2+\frac{1}{{\sin }^2\mathrm{\δ}}{\left(\mathrm{\Δ\δ}\right)}^2\end{array}----\left(3\right)$

Wherein Δ θ is the error of angle, and Δ δ is the error of phase place.When δ=111.5 °, minimum 28.80 (the Δ θ) of systematic error²+1.16(Δδ)²；When δ=141.7 °, systematic error is 38.83 (Δ θ) to the maximum²+2.60(Δδ)²When δ=120 °, when that is to say selection 1/3rd wave plate, systematic error is 32 (Δ θ)²+1.33(Δδ)².And use half-wave plate as shown in Figure 1 and quarter-wave plate combination to realize the error average out to 2.5 (Δ δ) of complete mutual unbiased base²+58(Δθ)².Can be seen that the half of the error that unicast sheet realizes the error ratio original device of complete unbiased base is the least when selecting δ=111.5 °.When selecting 1/3rd wave plates, error is close to the half of original error.The most on average, the half of the error of the error of unicast sheet the chances are original double wave plate combined.

So far, already in connection with accompanying drawing, the embodiment of the present invention has been described in detail.According to above description, those skilled in the art the unicast sheet of mutual unbiased base complete to list quantum bit of the present invention should realize device and had and clearly recognize.

Additionally, the above-mentioned definition to each element and method is not limited in various concrete structures, shape or the mode mentioned in embodiment, it can be changed or replace by those of ordinary skill in the art simply.

In sum, the present invention utilizes the device of the complete mutual unbiased base of a single quantum bit system of wave plate realization, not only save the use of wave plate, decrease wave plate number of revolutions, and only one of which wave plate introduces error, thus reducing error, this is the rareest in high-precision quantum information task.

Particular embodiments described above; the purpose of the present invention, technical scheme and beneficial effect are further described; it is it should be understood that; the foregoing is only the specific embodiment of the present invention; it is not limited to the present invention; all within the spirit and principles in the present invention, any modification, equivalent substitution and improvement etc. done, should be included within the scope of the present invention.

Claims (6)

1. the complete mutual unbiased base of single quantum bit realize device, it is characterised in that by one Individual wave plate (4) and polaroid (3) are constituted:

The phase place of described wave plate (4) is δ, wave plate (4) described in incident illumination normal incidence, wherein, and phase place In the range of δ is in k × 360 ° ± [111.5 °, 141.76 °], k is integer；

Described polaroid (3) is positioned at the light path rear end of described wave plate (4), and being used for will be by described wave plate (4) composition polarized in the horizontal direction in incident illumination separates with the composition vertically polarized；

Wherein, the optical axis defining described wave plate (4) is θ relative to the angle of horizontal direction, as this θ Value is respectively θ₁,θ₂,θ₃Time, it is achieved three measurement bases of single quantum bit, these three measurement base structures Become complete mutual unbiased base.

The most according to claim 1 realize device, it is characterised in that described wave plate (4) Phase place δ and optical axis meet the framework gesture making described wave plate (4) relative to the angle, θ of horizontal direction Φ (δ)=0, wherein:

$\mathrm{\Φ}\left(\mathrm{\δ}\right)=\underset{{\mathrm{\θ}}_1,{\mathrm{\θ}}_2,{\mathrm{\θ}}_3}{min}\{{\[\stackrel{\→}{r}(\mathrm{\δ},{\mathrm{\θ}}_1)\·\stackrel{\→}{r}(\mathrm{\δ},{\mathrm{\θ}}_2)\]}^2+{\[\stackrel{\→}{r}(\mathrm{\δ},{\mathrm{\θ}}_1)\·\stackrel{\→}{r}(\mathrm{\δ},{\mathrm{\θ}}_3)\]}^2+{\[\stackrel{\→}{r}(\mathrm{\δ},{\mathrm{\θ}}_2)\·\stackrel{\→}{r}(\mathrm{\δ},{\mathrm{\θ}}_3)\]}^2\}$

Wherein, definition horizontal polarization and vertical polarization direction are basic vector, and wave plate is real after being applied to polaroid The Bloch vector form now measuring base is:

$\stackrel{\→}{r}(\mathrm{\δ},\mathrm{\θ})=\left(\begin{array}{ccc}{\sin }^2\frac{\mathrm{\δ}}{2}sin4\mathrm{\θ}& -sin\mathrm{\δ}sin2\mathrm{\θ}& 1-2{\sin }^2\frac{\mathrm{\δ}}{2}{\sin }^{2}2\mathrm{\θ}\end{array}\right).$

The most according to claim 2 realize device, it is characterised in that k=0, described wave plate (4) Phase place δ be in [111.5 °, 141.76 °] in the range of.

The most according to claim 3 realize device, it is characterised in that for described wave plate (4):

Its phase place δ is: δ=120 °；

Its optical axis is respectively as follows: θ relative to three values of the angle, θ of horizontal direction₁=0 °, θ₂=27.37 °, θ₃=117.37 °.

The most according to claim 3 realize device, it is characterised in that for described wave plate (4):

Its phase place δ is: δ=126.32 °；

Its optical axis is respectively as follows: θ relative to three values of the angle, θ of horizontal direction₁=15.66 °, θ₂=45 °, θ₃=74.34 °.

The most according to claim 3 realize device, it is characterised in that for described wave plate (4):

Its phase place δ is: δ=141.76 °；

Its optical axis is respectively as follows: θ relative to three values of the angle, θ of horizontal direction₁=31.90 °, θ₂ =58.10 °, θ₃=135 °.