CN103746799A - Differential non-Gauss operation radioactivity continuous variable quantum key distribution method - Google Patents

Abstract

The invention discloses a differential non-Gaussian operation radioactive continuous variable quantum key distribution method. The trusted third party Fred uses a laser generator to prepare a dual-mode vacuum compression state, which is denoted as ρ _ab , and the photons pass through the dual-mode vacuum compression state After ρ _ab beam splitting, it undergoes double-mode unitary transformation co-generation of a dual-mode vacuum-squeezed state At this time, the output modes of the photons are denoted as a ₀ and b ₀ respectively, and the non-Gaussian operation is realized through the photon subtraction operation, and the modes a ₁ and b ₁ are respectively obtained. When Alice and Bob receive the photon output mode a from the noise channel ₂ , when b ₂ , use heterodyne detector to detect secret negotiation and error correction, and finally extract the key for data encryption. The beneficial effect of the invention is that the non-Gaussian modulation is realized in the signal source by using the photon number resolution detector, and the security of the key transmission in the quantum communication system is improved.

Description

A kind of non-Gauss of difference operates radioactivity continuous variable quantum key delivering method

Technical field

The invention belongs to Quantum Secure Communication field, the non-Gauss who relates to a kind of difference operates radioactivity continuous variable quantum key delivering method.

Background technology

Continuous variable quantum key distribution is encoded by the orthogonality of photon field x and p and is extracted it with homodyne or Heterodyne detect, for selecting initial discrete variable quantum key distribution, be that ready-made lasing light emitter is more conveniently integrated into current communication system because it provides higher detection efficiency.In 10 years of past, in continuous variable quantum key distribution, produced major progress.Many nearest results comprise that unconditional Security Proof resists any potential attack strategy.But the quantum channel that depends on that safety analysis is too much is linear hypothesis.Recently, Guo and other people have proposed one and have improved four state EBCVQKD schemes, and wherein legal participant calculates covariance matrix under channel is not linear hypothesis.Not fortunately, these two legitimate correspondence are rectified the correlation of handing over can not reach an EPR state, and this has significantly limited the key rate of the CVQKD in practice.

Summary of the invention

Object of the present invention operates radioactivity continuous variable quantum key delivering method in the non-Gauss that a kind of difference is provided, and has solved the distribution method of current quantum key the method under not modulating based on non-Gauss, has limited the problem of practical application.

The present invention carries out according to following steps:

The first step: the Fred of trusted third party utilizes laser generator to prepare a bimodulus squeezed vacuum state, is designated as ρ _ab, photon is through bimodulus squeezed vacuum state ρ _abafter beam splitting, then pass through bimodulus unitary transformation

the common bimodulus squeezed vacuum state that produces now the output mould of photon is designated as respectively a ₀and b ₀, photon through conversion after respectively at mould a ₀and b ₀upper transmission;

Second step: photon output mould a ₀and b ₀by photon subtraction, operate to realize non-Gauss's operation, obtain respectively mould a ₁with mould b ₁, Fred makes respectively mould a ₁with mould b ₁by becoming final photon output mould a after noisy communication channel ₂and b ₂;

The 3rd step: receive noisy communication channel at Alice and Bob and pass the photon output mould a coming ₂, b ₂time, with heterodyne detector, detect and carry out secret negotiation and error correction, finally extract and obtain key for data encryption.

Feature of the present invention is also: bimodulus unitary transformation in the first step

the common bimodulus squeezed vacuum state that produces

conversion process as follows:

${|\mathrm{\φ}>}_{a_0{b}_0}={\hat{U}}_{\mathrm{ab}}{|r>}_a{|-r>}_b=\underset{n=0}{\overset{\∞}{\mathrm{\Σ}}}{\mathrm{\α}}_n{|n>}_a{|n>}_b,$

Wherein represent beam splitter operation,

represent respectively to act on generation operator and the annihilations operator on mould a and b, n represents the number of photon;

If adopt α=sinh (r), r represents the compressed coefficient, factor alpha _ncan be expressed from the next:

${\mathrm{\α}}_n=\frac{{\mathrm{\α}}^n}{{(1+{\mathrm{\α}}^2)}^{\frac{n+1}{2}}}.$

Photon output mould a in second step ₀and b ₀by photon subtraction, obtain described final mould a ₂, b ₂process be:

Fred is at mould c ₀and s ₀upper preparation vacuum state, by mould a ₀with mould s ₀photon be input to efficiency of transmission μ _abeam splitter, by mould b ₀with mould c ₀photon be input to efficiency of transmission μ _bbeam splitter, in two beam splitters, wherein beam splitter has identical efficiency of transmission μ _a=μ _b=μ, its output is designated as mould c ₁and s ₁, respectively at mould c ₁and s ₁upper effect unitary operator

result state can be represented as:

$\begin{array}{c}{|\mathrm{\&phi;}>}_{a1b1s1\mathrm{<figure-callout\; id="1"\; label="c"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">c</figure-callout>}1}=[{\hat{U}}_{{\mathrm{as}}_0}\&CircleTimes;{\hat{U}}_{{\mathrm{bc}}_0}]{\mathrm{\&phi;}>}_{\mathrm{ab}}{|0>}_{{\mathrm{<figure-callout\; id="0"\; label="s"\; filenames="HDA0000446822670000032.png"\; state="\{\{state\}\}">s</figure-callout>}}_0}{|0>}_{{\mathrm{<figure-callout\; id="0"\; label="c"\; filenames="HDA0000446822670000032.png"\; state="\{\{state\}\}">c</figure-callout>}}_0}\\ =\underset{n}{\mathrm{\&Sigma;}}{\mathrm{\&alpha;}}_n\underset{k=0}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&xi;}}_{\mathrm{nk}}\left|{n-k>}_{a1}\right|{n-k>}_{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">b</figure-callout>}1}\left|{k>}_{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">s</figure-callout>}1}\right|{k>}_{\mathrm{<figure-callout\; id="1"\; label="c"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">c</figure-callout>}1}\end{array},$

Here at mould s ₁, c ₁, a ₁, b ₁represent respectively four output moulds of beam splitter, represent the direct product operation of two operators,

and

represent binomial coefficient, with photon number resolved detection device detection mould s ₁, c ₁in photon, when having k photon at pattern s ₁and c ₁on photon when being detected by photon number resolved detection device, bimodulus squeezed state can be represented as following non-Gauss's state:

$\begin{array}{c}{|{\mathrm{\&phi;}}^{\left(k\right)}>}_{a1\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">b</figure-callout>}1}={\mathrm{<figure-callout\; id="1"\; label="c"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">c</figure-callout>}}_1{<k|}_{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">s</figure-callout>}1}<k{|\mathrm{\&phi;}>}_{a1b1s1\mathrm{<figure-callout\; id="1"\; label="c"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">c</figure-callout>}1}\\ =\frac{1}{\sqrt{p_k}}\underset{n=k}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_n{\mathrm{\&xi;}}_{\mathrm{nk}}{|n-k>}_{a1}{|n-k>}_{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">b</figure-callout>}1}\end{array},$

Wherein p _krepresent normalization factor, represent to produce non-Gauss's probability of state,

$p_k=\underset{n=k}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA0000446822670000022.png"\; state="\{\{state\}\}">n</figure-callout>}}^2{\mathrm{\&xi;}}_{\mathrm{<figure-callout\; id="2"\; label="nk"\; filenames="HDA0000446822670000022.png"\; state="\{\{state\}\}">nk</figure-callout>}}^2,$

The photon finally generating, obtains noisy photon at receiving terminal and is designated as respectively mould a respectively to Alice and Bob transmission through noisy communication channel ₂and b ₂.

The invention has the beneficial effects as follows by utilizing photon number resolved detection device to realize non-Gauss's modulation in information source, improved the fail safe of cipher key delivery in quantum communication system.

Accompanying drawing explanation

Fig. 1 is the principle schematic of the radioactive EBCVQKD under non-Gauss of the present invention operates;

Fig. 2 is | φ > _abwith | φ ^(k)> _a1b1degree of entanglement comparison diagram;

Fig. 3 is the correlation contrast of z1 and z0;

Channel excess noise tolerance limit lower bound based on the scheme of tangling when Fig. 4 is use photon subtraction and original scheme efficiency of transmission function curve diagram;

Fig. 5 is efficiency of transmission while being T=0.268, μ and progressive key rate under different channels noise the function relation figure of (solid line);

Fig. 6 is in use photon difference and under high channel noise situations, based on progressive key rate in the scheme of tangling

function relation figure with transmission range.

Embodiment

The non-Gauss of a kind of difference of the present invention operates radioactivity continuous variable quantum key delivering method as shown in Figure 1, by neutral participant, Fred, a Gaussian source tangling, between two remote participants, set up the source of tangling, Alice and Bob, encrypt in order to produce a key rate in the quantum network of a radiation.

The first step: the Fred of trusted third party utilizes laser generator to prepare a bimodulus squeezed vacuum state and is designated as ρ _ab, α here ²=V ₀/ 2, and V ₀represent the modulation variance of single-mode squeezing state, this bimodulus squeezed state has variance 2 α ²+ 1.Bimodulus squeezed vacuum state ρ _abobject be by a mould by transmission efficiency T and additional noise ε optical fiber send to Alice simultaneously another one mould by the optical fiber with identical transmission efficiency T and additional noise ε, send to Bob, the vacuum state of two moulds is identical.Photon is respectively after beam splitter, through bimodulus unitary transformation

the common bimodulus squeezed vacuum state that produces

conversion process is as follows:

${|\mathrm{\&phi;}>}_{a_0{b}_0}={\hat{U}}_{\mathrm{ab}}{|r>}_a{|-r>}_b=\underset{n=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_n{|n>}_a{|n>}_b,$

Wherein

represent beam splitter operation,

represent respectively to act on generation operator and the annihilations operator on mould a and b, utilize two mark devices on two mould a and b, to realize respectively the operation to photon, n represents the number of photon.If adopt α=sinh (r), r represents the compressed coefficient, Schmidt factor alpha in formula (1) _ncan be expressed from the next:

${\mathrm{\&alpha;}}_n=\frac{{\mathrm{\&alpha;}}^n}{{(1+{\mathrm{\&alpha;}}^2)}^{\frac{n+1}{2}}};$

Second step: photon is by bimodulus unitary transformation

after, photon is designated as respectively a at output mould ₀and b ₀, by photon subtraction, operate to realize non-Gauss's operation, as shown in the dotted line picture frame in Fig. 1, in Fig. 1, PNRD represents photon number resolving detection, Fred is at mould c ₀and s ₀upper preparation vacuum state; By mould a ₀with mould s ₀photon be input to efficiency of transmission μ _abeam splitter; On the other hand by mould b ₀with mould c ₀photon be input to efficiency of transmission μ _bbeam splitter, in two beam splitters, (there is identical efficiency of transmission μ _a=μ _b=μ), its output is designated as mould c ₁and s ₁; Respectively at mould c ₁and s ₁upper effect unitary operator

result state can be represented as:

$\begin{array}{c}{|\mathrm{\&phi;}>}_{a1b1s1\mathrm{<figure-callout\; id="1"\; label="c"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">c</figure-callout>}1}=[{\hat{U}}_{{\mathrm{as}}_0}\&CircleTimes;{\hat{U}}_{{\mathrm{bc}}_0}]{\mathrm{\&phi;}>}_{\mathrm{ab}}{|0>}_{{\mathrm{<figure-callout\; id="0"\; label="s"\; filenames="HDA0000446822670000032.png"\; state="\{\{state\}\}">s</figure-callout>}}_0}{|0>}_{{\mathrm{<figure-callout\; id="0"\; label="c"\; filenames="HDA0000446822670000032.png"\; state="\{\{state\}\}">c</figure-callout>}}_0}\\ =\underset{n}{\mathrm{\&Sigma;}}{\mathrm{\&alpha;}}_n\underset{k=0}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&xi;}}_{\mathrm{nk}}\left|{n-k>}_{a1}\right|{n-k>}_{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">b</figure-callout>}1}\left|{k>}_{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">s</figure-callout>}1}\right|{k>}_{\mathrm{<figure-callout\; id="1"\; label="c"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">c</figure-callout>}1}\end{array},$ Here at mould s ₁, c ₁, a ₁, b ₁represent respectively four output moulds of beam splitter,

represent the direct product operation of two operators,

and represent binomial coefficient.With photon number resolved detection device detection mould s ₁, c ₁in photon, when having k photon at pattern s ₁and c ₁on photon when being detected by photon number resolved detection device, bimodulus squeezed state can be represented as following non-Gauss's state:

$\begin{array}{c}{|{\mathrm{\&phi;}}^{\left(k\right)}>}_{a1\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">b</figure-callout>}1}={\mathrm{<figure-callout\; id="1"\; label="c"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">c</figure-callout>}}_1{<k|}_{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">s</figure-callout>}1}<k{|\mathrm{\&phi;}>}_{a1b1s1\mathrm{<figure-callout\; id="1"\; label="c"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">c</figure-callout>}1}\\ =\frac{1}{\sqrt{p_k}}\underset{n=k}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_n{\mathrm{\&xi;}}_{\mathrm{nk}}{|n-k>}_{a1}{|n-k>}_{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">b</figure-callout>}1}\end{array},$

Wherein p _krepresent normalization factor, also represent to produce non-Gauss's probability of state,

$p_k=\underset{n=k}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA0000446822670000022.png"\; state="\{\{state\}\}">n</figure-callout>}}^2{\mathrm{\&xi;}}_{\mathrm{<figure-callout\; id="2"\; label="nk"\; filenames="HDA0000446822670000022.png"\; state="\{\{state\}\}">nk</figure-callout>}}^2,$

For example, when k ∈ 1,2}, it can be calculated by formula below:

${\mathrm{<figure-callout\; id="1"\; label="p"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">p</figure-callout>}}_1=\frac{{\mathrm{\&alpha;}}^2{(1-\mathrm{\&mu;})}^2(1+{\mathrm{\&alpha;}}^2+{\mathrm{\&alpha;}}^2{\mathrm{\&mu;}}^2)}{{(1+{\mathrm{\&alpha;}}^2-{\mathrm{\&alpha;}}^2{\mathrm{\&mu;}}^2)}^3},$

${\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA0000446822670000022.png"\; state="\{\{state\}\}">p</figure-callout>}}_2=\frac{{\mathrm{\&alpha;}}^4{(1-\mathrm{\&mu;})}^4({(1+{\mathrm{\&alpha;}}^2)}^2+{4\mathrm{\&alpha;}}^2{\mathrm{\&mu;}}^2(1+{\mathrm{\&alpha;}}^2)+{\mathrm{\&alpha;}}^4{\mathrm{\&mu;}}^4}{{(1+{\mathrm{\&alpha;}}^2-{\mathrm{\&alpha;}}^2{\mathrm{\&mu;}}^2)}^5},$

Wherein μ represents beam splitter efficiency of transmission.

The 3rd step: at Alice(or Bob) when receiving noisy communication channel and passing the photon of coming, its mould is designated as respectively a ₂, b ₂, to use identical ideal detector (comprise zero-difference detection device homodynedetector or heterodyne detector heterodynedetector, this programme is selected heterodyne detector) to detect to extract and obtain key, Heterodyne detect device efficiency equates, i.e. η _a=η _b.Thereby this is can be further used in extraction encrypted private key in order to share two relevant gaussian variables.Because actual quantum channel and detector are not desirable, the interchannel noise of increase is regarded as channel input and is designated as X _line=1/T-1+ ε, wherein 1/T-1 represents loss of signal channel, ε represents noisy communication channel.

By following process, the present invention is verified, verifies efficiency of the present invention:

In order to show to generate the non-Gauss's quantum state of bimodulus | φ ^(k)> _a1b1feature, we are using negative logarithm as tangling measurement.According to logarithmic function character, we can obtain original compression Entangled State | φ > _a0b0in the evolution based under the non-Gauss's operation of subtraction.Namely, | φ > _a0b0with | φ ^(k)> _a1b1negative logarithm represent to be expressed as the degree of entanglement of quantum state respectively:

${\mathrm{<figure-callout\; id="0"\; label="E"\; filenames="HDA0000446822670000032.png"\; state="\{\{state\}\}">E</figure-callout>}}_0=-2{\log }_2(\sqrt{1+{\mathrm{\&alpha;}}^2}-\mathrm{\&alpha;}){-\log }_2(1+{\mathrm{\&alpha;}}^2),$

$E_{k,u}=2{\mathrm{<figure-callout\; id="2"\; label="log"\; filenames="HDA0000446822670000022.png"\; state="\{\{state\}\}">log</figure-callout>}}_2\frac{{\mathrm{\&alpha;}}^k{(1-\mathrm{\&mu;})}^k}{{(\sqrt{1+{\mathrm{\&alpha;}}^2}-\mathrm{\&alpha;\&mu;})}^{k+1}}-{\log }_2{p}_k,$

E ₀and E _{k, μ}negative logarithm represent the degree of entanglement of quantum state, we notice that subtraction noted earlier operates for μ=1st, non-existent.Prove easily photon k state | φ ^(k)| _a1b1in k>=1 o'clock, than original input state | φ > _a0b0there is a larger degree of entanglement, have a lot of Entangled States to use a suitable transmission conditions μ ∈ (0,1).In addition, larger quantity deducts larger the tangling that photon k means two-way state.It shows that the non-Gauss operation based on subtraction can increase the correlation of two kinds of patterns of two-way state.We also notice that photon number resolving detection should be used as effective detection, and this has caused one to tangling after process in source, to produce non-Gaussian Mixture state, and has good degree of entanglement.As Fig. 2 | φ > _abwith | φ ^(k)> _a1b1degree of entanglement comparison diagram, here because μ ∈ 0.5,0.8} and k ∈ 1,2}, so | φ > _a0b0and E _{k, μ}=| φ ^(k)> _a1b1there is α ∈ (0,1).

We attempt the raising of tangling performance that proof is brought by Fred, Fred is transmitting Entangled State to the non-Gauss's operation having completed between remote participant based on photon subtraction, and it can improve the performance of the continuous variable quantum key distribution based on tangling in radiating light sub-network, the key rate of radioactive continuous variable quantum key distribution based on tangling.

In this section, we show the key rate of the EBCVQKD that how to estimate the radiation under the non-Gauss's operation based on photon subtraction, so we concentrate on, analyze direct negotiation, simultaneously because oppositely negotiation can obtain by similar method.Based on the EBCVQKD of Gauss's modulation, this key rate K _gcan have formula below to calculate:

$K_G={\mathrm{\&beta;I}}_{\mathrm{ab}}^G-X_{\mathrm{ae}}^G,$

Wherein β is negotiation efficiency,

shannon information total between Alice and Bob,

represent Holevo number, it is illustrated on the key of Alice to listener-in's maximum effective information, and formula is as follows:

$X_{\mathrm{ae}}^G=S\left({\mathrm{\&rho;}}_e\right)-\underset{\mathrm{ma}}{\mathrm{\&Sigma;}}p\left(\mathrm{ma}\right)S\left({\mathrm{\&rho;}}_e^{\mathrm{ma}}\right),$

Wherein S is variational OR entropy, be listener-in's localized state, ma represents the measurement result that Alice realizes along with the possibility of p (ma), represent listener-in's auxiliary state, it is to obtain under the ma of the measurement result of Alice condition.Because listener-in can provide the pure state of an Alice and the density matrix of Bob, we obtain S (ρ _e)=S (ρ _a2b2) and

therefore, Holevo quantity can be written as:

$X_{\mathrm{ae}}^G=S\left({\mathrm{\&rho;}}_{a2b2}\right)-S\left({\mathrm{\&rho;}}_{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA0000446822670000022.png"\; state="\{\{state\}\}">b</figure-callout>}2}^{\mathrm{ma}}\right),$

And, as follows according to a useful covariance matrix:

Wherein

ρ is that density matrix and { } represent anticommutator, uses the result state ρ of the non-Gauss's operation based on photon subtraction in the continuous variable quantum key distribution based on tangling of radiation _a1b1=| φ ^(k)> _a1b1< φ ^(k)| covariance matrix can be calculated by following formula:

${\mathrm{\&Gamma;}}_{a1b1}^N=\left(\begin{array}{cc}\mathrm{<figure-callout\; id="2"\; label="aI"\; filenames="HDA0000446822670000022.png"\; state="\{\{state\}\}">aI</figure-callout>}2& \mathrm{c\&sigma;z}\\ \mathrm{c\&sigma;z}& \mathrm{<figure-callout\; id="2"\; label="bI"\; filenames="HDA0000446822670000022.png"\; state="\{\{state\}\}">bI</figure-callout>}2\end{array}\right),$

Wherein I ₂be 2 × 2 unit matrix, σ Z is take 1 and-1 as cornerwise diagonal matrix, while a, and b and c represent effective parameter, by formula below, are drawn:

$a=a1b1{<{\mathrm{\&phi;}}^{\left(k\right)}\left|{1+2\hat{a}}^{\&DownArrow;}\hat{a}\right|{\mathrm{\&phi;}}^{\left(k\right)}>}_{a1b1}=V_0{p}_k+1,$

$b=a1b1{<{\mathrm{\&phi;}}^{\left(k\right)}\left|{1+2\hat{b}}^{\&DownArrow;}\hat{b}\right|{\mathrm{\&phi;}}^{\left(k\right)}>}_{a1b1}=V_0{p}_k+1,$

$c=a1b1{<{\mathrm{\&phi;}}^{\left(k\right)}\left|{\hat{a}\hat{b}+\hat{a}}^{\&DownArrow;}{\hat{b}}^{\&DownArrow;}\right|{\mathrm{\&phi;}}^{\left(k\right)}>}_{a1b1}=V_0{z}_k,$

Wherein

with representing that photon subtraction operates in pattern separately carries out, p _kin equation, calculate correlation Z _kby formula below, obtained:

$\mathrm{zk}=\underset{n=k}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{{\mathrm{\&alpha;}}_n^3{\mathrm{\&xi;}}_{\mathrm{nk}}^3}{\mathrm{pk\&alpha;n}+1\mathrm{\&xi;}(n+1)k},$

For example, when k=1, we obtain:

$\mathrm{<figure-callout\; id="1"\; label="z"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">z</figure-callout>}1=\frac{\sqrt{{\mathrm{\&alpha;}}^2+1}{(1+{\mathrm{\&alpha;}}^2-{\mathrm{\&alpha;}}^2{\mathrm{\&mu;}}^2)}^3}{{\mathrm{\&alpha;}}^2{\mathrm{\&mu;}}^3(1+{\mathrm{\&alpha;}}^2-{\mathrm{\&alpha;}}^2{\mathrm{\&mu;}}^2)}\&CenterDot;(\frac{1}{{\mathrm{\&alpha;}}^2{\mathrm{\&mu;}}^2}\mathrm{<figure-callout\; id="1"\; label="ln"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">ln</figure-callout>}\frac{1+{\mathrm{\&alpha;}}^2-{\mathrm{\&alpha;}}^2{\mathrm{\&mu;}}^2}{1+{\mathrm{\&alpha;}}^2}+\frac{1}{1+{\mathrm{\&alpha;}}^2-{\mathrm{\&alpha;}}^2{\mathrm{\&mu;}}^2}+\frac{{2\mathrm{\&alpha;}}^4{\mathrm{\&mu;}}^4}{{(1+{\mathrm{\&alpha;}}^2-{\mathrm{\&alpha;}}^2{\mathrm{\&mu;}}^2)}^3}),$

We notice that the continuous variable quantum key distribution based on subtraction does not exist when μ=1, and this is with original to have the source of tangling relevant in middle continuous variable quantum key distribution.Being familiar with the relation of matrix can make larger than Gauss EPR state for suitable parameter μ and α.It shows that the Unconditional security of the scheme proposing for the two-way state being subtracted is subject to the restriction of μ and α value, and when α levels off to 0 time, it becomes the lower bound of Gauss EPR state.But when the value of α becomes very little, it just becomes invalid in practice.Our reason of having to select suitable μ and α value that will complete in the scheme proposing that Here it is.

Because being that the strategy most with attack of being carried out by listener-in is that Gauss attacks, we suppose to exist Gauss's state ρ ' of equal value _a1b1there is identical covariance matrix as the continuous variable quantum key distribution based on tangling take photon subtraction as basis, for example,

ρ ' _a1b1and ρ _a1b1between pass tie up in Ref. and come into question.Next, we consider with ρ ' _a1b1replacement is subtracted state ρ _a1b1for the efficiency of the scheme of basic proposition.

In the photon channel of radiation after the transmission of photon, transmitting state ρ ' _a1b1covariance matrix

can be represented as:

${\mathrm{\&Gamma;}}_{a2b2}^G=\left(\begin{array}{cc}T(a+X_{\mathrm{line}})I_2& \sqrt{T}c{\mathrm{\&sigma;}}_z\\ \sqrt{T}c{\mathrm{\&sigma;}}_z& T(b+X_{\mathrm{line}})I_2\end{array}\right),$

After this, we are due to this true symbol that uses of a=b

Α=T (a+X _line)=T (b+X _line), covariance matrix

pungent eigenvalue by following formula, provided:

${\mathrm{\&lambda;}}_{\mathrm{1,2}}^2=\&PlusMinus;\sqrt{12T{A}^2{c}^2-{3T}^2{c}^4}+{\mathrm{<figure-callout\; id="2"\; label="Tc"\; filenames="HDA0000446822670000022.png"\; state="\{\{state\}\}">Tc</figure-callout>}}^2+{2\mathrm{<figure-callout\; id="2"\; label="A"\; filenames="HDA0000446822670000022.png"\; state="\{\{state\}\}">A</figure-callout>}}^2,$

The correlation comparison diagram of z1 and z0 as shown in Figure 3.Here z0 represents the relation of Gauss EPR state, simultaneously z1, μ for

along with α ∈ { variation μ ∈ { 0.5, the 0.8} of 0,1}.

Suppose that Alice carries out Heterodyne detect, produces coherent states by mould and the attached d0 of vacuum in conjunction with her in the beam splitter of a balance.At former primary state ρ _a1d0b1on mutual execution can be described to

bob continues to adopt homodyne or Heterodyne detect by his pattern.Subsequently, for homodyne detection shared information between Alice and Bob can be calculated by following formula:

$I_{\mathrm{ab}}^{\mathrm{hom}}=\frac{1}{2}{\log }_2\frac{(A+1)A}{(A+1)A-{\mathrm{<figure-callout\; id="2"\; label="Tc"\; filenames="HDA0000446822670000022.png"\; state="\{\{state\}\}">Tc</figure-callout>}}^2},$

Heterodyne detect is provided by following formula:

$I_{\mathrm{ab}}^{\mathrm{het}}={\log }_2\frac{{(A+1)}^2}{{(A+1)}^2-{\mathrm{<figure-callout\; id="2"\; label="Tc"\; filenames="HDA0000446822670000022.png"\; state="\{\{state\}\}">Tc</figure-callout>}}^2},$

Next, we consider be calculate listener-in and Alice total information, for example,

as the condition of Alice homodyne detection.Use method of purification, we have

it is the pungent eigenvalue λ of relevant variance matrix _3,4function, concrete formula is as follows:

${\mathrm{\&rho;}}_{b2\mathrm{<figure-callout\; id="1"\; label="d"\; filenames="HDA0000446822670000022.png,HDA0000446822670000031.png"\; state="\{\{state\}\}">d</figure-callout>}1}^{\mathrm{ma}}=\left(\begin{array}{cccc}A-\frac{{\mathrm{Tc}}^2}{A+1}& 0& \frac{\sqrt{2T}c}{A+1}& 0\\ 0& A& 0& -\frac{\sqrt{2\mathrm{<figure-callout\; id="2"\; label="T\; c"\; filenames="HDA0000446822670000022.png"\; state="\{\{state\}\}">T</figure-callout>}}c}{2}\\ 0& -\frac{\sqrt{2\mathrm{<figure-callout\; id="2"\; label="T\; c"\; filenames="HDA0000446822670000022.png"\; state="\{\{state\}\}">T</figure-callout>}}c}{2}& 0& \frac{A+1}{2}\end{array}\right),$

Therefore, variational OR entropy with good conditionsi by

provide, wherein G (x)=(x+1) log ₂(x+1)-xlog ₂x and pungent eigenvalue λ _3,4can be calculated by following formula:

${\mathrm{\&lambda;}}_{\mathrm{3,4}}^2=\frac{1}{2}(A\&PlusMinus;\sqrt{A^2-4B}),$

Wherein A=A (A+1)-Tc ²(A-1)/(A+1) and B=(A ²-Tc ²) [A-Tc ²/ (A+1)].

Notice a Heterodyne detect be in itself one by the supposition of coherent states, heterodyne measurement result is as follows in covariance matrix:

Then, pungent eigenvalue is directly expressed as:

λ ₅＝A-Tc ²/(A+1)，

Therefore, by the variational OR entropy of Heterodyne detect, just can represent as follows:

$S\left({\mathrm{\&rho;}}_e^{\mathrm{ma}}\right)=G\left(\frac{{\mathrm{\&lambda;}}_5-1}{2}\right)$

Thus, for the lower bound of the radioactive continuous variable quantum key rate based on photon subtraction, by following formula, provided:

$K_N=p_k({\mathrm{\&beta;I}}_{\mathrm{ab}}^G-x_{\mathrm{be}}^G),$

Wherein pk has represented the possibility of successful implementation under non-Gauss's operation.Namely, the key rate under homodyne detection is calculated by following formula:

$K_N^{\mathrm{ho}}=p_k[{\mathrm{\&beta;I}}_{\mathrm{ab}}^{\mathrm{ho}}-\underset{i=\mathrm{1,2}}{\mathrm{\&Sigma;}}G\left(\frac{{\mathrm{\&lambda;}}_i-1}{2}\right)+\underset{i=\mathrm{3,4}}{\mathrm{\&Sigma;}}G\left(\frac{{\mathrm{\&lambda;}}_i-1}{2}\right)],$

Heterodyne detect is drawn by following formula simultaneously:

$K_N^{\mathrm{he}}=p_k[{\mathrm{\&beta;I}}_{\mathrm{ab}}^{\mathrm{he}}-\underset{i=\mathrm{1,2}}{\mathrm{\&Sigma;}}G\left(\frac{{\mathrm{\&lambda;}}_i-1}{2}\right)+G\left(\frac{{\mathrm{\&lambda;}}_5-1}{2}\right)],$

In the continuous variable quantum key distribution based on tangling of radiation, in photon channel, do not there is non-Gauss's operation, original Gauss's state ρ ₀covariance matrix

${\mathrm{\&Gamma;}}_0^G\left(\begin{array}{cc}(1+{\mathrm{<figure-callout\; id="0"\; label="TV"\; filenames="HDA0000446822670000032.png"\; state="\{\{state\}\}">TV</figure-callout>}}_0+x_{\mathrm{line}})I2& \sqrt{\mathrm{<figure-callout\; id="0"\; label="T\; z"\; filenames="HDA0000446822670000032.png"\; state="\{\{state\}\}">T</figure-callout>}}{z}_{}{}_0{\mathrm{\&sigma;}}_{\mathrm{<figure-callout\; id="0"\; label="z\; T\; z"\; filenames="HDA0000446822670000032.png"\; state="\{\{state\}\}">z</figure-callout>}}& \\ \sqrt{T}{z}_{}{}_0{\mathrm{\&sigma;}}_z& (1+{\mathrm{<figure-callout\; id="0"\; label="TV"\; filenames="HDA0000446822670000032.png"\; state="\{\{state\}\}">TV</figure-callout>}}_0+x_{\mathrm{line}})I2\end{array}\right),$

Wherein

with similar method, covariance matrix

pungent eigenvalue by following formula, calculated:

${\mathrm{\&lambda;}}_{\mathrm{1,2}}^0=(1+{\mathrm{<figure-callout\; id="0"\; label="TV"\; filenames="HDA0000446822670000032.png"\; state="\{\{state\}\}">TV</figure-callout>}}_0+x_{\mathrm{line}})\&PlusMinus;\sqrt{\mathrm{<figure-callout\; id="0"\; label="T\; z"\; filenames="HDA0000446822670000032.png"\; state="\{\{state\}\}">T</figure-callout>}}{z}_{}{}_0,$

And covariance matrix

pungent eigenvalue be with good conditionsi on the heterodyne measurement of Bob again:

${\mathrm{\&lambda;}}_3^0=(1+{\mathrm{<figure-callout\; id="0"\; label="TV"\; filenames="HDA0000446822670000032.png"\; state="\{\{state\}\}">TV</figure-callout>}}_0+x_{\mathrm{line}})(1+{\mathrm{TZ}}_c^2)-{\mathrm{<figure-callout\; id="0"\; label="Tz"\; filenames="HDA0000446822670000032.png"\; state="\{\{state\}\}">Tz</figure-callout>}}_0^2,$

In order to estimate to propose a plan, use the performance of emulation, we suppose to modulate variance V0=0.7, negotiation efficiency β=80%, participant's detection efficiency η=0.5.Fig. 4 is while using photon subtraction, the channel excess noise tolerance limit lower bound curve based on the scheme of tangling, i.e. and the curve of top in figure, original scheme efficiency of transmission function is the curve of below in figure.Expression is for different channel additional noise, and the scheme after upgrading under Gauss's operation allows than the original scheme of using Gauss's modulation the secure communication distance of more growing.

In order to create the relation subtracting as non-Gauss's operating value on basis in the performance of radioactive continuous variable quantum key distribution based on tangling with take subtraction, it is consistent with the efficiency of photon subtraction operation as the function of parameter μ that we describe key rate KG, when being illustrated in figure 5 efficiency of transmission and being T=0.268, u and progressive key rate under different channels noise

the functional relation of (solid line part), and given channel excess noise ε=0.01 o'clock, under different distance

(dotted portion).Be followed successively by from top to bottom, ε=0.01,0.02,0.03 and 0.05, apart from d=20,0,100 and 150 kms.Represent to exist a suitable μ to go the maximum best key rate of getting.In addition, the continuous variable quantum key distribution based on tangling of proposition is unsafe in the time of less for μ value, because nearly all light field is all tackled this fact of less μ, so this increases along with channel additional noise compared with little μ.In addition, modulation V0 is another key factor that affects key rate.We propose best key rate KN in a given distance, and Here it is modulates a function of variance in Fig. 6.Fig. 6 is in use photon difference and under high channel noise situations, based on progressive key rate in the scheme of tangling

with the functional relation of transmission range, be followed successively by from top to bottom ε=0.1,0.12,0.14,0.15.In fact, in proposing a plan along with modulation variance increase performance also gradually improve.Namely, photon subtraction can well improve the performance of the continuous variable quantum key distribution based on tangling, and this depends on the realization of non-Gauss's operation in itself.

Claims (3)

1. a differential non-Gaussian operation radioactive continuous variable quantum key distribution method, characterized in that it proceeds in accordance with the following steps: Step 1: Fred, a trusted third party, uses _a laser generator to prepare a dual-mode vacuum squeezed state, which is denoted as ρ _ab .

co-generation of a dual-mode vacuum-squeezed state

At this time, the output modes of the photons are recorded as a ₀ and b ₀ respectively, that is, the photons are transmitted on the modes a ₀ and b ₀ respectively after transformation; Step 2: The photon output mode a ₀ and b ₀ realizes the non-Gaussian operation through photon subtraction operation, respectively obtains the mode a ₁ and the mode b ₁ , and Fred respectively makes the mode a ₁ and the mode b ₁ pass through the noise channel and becomes the final Photon output modes a ₂ and b ₂ ; Step 3: When Alice and Bob receive the photon output modules a ₂ and b ₂ from the noise channel, they use heterodyne detectors to detect, conduct secret negotiation and error correction, and finally extract the key for data encryption. 2. according to the non-Gaussian operation radioactivity continuous variable quantum key distribution method of a kind of difference described in claim 1, it is characterized in that: in the first step, double-mode unitary transformation

co-generation of a dual-mode vacuum-squeezed state

The transformation process is as follows: ${<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; \&phi;</span>}<span\; class="notranslate">\; ></span>}_{{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}<span\; class="notranslate">\; =</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}}_{\mathrm{<span\; class="notranslate">\; ab</span>}}{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; ></span>}_{\mathrm{<span\; class="notranslate">\; a</span>}}{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; ></span>}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{<span\; class="notranslate">\; \&infin;</span>}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; ></span>}_{\mathrm{<span\; class="notranslate">\; a</span>}}{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; ></span>}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; ,</span>$ in

represents the beam splitter operation, represent the generation operator and annihilation operator acting on the modules a and b, respectively, and n represents the number of photons; If α=sinh(r) is adopted, r represents the compression coefficient, and the coefficient α _n can be expressed by the following formula: ${\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}^{\mathrm{<span\; class="notranslate">\; no</span>}}}{{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}^{\frac{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}}}<span\; class="notranslate">\; .</span>$ 3. according to the non-Gaussian operation radioactive continuous variable quantum key distribution method of a kind of difference described in claim 1, it is characterized in that: in the second step, the photon output modulus a ₀ and b ₀ obtain the described final by photon subtraction The process of modulo a ₂ and b ₂ is: Fred prepares vacuum state on mode c ₀ and s ₀ , input photons of mode a ₀ and mode s ₀ to beam splitter with transmission efficiency μ _a , input photons of mode b ₀ and mode c ₀ to transmission efficiency μ _b In the two beam splitters where the beam splitters have the same transmission efficiency μ _a = μ _b = μ, their outputs are denoted as modulus c ₁ and s ₁ , respectively on the modulus c ₁ and s ₁ action unitary operator The resulting state can be expressed as: $\begin{array}{c}{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; \&phi;</span>}<span\; class="notranslate">\; ></span>}_{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; c</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}}_{{\mathrm{<span\; class="notranslate">\; as</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}<span\; class="notranslate">\; \&CircleTimes;</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}}_{{\mathrm{<span\; class="notranslate">\; bc</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}<span\; class="notranslate">\; ]</span>{\mathrm{<span\; class="notranslate">\; \&phi;</span>}<span\; class="notranslate">\; ></span>}_{\mathrm{<span\; class="notranslate">\; ab</span>}}{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ></span>}_{{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ></span>}_{{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}\\ <span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}\underset{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&xi;</span>}}_{\mathrm{<span\; class="notranslate">\; nk</span>}}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; ></span>}_{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; ></span>}_{\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; ></span>}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; ></span>}_{\mathrm{<span\; class="notranslate">\; c</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}\end{array}<span\; class="notranslate">\; ,</span>$ Here, the modes s ₁ , c ₁ , a ₁ , and b ₁ respectively represent the four output modes of the beam splitter,

Represents the direct product operation of two operators,

and

Represents the binomial coefficient, with a photon number resolving detector to detect photons in mode s ₁ , c ₁ , when there are k photons on mode s ₁ and c ₁ photons are detected by photon number resolving detector, dual mode The squeezed state can be represented as a non-Gaussian state as follows: $\begin{array}{c}{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; \&phi;</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; ></span>}_{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{<span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; k</span>}{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; \&phi;</span>}<span\; class="notranslate">\; ></span>}_{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; c</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}\\ <span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\sqrt{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}}}\underset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; k</span>}}{\overset{<span\; class="notranslate">\; \&infin;</span>}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&xi;</span>}}_{\mathrm{<span\; class="notranslate">\; nk</span>}}{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; ></span>}_{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; ></span>}_{\mathrm{<span\; class="notranslate">\; b</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}\end{array}<span\; class="notranslate">\; ,</span>$ where p _k represents the normalization factor, which represents the probability of producing a non-Gaussian state, that is ${\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; k</span>}}{\overset{<span\; class="notranslate">\; \&infin;</span>}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; \&xi;</span>}}_{\mathrm{<span\; class="notranslate">\; nk</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>$ Finally, the generated photons are respectively transmitted to Alice and Bob through the noise channel, and the noisy photons obtained at the receiving end are denoted as modulus a ₂ and b ₂ respectively.

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