• Photonics Research
  • Vol. 9, Issue 6, 992 (2021)
Tianfeng Feng1、†, Changliang Ren2、5、†,*, Qin Feng1、†, Maolin Luo1, Xiaogang Qiang3, Jing-Ling Chen4、6、*, and Xiaoqi Zhou1、7、*
Author Affiliations
  • 1State Key Laboratory of Optoelectronic Materials and Technologies and School of Physics, Sun Yat-sen University, Guangzhou 510275, China
  • 2Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Key Laboratory for Matter Microstructure and Function of Hunan Province, Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China
  • 3National Innovation Institute of Defense Technology, AMS, Beijing 100071, China
  • 4Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071, China
  • 5e-mail: renchangliang@hunnu.edu.cn
  • 6e-mail: chenjl@nankai.edu.cn
  • 7e-mail: zhouxq8@mail.sysu.edu.cn
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    DOI: 10.1364/PRJ.411033 Cite this Article Set citation alerts
    Tianfeng Feng, Changliang Ren, Qin Feng, Maolin Luo, Xiaogang Qiang, Jing-Ling Chen, Xiaoqi Zhou. Steering paradox for Einstein–Podolsky–Rosen argument and its extended inequality[J]. Photonics Research, 2021, 9(6): 992 Copy Citation Text show less

    Abstract

    The Einstein–Podolsky–Rosen (EPR) paradox is one of the milestones in quantum foundations, arising from the lack of a local realistic description of quantum mechanics. The EPR paradox has stimulated an important concept of “quantum nonlocality,” which manifests itself in three types: quantum entanglement, quantum steering, and Bell’s nonlocality. Although Bell’s nonlocality is more often used to show “quantum nonlocality,” the original EPR paradox is essentially a steering paradox. In this work, we formulate the original EPR steering paradox into a contradiction equality, thus making it amenable to experimental verification. We perform an experimental test of the steering paradox in a two-qubit scenario. Furthermore, by starting from the steering paradox, we generate a generalized linear steering inequality and transform this inequality into a mathematically equivalent form, which is friendlier for experimental implementation, i.e., one may measure the observables only in the x, y, or z axis of the Bloch sphere, rather than other arbitrary directions. We also perform experiments to demonstrate this scheme. Within the experimental errors, the experimental results coincide with theoretical predictions. Our results deepen the understanding of quantum foundations and provide an efficient way to detect the steerability of quantum states.
    |Ψ(α,φ)=cosα|00+eiφsinα|11,

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    |Ψ(α,φ)=|+n^|χ+n^+|n^|χn^,

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    ρ˜0z^=cos2α|00|=1ρ1,(3a)

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    ρ˜1z^=sin2α|11|=2ρ2,(3b)

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    ρ˜0x^=(1/2)|χ+χ+|=3ρ3,(3c)

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    ρ˜1x^=(1/2)|χχ|=4ρ4,(3d)

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    ρ˜0n^j=ξ(0|n^j,ξ)ξρξ,(4a)

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    ρ˜1n^j=ξ(1|n^j,ξ)ξρξ,j=1,2,,k.(4b)

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    Sk(θ,ϕ)=j=1k[a=01P(Anj=a)ρan^j(θ,ϕ)]CLHS,

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    S3=P(Ax=0)|χ+χ+|+P(Ax=1)|χχ|+P(Ay=0)|χ+χ+|+P(Ay=1)|χχ|+P(Az=0)|00|+P(Az=1)|11|CLHS,

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    S3(θ,ϕ)=sin2θcosϕAxσxsin2θcosϕAyσy+sin2θsinϕAxσy+sin2θsinϕAyσx+Azσz+2cos2θσzCLHS,

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    S3(π/4,0)=AxσxAyσy+Azσz3.

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    ρ1=V|Ψ(α)Ψ(α)|+1V411,(9a)

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    ρ2=V|Ψ(α)Ψ(α)|+(1V)|Φ(α)Φ(α)|,(9b)

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    P^0n^j=1+n^j·σ2=|+n^+n^|,(A1a)

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    P^1n^j=1n^j·σ2=|n^n^|.(A1b)

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    ρ˜0n^j=trA[(P^0n^j1)|ΨΨ|]=|χ+n^jχ+n^j|,(A2a)

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    ρ˜1n^j=trA[(P^1n^j1)|ΨΨ|]=|χn^jχn^j|,(A2b)

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    ρ0n^j=ρ˜0n^jtr(ρ˜0n^j)=|χ+jχ+j|,(A3a)

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    ρ1n^j=ρ˜1n^jtr(ρ˜1n^j)=|χjχj|,(A3b)

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    |χ+j=|χ+n^jtr[|χ+n^jχ+n^j|],(A4)

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    |χj=|χn^jtr[|χn^jχn^j|](A5)

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    tr[(P^0n^j|χ+jχ+j|)|ΨΨ|]+tr[(P^1n^j|χjχj|)|ΨΨ|]=tr[|χ+n^jχ+n^j|]+tr[|χn^jχn^j|]1.(A6)

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    Sk=j=1k[a=01P(Anj=a)ρan^j]CLHS,(A7)

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    ρ0n^j=|χ+jχ+j|,ρ1n^j=|χjχj|(A8)

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    SkLSI=j=1kAjm^j·σCLHSLSI,(A9)

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    ρ0n^j=|χ+jχ+j|=12(1+m^+j·σ),ρ1n^j=|χjχj|=12(1+m^j·σ),(A10)

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    P(Aj=0)=1+Aj2,P(Aj=1)=1Aj2.(A11)

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    Sk=j=1k[1+Aj212(1+m^+j·σ)+1Aj212(1+m^j·σ)]CLHS.(A12)

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    n^j=(sinτcosγ,sinτsinγ,cosτ),(A13)

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    |+n^j=(cosτ2sinτ2eiγ),|n^j=(sinτ2cosτ2eiγ),(A14)

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    |χ+n^j=+n^j|Ψ(θ,ϕ)=cosτ2cosθ|0+ei(ϕγ)sinτ2sinθ|1,(A15)

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    |χn^j=n^j|Ψ(θ,ϕ)=sinτ2cosθ|0ei(ϕγ)cosτ2sinθ|1,(A16)

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    χn^j|χ+n^j=cosτ2sinτ2(cos2θsin2θ).(A17)

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    m^+j=m^jm^j.(A18)

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    SkLSI=j=1kAjm^j·σ2CLHSk,(A19)

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    m^+j=(m+xj,m+yj,m+zj),m^j=(mxj,myj,mzj),(A20)

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    Sk=j=1k[1+Aj212(1+m^+j·σ)+1Aj212(1+m^j·σ)]=j=1k[12+1+Aj4(m^+xjσx+m^+yjσy+m^+zjσz)+1Aj4(m^xjσx+m^yjσy+m^zjσz)]CLHS.(A21)

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    S3=P(Ax=0)|χ+χ+|+P(Ax=1)|χχ|+P(Ay=0)|χ+χ+|+P(Ay=1)|χχ|+P(Az=0)|00|+P(Az=1)|11|CLHS,(B1)

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    |χ±=cosθ|0±eiϕsinθ|1,|χ±=cosθ|0ieiϕsinθ|1.(B2)

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    |χ+χ+|=12(1+m^+·σ),|χχ|=12(1+m^·σ),|χ+χ+|=12(1+m^+·σ),|χχ|=12(1+m^·σ),|00|=12(1+σz),|11|=12(1σz),(B3)

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    m^+=(sin2θcosϕ,sin2θsinϕ,cos2θ),m^=(sin2θcosϕ,sin2θsinϕ,cos2θ),m^+=(sin2θsinϕ,sin2θcosϕ,cos2θ),m^=(sin2θsinϕ,sin2θcosϕ,cos2θ).(B4)

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    |χ+χ+|+|χ+χ+|+|00|,(B5)

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    3+44cos2θ+cos4θ2,(B6)

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    344cos2θ+cos4θ2.(B7)

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    |χ+χ+|+|χ+χ+|+|11|,(B8)

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    3+4+4cos2θ+cos4θ2,(B9)

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    34+4cos2θ+cos4θ2.(B10)

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    CLHS=Max{3+C+2,3+C2},(B11)

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    C±=4±4cos2θ+cos4θ.(B12)

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    P(Ai=0)=1+Ai2,P(Ai=1)=1Ai2,(B13)

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    S3=sin2θcosϕAxσxsin2θcosϕAyσy+sin2θsinϕAxσy+sin2θsinϕAyσx+Azσz+2cos2θσzCLHS,(B14)

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    ρ1=ρAB(α,V)=V|Ψ(α)Ψ(α)|+1V411,(B15)

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    ρ2=V|Ψ(α)Ψ(α)|+(1V)|Φ(α)Φ(α)|,(B16)

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    |Ψ=sinβ2|HA|HBucosβ2|HA|VBu+12|VA|VBl,(C1)

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    (sinβ001),(C2)

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    |Ψ=sinβ2|HA|HB+12|VA|VB.(C3)

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    |Ψ=sinβ(sinβ)2+1|HA|HB+1(sinβ)2+1|VA|VB.(C4)

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    |Ψ=cosα|HA|HB+sinα|VA|VB.(C5)

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    ρ1=V|ΨΨ|+(1V)114,(C6)

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    ρ2=V|ΨΨ|+(1V)|ΦΦ|,(C7)

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    Tianfeng Feng, Changliang Ren, Qin Feng, Maolin Luo, Xiaogang Qiang, Jing-Ling Chen, Xiaoqi Zhou. Steering paradox for Einstein–Podolsky–Rosen argument and its extended inequality[J]. Photonics Research, 2021, 9(6): 992
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