• Photonics Research
  • Vol. 11, Issue 9, 1484 (2023)
Nicolò Leone1、*, Stefano Azzini1, Sonia Mazzucchi2, Valter Moretti2, Matteo Sanna1, Massimo Borghi3, Gioele Piccoli4, Martino Bernard4, Mher Ghulinyan4, and Lorenzo Pavesi1
Author Affiliations
  • 1Department of Physics, University of Trento, 38122 Trento, Italy
  • 2Department of Mathematics and TIFPA, University of Trento, 38122 Trento, Italy
  • 3Department of Physics, University of Pavia, 27100 Pavia, Italy
  • 4Centre for Sensors and Devices, Fondazione Bruno Kessler, 38123 Povo, Italy
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    DOI: 10.1364/PRJ.488875 Cite this Article Set citation alerts
    Nicolò Leone, Stefano Azzini, Sonia Mazzucchi, Valter Moretti, Matteo Sanna, Massimo Borghi, Gioele Piccoli, Martino Bernard, Mher Ghulinyan, Lorenzo Pavesi. Generation of quantum-certified random numbers using on-chip path-entangled single photons from an LED[J]. Photonics Research, 2023, 11(9): 1484 Copy Citation Text show less

    Abstract

    Single-photon entanglement is a peculiar type of entanglement in which two or more degrees of freedom of a single photon are correlated quantum-mechanically. Here, we demonstrate a photonic integrated chip able to generate and manipulate single-photon path-entangled states, using a commercial red LED as light source. A Bell test, in the Clauser, Horne, Shimony, and Holt (CHSH) form, is performed to confirm the presence of entanglement, resulting in a maximum value of the CHSH correlation parameter equal to 2.605±0.004. This allows us to use it as an integrated semi-device independent quantum random number generator able to produce certified random numbers. The certification scheme is based on a Bell’s inequality violation and on a partial characterization of the experimental setup, without the need of introducing any further assumptions either on the input state or on the particular form of the measurement observables. In the end a min-entropy of 33% is demonstrated.
    |ϕ±=12(|UF±|DN),

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    |ψ±=12(|UN±|DF).

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    |ψ=12(|UF+ieiξ|DN),

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    E(x,y)=P(a=b|x,y)P(ab|x,y),

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    χ=(1)xyE(x,y).

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    A(θ)=U1(θ)σzU1(θ),

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    B(ϕ)=U2(ϕ)σzU2(ϕ),

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    P(a=b|x,y)=P(|UF|ϕ,θ)+P(|DN|ϕ,θ),

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    P(ab|x,y)=P(|UN|ϕ,θ)+P(|DF|ϕ,θ),

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    P(|μν|ϕ,θ)=Tr[U(ϕ,θ)ρU(ϕ,θ)PμPν],

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    E(ϕ,θ)=P(|UF|ϕ,θ)+P(|DN|ϕ,θ)P(|UN|ϕ,θ)P(|DF|ϕ,θ).

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    UMMI=12(1ii1),UPS(ζ1,ζ2)=(e2iζ100e2iζ2).

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    UMZI(ζ1,ζ2)=iei(ζ1+ζ2)(sin(ζ1ζ2)cos(ζ1ζ2)cos(ζ1ζ2)sin(ζ1ζ2))=ieiZ(sin(ζ)cos(ζ)cos(ζ)sin(ζ)).

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    E(ϕ,θ)=cos(2(ϕθ)),

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    χ(ϕ,ϕ,θ,θ)=cos(2(ϕθ))cos(2(ϕθ))+cos(2(ϕθ))+cos(2(ϕθ)).

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    χ(α)=3cos(α)cos(3α).

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    UMMI(λ)=(t(λ)ir(λ)ir(λ)t(λ)),

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    UMZI(λ,ζ1,ζ2)=(t(λ)2e2iζ1(λ)r(λ)2e2iζ2(λ)ir(λ)t(λ)(e2iζ1(λ)+e2iζ2(λ))ir(λ)t(λ)(e2iζ1(λ)+e2iζ2(λ))t(λ)2e2iζ2(λ)r(λ)2e2iζ1(λ)).

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    U(ϕ1,ϕ2,ϕ3,ϕ4)=P1UMZI(ϕ1,ϕ2)+P2UMZI(ϕ3,ϕ4);U(θ1,θ2,θ3,θ4)=UMZI(θ1,θ2)P1+UMZI(θ3,θ4)P2;P1=(1000),P2=(0001).

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    U(ϕ1,ϕ2,ϕ3,ϕ4)=IUMZI(ϕ1,ϕ2),

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    U(θ1,θ2,θ3,θ4)=UMZI(θ1,θ2)I,

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    ρ=ρωdμ(ω),

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    ρωU(ϕ,θ)ωρωU(ϕ,θ)ω,

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    ρU(ϕ,θ)ωρωU(ϕ,θ)ωdμ(ω).

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    P(a,b|ϕ,θ)=P(a,b|ϕ,θ)ωdμ(ω),

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    P(a,b|ϕ,θ)ω=Tr[U(ϕ,θ)ωρωU(ϕ,θ)ωPaPb].

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    P(a,b|ϕ,θ)ω=Tr[ρωPaθ,ωPbϕ,ω]=Tr[U(ϕ,θ)ωρωU(ϕ,θ)ωPaPb].

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    χ=χωdμ(ω),

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    χω=Eω(ϕ,θ)Eω(ϕ,θ)+Eω(ϕ,θ)+Eω(ϕ,θ),

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    Eω(ϕ,θ)=P(a=b|ϕ,θ)ωP(ab|ϕ,θ)ω.

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    P(a,b|ϕ,θ)=Tr[U(ϕ,θ)LρLU(ϕ,θ)PaPb]Tr[U(ϕ,θ)LρLU(ϕ,θ)],

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    P(a,b|ϕ,θ)=Tr[U(ϕ,θ)ρU(ϕ,θ)PxPy],

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    Ureal(ϕ1,ϕ2,δϕ1,δϕ2,δϕ3,δϕ4)=P1UMZI(ϕ1+δϕ1,ϕ2+δϕ2)+P2UMZI(ϕ1+δϕ3,ϕ2+δϕ4);Ureal(θ1,θ2,δθ1,δθ2,δθ3,δθ4)=UMZI(θ1+δθ1,θ2+δθ2)P1+UMZI(θ1+δθ3,θ2+δθ4)P2.

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    UPS(ζ1,ζ2)real=(e2i(ζ1+δζ1)00e2i(ζ2+δζ2)).

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    Ureal(ϕ1,ϕ2,δϕ1,δϕ2,δϕ3,δϕ4)=P1UMMIUPS(ϕ1,ϕ2,δϕ1,δϕ2)realUMMI+P2UMMIUPS(ϕ1,ϕ2,δϕ3,δϕ4)realUMMI.

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    Ureal(ϕ1,ϕ2,δϕ1,δϕ2,δϕ3,δϕ4)=(IUMMI)(IUPSideal(ϕ1,ϕ2))D(δϕ1,δϕ2,δϕ3,δϕ4)(IUMMI),

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    D(δϕ1,δϕ2,δϕ3,δϕ4)=(e2iδϕ10000e2iδϕ20000e2iδϕ30000e2iδϕ4).

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    Uideal=I(UMMIUPS(ϕ1,ϕ2)VidealUMMI)

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    Videal=eiφeiϑn^·σ=eiφ(cosϑI+isinϑn^·σ),

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    n^=(0,0,1),φ=δϕ1+δϕ3+δϕ2+δϕ42,ϑ=δϕ1+δϕ3δϕ2δϕ42,

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    eχ=max{ϕ,ϕ,θ,θ,ρ}|χideal(ϕ,ϕ,θ,θ)χreal(ϕ,ϕ,θ,θ)|,

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    Iout1=acos(bW+d)2+c,

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    Iout2=asin(bW+d)2+c,

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    P(|μν|ϕ,θ)=N|μνN|μν,

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    E(ϕ,θ)=η(5+(48)6+(24)(5+26)cos(2ϕ)+(24)(5+26)cos(2θ)+48(30+(13)6)cos(2(ϕ+θ))+288(5+26)cos(2(ϕθ)),

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    Pguess(a,b|ϕi,θj)12+122(|χreal|eχ)2/4+eP,

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    Hmin=log2(Pguess(a,b|ϕx,θy)),

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    Tr[(UrealUideal)(UrealUideal)](B1)

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    =8Tr[D(δϕ1,δϕ2,δϕ3,δϕ4)(IVideal)+h.c.].(B2)

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    Tr[VidealU1+U1(Videal)]+Tr[VidealU2+U2(Videal)],(B3)

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    U1=(e2iδϕ100e2iδϕ2),(B4)

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    U2=(e2iδϕ300e2iδϕ4).(B5)

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    U(α,β)=(e2iα00e2iβ)=ei(α+β)ei(αβ)σz(B6)

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    Tr[U(α,β)V+VU(α,β)]=4cos(φαβ)(cosϑcos(αβ)+nzsinϑsin(αβ)).(B7)

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    Tr[VidealU1+U1(Videal)]+Tr[VidealU2+U2(Videal)]=4cos(φδϕ1δϕ2)(cosϑcos(δϕ1δϕ2)+nzsinϑsin(δϕ1δϕ2))+4cos(φδϕ3δϕ4)(cosϑcos(δϕ3δϕ4)+nzsinϑsin(δϕ3δϕ4)),(B8)

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    minφ,ϑ,n^UidealUrealHS=(88cos(δϕ1δϕ32+δϕ2δϕ42)×cos(δϕ1δϕ32δϕ2δϕ42))1/2,(B9)

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    Uideal=I(UMMIUPh(ϕ1,ϕ2)VidealUMMI)(B10)

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    F(φ,ϑ,nz)4cos(φδϕ1δϕ2)(cosϑcos(δϕ1δϕ2)+nzsinϑsin(δϕ1δϕ2))+4cos(φδϕ3δϕ4)(cosϑcos(δϕ3δϕ4)+nzsinϑsin(δϕ3δϕ4)),(B11)

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    F(φ,ϑ,nz)=f(φ)cosϑ+nzg(φ)sinϑ,(B12)

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    f(φ)=4(cos(φδϕ1δϕ2)cos(δϕ1δϕ2)+cos(φδϕ3δϕ4)cos(δϕ3δϕ4)),(B13)

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    g(φ)=4(cos(φδϕ1δϕ2)sin(δϕ1δϕ2)+cos(φδϕ3δϕ4)sin(δϕ3δϕ4)).(B14)

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    Fnz(φ,ϑ,nz)=g(φ)sinϑ.(B15)

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    G±(φ,ϑ)=f(φ)cosϑ±g(φ)sinϑ,(B20)

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    G+(φ,ϑ)=4(cos(φδϕ1δϕ2)cos(δϕ1δϕ2ϑ)+cos(φδϕ3δϕ4)cos(δϕ3δϕ4ϑ)).(B21)

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    {G+φ=0G+ϑ=0.(B22)

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    {sin(φδϕ1δϕ2)cos(δϕ1δϕ2ϑ)sin(φδϕ3δϕ4)cos(δϕ3δϕ4ϑ)=0,cos(φδϕ1δϕ2)sin(δϕ1δϕ2ϑ)+cos(φδϕ3δϕ4)sin(δϕ3δϕ4ϑ)=0.(B23)

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    {sin(φ2δϕ1+ϑ)sin(φ2δϕ3+ϑ)=0,sin(φ2δϕ2ϑ)+sin(φ2δϕ4ϑ)=0,(B24)

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    {φ+ϑ=δϕ1+δϕ3+kπ,φϑ=δϕ2+δϕ4+hπ,(B25)

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    {φ=δϕ1+δϕ3+δϕ2+δϕ42+k+h2π,ϑ=δϕ1+δϕ3δϕ2δϕ42+kh2π.(B26)

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    G+(δϕ1+δϕ3+δϕ2+δϕ42,δϕ1+δϕ3δϕ2δϕ42)=8cos(δϕ1δϕ32+δϕ2δϕ42)×cos(δϕ1δϕ32δϕ2δϕ42).(B27)

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    |δζ1|ϵ,|δζ2|ϵ,|δζ3|ϵ,|δζ4|ϵ,(B28)

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    maxδζ1,δζ2,δζ3,δζ4minφ,ϑ,n^UidealUrealHS22(1cos(2ϵ))1/2.(B29)

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    maxδζ1,δζ2,δζ3,δζ4minφ,ϑ,n^UidealUrealHS4ϵ.(B30)

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    Ureal(ϕ1,ϕ2,δϕ1,δϕ2,δϕ3,δϕ4)=Uϕreal(B31)

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    Ureal(θ1,θ2,δθ1,δθ2,δθ3,δθ4)=Uθreal.(B32)

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    UϕrealUθrealUϕidealUθidealHS(UϕrealUϕideal)(UφrealUφideal)HS+Uθideal(UϕrealUϕideal)HS+(UφrealUφideal)UϕidealHS(UϕrealUϕideal)(UφrealUφideal)HS+Uθideal(UϕrealUϕideal)HS+(UφrealUφideal)HSUϕideal(UϕrealUϕideal)HS(UφrealUφideal)HS+(UϕrealUϕideal)HS+(UφrealUφideal)HS.(B33)

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    maxδϕi,δθjminφ,ϑ,n^UidealUrealHS82ϵ.(B34)

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    G(P)sup{ρ˜,A,B}R(P)G(ρ˜,A,B),(C1)

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    P(x,y)=Tr[ρ˜PxAPyB],x,y=±1,(C2)

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    G(ρ˜,A,B)=supG(ψλ,A,B)dν(λ),(C3)

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    G(ψλ,A,B)max(x,y)Tr[|ψλψλ|PxAPyB].(C4)

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    χ=A1B1ψ+A1B2ψ+A2B1ψA2B2ψ,(C5)

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    χλ=A1λB1λψλ+A1λB2λψλ+A2λB1λψλA2λB2λψλ.(C6)

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    max(x,y)Tr[|ψλψλ|PxA(λ)iPyB(λ)j]dν(λ)(C7)

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    max(x,y)Tr[|ψλψλ|PxA(λ)iPyB(λ)j]dν(λ)(C8)

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    f(χλ)dν(λ)f(χλdν(λ))=f(χ).(C9)

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    G(Pi,j)f(χ),  i,j=1,2.(C10)

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    P(x,y)=Tr[|ψλψλ|PxA(λ)PyB(λ)].(C11)

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    Nicolò Leone, Stefano Azzini, Sonia Mazzucchi, Valter Moretti, Matteo Sanna, Massimo Borghi, Gioele Piccoli, Martino Bernard, Mher Ghulinyan, Lorenzo Pavesi. Generation of quantum-certified random numbers using on-chip path-entangled single photons from an LED[J]. Photonics Research, 2023, 11(9): 1484
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