Weak Value and Correlation Function

0.06 MB·English

by  Takahiro Sagawa

Weak Value and Correlation Function
Takahiro Sagawa
Department of Physics, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-8654, Japan
(Dated: January 27, 2009)
We show that there exists, in quantum theory, a close relationship between the weak value and
thecorrelation function, which sheds new lights on theconcept of theweak value.
9
PACSnumbers: 03.65.Ta,03.67.-a
0
0
2
In this report, we prove an equality which connects tain by performing the successive projection measure-
n two fundamental concepts in quantum theory: the weak ments of Aˆ at t1 and Bˆ at t2. The quasi-probability
Ja value and the correlation function. The weak value has Prψ(b,a) satisfies that PaPrψ(b,a) = hψ|PˆbB(t2)|ψi ≡
7 breeleanteadttooptihceoffoaucntdivaetiroenseoafrcqhueasn[t1u,m2,m3e,c4h,a5n,ic6s],[7a]n.dOins Prψ(bP),r P(b,baP)r=ψ(b1,.aW) e=nohtψe|tPhˆaaAt(,ti1n)|tψhies≡pecPiarψlc(aas)e, tahnadt
2 Pa,b ψ
the other hand, the (symmetric) correlation function is Aˆ(t1) and Bˆ(t2) are commuting, the quasi-probability
] defined as Rehψ|Bˆ(t2)Aˆ(t1)|ψi, where “Re” means “the Pr (b,a) reduces to the true probability.
h realpart”, |ψi is a state vector, and Aˆ(t1) and Bˆ(t2) are Wψ e then calculate the quasi-probability under the
p
- observables at time t1 and t2 in the Heisenberg picture. condition of b as Prψ(a|b) ≡ Prψ(b,a)/Prψ(b) =
nt Ttehriezecotrhreeldaytinoanmfiucnscotfioqnuaisntaumusesyfuslteqmusa,natintdyhtoascphlaaryaecd- Rehψ|Uˆ1†Uˆ2†|bihb|Uˆ2PˆaAUˆ1|ψi/hψ|Uˆ1†Uˆ2†|bihb|Uˆ2Uˆ1|ψi, and
a obtain
u an important role in quantum statistical mechanics such
[q areslatthieonlinfueanrctrieosnpoisnsreeltahteeodryto[8t]h.eFsoursceexpatmibpilleit,ythbeyctohre- Prψ(a|b)=bhPaAiw. (3)

fluctuation-dissipation theorem.
1 Averaging the eigenvalue a over the conditional quasi-
v To introduce the weak value, we consider two observ- probability, we have
2 ables Aˆ and Bˆ of a quantum system. We denote the
1 spectrumdecompositionsofthemasAˆ= a|aiha|and
2 Bˆ = b|bihb|, where a’s (b’s) are the eiPgenavalues of Aˆ bhAiw =XaPrψ(a|b), (4)
4 Pb a
. (Bˆ), and |ai’s (|bi’s) are the corresponding eigenvectors.
1 Let PˆA ≡ |aiha| and PˆB ≡ |bihb| be the projection op- which is the main result of this report. The left-hand
0 a b
side ofEq.(4) is the weakvalue, andthe right-handside
erators. We consider the unitary evolutionof the system
9
is related to the correlationfunction via Eq. (2).
0 from time 0 to t2, and suppose that the projection mea-
: surement of Bˆ is performed at time t2, and the outcome Inconclusion,we havederivedanequalitywhichgives
Xiv bbeisaonbitnatienremd.edLieatte|ψtiimbee,tahnedinUˆi1ti(aUlˆ2st)abtee,tth1e(u0n<itat1ry<evt2o)- uthseaavneerwagienotefrApˆruetnadteiornthoefctohnediwtieoankovfabluoevebrhAthiwe:quiatsiis-
r lution from 0 to t1 (t1 to t2). Then the definition of the probability which is defined via the correlationfunction.
a (real) weak value of Aˆ at time t1 with the post-selection While we have only considered the real weak value and
the symmetric correlation function, we can straightfor-
|bi is given by
wardly generalize our result to the complex weak value
hb|Uˆ2AˆUˆ1|ψi and the complex correlation function by removing the
bhAiw ≡Re hb|Uˆ2Uˆ1|ψi . (1) notation “Re”.
To relate the weak value to the correlation function,
Acknowledgments
we first define a quasi-probability distribution based on
the correlation function:
This work was supported by a Grant-in-Aid for Scien-
Prψ(b,a)≡Rehψ|PˆbB(t2)PˆaA(t1)|ψi, (2) tific Research (Grant No. 17071005), and by a Global
COE program “Physical Science Frontier” of MEXT,
where PˆbB(t2) ≡ Uˆ1†Uˆ2†PˆbBUˆ2Uˆ1 and PˆaA(t1) ≡ Uˆ1†PˆaAUˆ1 Japan. TS acknowledges JSPS Research Fellowships for
are described in the Heisenberg picture. We stress Young Scientists (Grant No. 208038). TS would like to
that this quasi-probability distribution does not corre- thank Prof. MasahitoUeda and Mr. Yutaka Shikano for
spond to the probability distribution which we can ob- valuable discussions.
2
[1] Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. [5] Michael J. W. Hall, Phys.Rev.A 69, 052113 (2004).
Lett. 60, 1351 (1988). [6] G.J.Pryde,J.L.O’Brien,A.G.White,T.C.Ralph,and
[2] N.W.M.Ritchie,J.G.Story,andRandallG.Hulet,Phys. H. M. Wiseman, Phys.Rev.Lett. 94, 220405 (2005).
Rev.Lett. 66, 1107 (1991). [7] L. Vaidman, Found.Phys.26, 895 (1996).
[3] H. M. Wiseman, Phys.Rev.A 65, 032111 (2002). [8] RKubo,MToda,andNHashitsume“Statistical Mechan-
[4] D. R. Solli, C. F. McCormick, and R. Y. Chiao, S. ics II, Nonequilibrium Statistical Mechanics” (Springer,
Popescu, J. M. Hickmann, Phys. Rev. Lett. 92, 043601 Berlin, 1985).
(2004).