Open access peer-reviewed chapter

Uncertainty Relations

By Kenjiro Yanagi

Submitted: November 8th 2019Reviewed: March 16th 2020Published: April 28th 2020

DOI: 10.5772/intechopen.92137

Downloaded: 52

Abstract

Uncertainty relations are inequalities representing the impossibility of simultaneous measurement in quantum mechanics. The most well-known uncertainty relations were presented by Heisenberg and Schrödinger. In this chapter, we generalize and extend them to produce several types of uncertainty relations.

Keywords

  • trace inequality
  • variance
  • covariance
  • skew information
  • metric adjusted skew information
  • noncommutativity
  • observable
  • operator inequality

1. Introduction

Let MnC(resp. Mn,saC) be the set of all n×ncomplex matrices (resp. all n×nself-adjoint matrices), endowed with the Hilbert-Schmidt scalar product AB=TrAB. Let Mn,+Cbe the set of strictly positive elements of MnCand Mn,+,1CMn,+Cbe the set of strictly positive density matrices, that is Mn,+,1C=ρMnCTrρ=1ρ>0. If not otherwise specified, hereafter, we address the case of faithful states, that is ρ>0. It is known that the expectation of an observable AMn,saCin state ρMn,+,1Cis defined by

EρA=TrρA,

and the variance of an observable AMn,saCin state ρMn,+,1Cis defined by

VρA=TrρAEρAI2=TrρA2EρA2=TrρA02,

where A0=AEρAI.

In Section 2, we introduce the Heisenberg and Schrödinger uncertainty relations. In Section 3, we present uncertainty relations with respect to the Wigner-Yanase and Wigner-Yanase-Dyson skew information. To represent the degree of noncommutativity between ρMn,+,1Cand AMn,saC, the Wigner-Yanase skew information IρAis defined by

IρA=12Triρ1/2A2=TrρA2Trρ1/2Aρ1/2A,

where XY=XYYX. Furthermore, the Wigner-Yanase-Dyson skew information Iρ,αAis defined by

Iρ,αA=12TriραAiρ1αA=TrρA2TrραAρ1αA,α01.

The convexity of Iρ,αAwith respect to ρwas famously demonstrated by Lieb [1], and the relationship between the Wigner-Yanase skew information and the uncertainty relation was originally developed by Luo and Zhang [2]. Subsequently, the relationship between the Wigner-Yanase-Dyson skew information and the uncertainty relation was provided by Kosaki [3] and Yanagi-Furuichi-Kuriyama [4]. In Section 4, we discuss the metric adjusted skew information defined by Hansen [5], which is an extension of the Wigner-Yanase-Dyson skew information. The relationship between metric adjusted skew information and the uncertainty relation was provided by Yanagi [6] and generalized by Yanagi-Furuichi-Kuriyama [7] for generalized metric adjusted skew information and the generalized metric adjusted correlation measure. In Sections 5 and 6, we provide non-Hermitian extensions of Heisenberg-type and Schrödinger-type uncertainty relations related to generalized quasi-metric adjusted skew information and the generalized quasi-metric adjusted correlation measure. As a result, we obtain results for non-Hermitian uncertainty relations provided by Dou and Du as corollaries of our results. Finally, in Section 7, we present the sum types of uncertainty relations.

2. Heisenberg and Schrödinger uncertainty relations

Theorem 1.1 (Heisenberg uncertainty relation). For A,BMn,saC, ρMn,+,1C,

VρAVρB14Tr[ρ[AB]]2,E1

where AB=ABBAis the commutator.

Theorem 1.2 (Schrödinger uncertainty relation). For A,BMn,saC, ρMn,+,1C,

VρAVρBReTrρA0B0214Tr[ρ[AB]]2.

Proof of Theorem 1.2. By the Schwarz inequality

TrρA0B02=Trρ1/2B0ρ1/2A02Trρ1/2B0ρ1/2B0Trρ1/2A0ρ1/2A0=TrρA02TrρB02=VρAVρB.

Since

TrρA0B0=TrρA0B0TrρB0A0=TrρA0B0TrA0B0ρ¯=TrρA0B0TrρA0B0¯=2iImTrρA0B0,

we have

TrρA0B02=ReTrρA0B02+ImTrρA0B02=ReTrρA0B02+14TrρA0B02.

Since TrρA0B0=TrρAB, we obtain

VρAVρBReTrρA0B0214Tr[ρ[AB]]2.

3. Uncertainty relation for Wigner-Yanase-Dyson skew information

3.1 Wigner-Yanase skew information

To represent the degree of noncommutativity between ρMn,+,1Cand AMn,saC, the Wigner-Yanase skew information IρAand related quantity JρAare defined as

IρA=12Triρ1/2A02=TrρA02Trρ1/2A0ρ1/2A0.
JρA=12TrρA0B02=TrρA02+Trρ1/2A0ρ1/2A0,

where AB=AB+BA. The quantity UρArepresenting a quantum uncertainty excluding the classical mixture is defined as

UρA=IρAJρA=VρA2VρAIρA2.

We note the following relation:

0IρAUρAVρA.E2

Luo [8] then derived the uncertainty relation of UρA.

Theorem 1.3. For A,BMn,saC, ρMn,+,1C,

UρAUρB14Tr[ρ[AB]]2.E3

Inequality (3) is a refinement of (1) in terms of (2).

3.2 Wigner-Yanase-Dyson skew information

Here, we introduce a one-parameter inequality extended from (3). For 0α1,A,BMn,saCand ρMn,+,1C, we define the Wigner-Yanase-Dyson skew information as follows:

Iρ,αA=12TriραA0iρ1αA0=TrρA02TrραA0ρ1αA0.

We also define

Jρ,αA=12TrραA0ρ1αA0=TrρA2+TrραA0ρ1αA0.

We note that

12TriραA0iρ1αA0]=12ReiραAiρ1αA;

however, we have

12TrραA0ρ1αA012TrραAρ1αA.

We then have the following inequalities:

Iρ,αAIρAJρAJρ,αA,E4

because Trρ1/2Aρ1/2ATrραAρ1αA. We define

Uρ,αA=Iρ,αAJρ,αA=VρA2(VρAIρ,αA.E5

From (2), (4), and (5), we have

0Iρ,αAIρAUρA

and

0Iρ,αAUρ,αAUρA.

We provide the following uncertainty relation with respect to Uρ,αAas a direct generalization of (3).

Theorem 1.4 ([9]). For A,BMn,saC, ρMn,+,1C,

Uρ,αAUρ,αBα1αTr[ρ[AB]]2.E6

Proof of Theorem 1.4. By spectral decomposition, there exists an orthonormal basis ϕ1ϕ2ϕnconsisting of eigenvectors of ρ. Let λ1,λ2,,λnbe the corresponding eigenvalues, where i=1nλi=1and λi0. Thus ρhas a spectral representation ρ=i=1nλiϕiϕi. We can obtain the following representations of Iρ,αAand Jρ,αA:

Iρ,αA=i<jλi+λjλiαλj1αλi1αλjαϕiA0ϕj2.
Jρ,αAi<jλi+λj+λiαλj1α+λi1αλjαϕiA0ϕj2.

Since 12α2t12tαt1α20for any t>0and 0α1, we define t=λiλjand have

12α2λiλj12λiλjαλiλj1α20.

Then,

λi+λj2λiαλj1α+λi1αλjα24α1αλiλj2.E7

Since

TrρAB=TrρA0B0=2iImTrρA0B0=2iImi<jλiλjϕiA0ϕjϕjB0ϕi=2ii<jλiλjImϕiA0ϕjϕjB0ϕi,
TrρAB=2i<jλiλjImϕiA0ϕjϕjB0ϕi2i<jλiλjImϕiA0ϕjϕjB0ϕi.

We then have

TrρAB24i<jλiλjImϕiA0ϕjϕjB0ϕi2.

By (7) and the Schwarz inequality,

α1αTr[ρ[AB]]24α1αi<jλiλjImϕiA0ϕjϕjB0ϕi2=i<j2α1αλiλjImϕiA0ϕjϕjB0ϕi2i<j2α1αλiλjϕiA0ϕjϕjB0ϕi2i<jλi+λj2λiαλj1α+λi1αλjα21/2ϕiA0ϕjϕjB0ϕi2i<jλi+λjλiαλj1αλi1αλjαϕiA0ϕj2×i<jλi+λj+λiαλj1α+λi1αλjαϕiB0ϕj2.

Then, we have

Iρ,αAJρ,αBα1αTr[ρ[AB]]2.

We also have

Iρ,αBJρ,αAα1αTr[ρ[AB]]2.

Thus, we have the final result, (6).

When α=12, we obtain the result in Theorem 1.3.

4. Metric adjusted skew information and metric adjusted correlation measure

4.1 Operator monotone function

A function f:0+Ris considered operator monotone if, for any n, and A,BMnsuch that 0AB, the inequalities 0fAfBhold. An operator monotone function is said to be symmetric if fx=xfx1and normalized if f1=1.

Definition 1 Fopis the class of functions f:0+0+such that:

  1. f1=1.

  2. tft1=ft.

  3. fis operator monotone.

Example 1. Examples of elements of Fopare given by the following:

fRLDx=2xx+1,fWYx=x+122,fBKMx=x1logx,
fSLDx=x+12,fWYDx=α1αx12xα1x1α1,α01.

Remark 1. Any fFopsatisfies

2xx+1fxx+12,x>0.

For fFop, we define f0=limx0fx. We introduce the sets of regular and non-regular functions

Fopr=fFopf00,FopnfFopf0=0

and notice that trivially Fop=FoprFopn.

Definition 2. For fFopr, we set

f˜x=12x+1x12f0fx,x>0.

Theorem 1.5 ([10]). The correspondence ff˜is a bijection between Foprand Fopn.

4.2 Metric adjusted skew information

In the Kubo-Ando theory [11] of matrix means, a mean is associated with each operator monotone function fFopby the following formula:

mfAB=A1/2fA1/2BA1/2A1/2,

where A,BMn,+C. Using the notion of matrix means, the class of monotone metrics can be defined by the following formula:

ABρ,f=TrAmfLρRρ1B,

where LρA=ρA,RρA=.

Definition 3. For AMn,saC, we define as follows:

IρfA=f02iρAiρAρ,f,
CρfA=TrmfLρRρAA,
UρfA=VρA2VρAIρfA2.

Quantity IρfAis referred to as the metric adjusted skew information, and ABρ,fis referred to as the metric adjusted correlation measure.

Proposition 1. The following holds:

  1. IρfA=IρfA0=TrρA02Trmf˜LρRρA0A0=VρACρf˜A0.

  2. JρfA=TrρA02+Trmf˜LρRρA0A0=VρA+Cρf˜A0.

  3. 0IρfAUρfAVρA.

  4. UρfA=IρfAJρfA.

Theorem 1.6 ([6]). For fFopr, if

x+12+f˜x2fx,E8

then it holds that

UρfAUρfBf0Tr(ρ[AB])2,E9

where A,BMn,saC.

To prove Theorem 1.6, several lemmas are used.

Lemma 1. If (8) holds, then the following inequality is satisfied:

x+y22mf˜xy2f0xy2.

Proof of Lemma 1. By (8), we have

x+y2+mf˜xy2mfxy.E10

Since

mf˜xy=yf˜xy=y2xy+1xy12f0fx/y=x+y2f0xy22mfxy,

we have

x+y22mf˜xy2=x+y2mf˜xyx+y2+mf˜xy=f0xy22mfxyx+y2+mf˜xyf0xy2.by10

Lemma 2. Let ϕ1ϕ2ϕnbe a basis of eigenvectors of ρ, corresponding to the eigenvalues λ1λ2λn. We set ajk=ϕjA0ϕk,bjk=ϕjB0ϕk. Then, we have

IρfA=12j,kλj+λkajkakjj,kmf˜λjλkajkakj,
JρfA=12j,kλj+λkajkakj+j,kmf˜λjλkajkakj,
UρfA2=14j,kλj+λkajk22j,kmf˜λjλkajk22.

Proof of Theorem 1.6. Since

TrρAB=TrρA0B0=j,kλjλkajkbkj,

we have

f0Tr(ρ[AB])2j,kf01/2λjλkajkbkj2j,kλj+λk22mf˜λjλk21/2ajkbkj2j,kλj+λk2mf˜λjλkajk2×j,kλj+λk2+mf˜λjλkbkj2=IρfAJρfB.

We also have

IρfBJρfAf0Tr(ρ[AB])2.

Thus, we have the final result (9).

5. Generalized metric adjusted skew information

We assume that fFoprsatisfies the following condition (A):

gxkx12fx,for somek>0.

Let

Δgfx=gxkx12fxFop.

Definition 4. For A,BMn,saC,ρMn,+,1Cwe define the following:

CorrρgfAB=kiρA0iρB0f=TrA0mgLρRρB0TrA0mΔgfLρRρB0.
IρgfA=CorrρgfAA
=TrA0mgLρRρA0TrA0mΔgfLρRρA0TrA0mΔgfLρRρA0.
JρgfA=TrA0mgLρRρA0TrA0mΔgfLρRρA0+TrA0mΔgfLρRρA0.
UρgfA=IρgfAJρgfA.

IρgfAis referred to as the generalized metric adjusted skew information, and CorrρgfABis referred to as the generalized metric adjusted correlation measure.

Theorem 1.7 ([7]). Under condition (A), the following holds:

  1. 1. (Schrödinger type) For A,BMn,saC,ρMn,+,1C,

IρgfAIρgfBCorrρgf(AB)2.

  1. 2. (Heisenberg type) For A,BMn,saC,ρMn,+,1C, we assume the following condition (B):

gx+Δgfxfxfor some>0.

Then,

UρgfAUρgfBkTr[ρ[AB]]2.

6. Generalized quasi-metric adjusted skew information

In this section, we present general uncertainty relations for non-Hermitian observables X,YMnC.

Definition 5. For X,YMnC,A,BMn,+Cwe define the following:

ΓA,BgfXY=kLARBXLARBYf=kTrXLARBmfLARB1LARBY=TrXmgLARBYTrXmΔbfLARBY,
ΨA,BgfXY=TrXmgLARBY+TrXmΔgfLARBY,
IA,BgfX=ΓA,BgfXX,JA,BgfX=ΨA,BgfXX,UA,BgfX=IA,BgfXJA,BgfX.

IA,BgfXis referred to as the generalized quasi-metric adjusted skew information, and ΓA,BgfXYis referred to as the generalized quasi-metric adjusted correlation measure.

Theorem 1.8 ([12]). Under condition (A), the following holds:

  1. 1. (Schrödinger type) For X,YMnC,A,BMn,+C,

IA,BgfXIA,BgfYΓA,Bgf(XY)2116IA,BgfX+YIA,BgfXY2.

  1. 2. (Heisenberg type) For X,YMnC,A,BMn,+C, we assume condition (B). Then,

UA,BgfXUA,BgfYkTr[XLARBY]2.

In particular,

kTrXLARBX2TrXmgLARBmΔgfLARBX×TrXmgLARB+mΔgfLARBX,E11

where XMnCand A,BMn,+C.

Proof of 1 in Theorem 1.8. By the Schwarz inequality, we have

IA,BgfXIA,BgfY=ΓA,BgfXXΓA,BgfYYΓA,Bgf(XY)2.

Now, we prove the second inequality. Since

IA,BgfX+Y=TrX+YmgLARBX+YTrX+YmΔgfLARBX+Y,
IA,BgfXY=TrXYmgLARBXYTrXYmΔgfLARBXY,

we have

IA,BgfX+YIA,BgfXY=2TrXmgLARBY+2TrYmgLARBX]2TrXmΔgfLARBY2TrYmΔgfLARBX=2ΓA,BgfXY+2ΓA,BgfYX=4ReΓA,BgfXY.

Similarly, we have

IA,BgfX+Y+IA,BgfXY=2IA,BgfX+IA,BgfY.

Then,

ΓA,Bg,f)XY=ReΓA,BgfXY+iImΓA,BgfXY
=14IA,BgfX+YIA,BgfXY+iIm{ΓA,BgfXY.

Thus,

ΓA,BgfXY2=116IA,BgfX+YIA,BgfXY2+ImΓA,BgfXY2
116IA,BgfX+YIA,BgfXY2.

We use the following lemma to prove 2:

Lemma 3

mgxy2mΔgfxy2kxy2.

Proof of Lemma 3. By conditions (A) and (B), we have

mΔgfxy=mgxykxy2mfxy,
mgxy+mΔgfxymfxy.

We then have

mgxy2mΔgfxy2=mgxymΔgfxymgxy+mΔgfxy
kxy2mfxymfxy=kxy2.

Proof of 2 in Theorem 1.8. Let

A=i=1nλiϕiϕiB=i=1nμiψiψi

be the spectral decompositions of Aand B, respectively. Then, we have

IA,BgfX=i,jmgλiμjmΔgfλiμjϕiXψj2,
JA,BgfY=i,jmgλiμj+mΔgfλiμjϕiYψj2,

Since

LARB=i=1nj=1nλiμjLϕiϕiRψjψj,

we have

TrXLARBY=i=1nj=1nλiμjϕiXψj¯ϕiYψj.

Then, by Lemma 3, we have

kTrXLARBY2i=1nj=1nkλiμjϕiXψjϕjYψi2i=1nj=1n(mgλiμj2mΔgf(λiμj)2)ϕiXψjϕjYψi2i=1nj=1n(mgλiμjmΔgf(λiμj))ϕiXϕj2i=1nj=1n(mgλiμj+mΔgf(λiμj))ϕjYψi2=IA,BgfXJA,BgfY.

Similarly, we have kTrXLARBY2IA,BgfYJA,BgfX. Therefore,

UA,BgfXUA,BgfYkTr[XLARBY]2.

When A=B=ρMn,+,1C,X=AMnC), and Y=BMnC, we obtain the result in Theorem 1.7.

We assume that

gx=x+12,fx=α1αx12xα1x1α1,k=f02,=2.

We then obtain the following trace inequality by substituting X=Iin (11).

α1αTrLARBI212TrA+B212TrAαB1α+A1αBα2.E12

This is a generalization of the trace inequality provided in [13]. In addition, we produce the following new inequality by combining a Chernoff-type inequality with Theorem 1.8.

Theorem 1.9 ([14]). We have the following:

12TrA+BLARBIinf0α1TrA1αBαTrA1/2B1/2
12TrAαB1α+A1αBα12TrA+B2α1α(Tr[LARBI)2.

The following lemma is necessary to prove Theorem 1.9.

Lemma 4. Let fs=TrA1sBsfor A,BMnCand 0s1. Then fsis convex in s.

Proof of Lemma 4. f's=TrA1slogABs+A1sBslogB. And then

f''s=TrA1slogA2BsA1slogABslogBTrA1slogABslogBA1sBslogB2=TrA1slogA2BsTrA1slogAlogBBsTrlogBlogAA1sBs+TrA1slogB2Bs=TrA1slogAlogAlogBBsTrA1slogAlogBlogBBs=TrA1slogAlogBBslogATrA1slogAlogBlogBBs=TrA1slogAlogBBslogAlogB=TrA1s/2logAlogBBslogAlogBA1s/20.

fsis convex in s.

Proof of Theorem 1.9. The third and fourth inequalities follow from Lemma 4 and (12), respectively. Thus, we only prove the following inequality:

TrA+BLARBI2TrA1αBα0α1.

Let

A=iλiϕiϕi=i,jλiϕiϕiψjψj,
B=jμjψjψj=i,jμjϕiϕiψjψj.

Then, we have

TrA=i,jλiϕiψj2,TrB=i,jμjϕiψj2.

And since

LARB=i,jλiμjLϕiϕiRψjψj,

we have

LARBI=i,jλiμjϕiϕiψjψj.

Then, we have

TrLARBI=i,jλiμjϕiψj2.

Therefore,

TrA+BLARBI=i,jλi+μjλiμjϕiψj2.

However, since we have

Aα=iλiαϕiϕi=i,jλiαϕiϕiψjψj,
B1α=jμj1αψjψj=i,jμj1αϕiϕiψjψj,
AαB1α=i,jλiαμj1αϕiϕiψjψj.

Then,

TrAαB1α=i,jλiαμj1αϕiψj2.

Thus,

2TrAαB1αTrA+BLARBI=i,j2λiαμj1αλi+μjλiμjϕiψj2.

Since 2xαy1αx+yxy0for x,y>0,0α1in general, we can obtain Theorem 1.9.

Remark 2. We note the following 1, 2:

  1. 1. 12TrA+BABinf0α1TrA1αBαTrA1/2B1/2

12TrA+B214TrAB2.

  1. 2. There is no relationship between TrLARBIand TrAB. When

A=32121232,B=4001,

we have TrLARBI=3,TrAB=10. When

A=1327272132,B=2005,

we have TrLARBI=8,TrAB=58.

7. Sum type of uncertainty relations

Let A,BMn,saChave the following spectral decompositions:

A=i=1nλiϕiϕiB=i=1nμiψiψi.

For any quantum state ϕ, we define the two probability distributions

P=p1p2pn,Q=qiq2qn,

where pi=ϕiϕ2,qj=ψjϕ2. Let

HP=i=1npilogpi,HQ=j=1nqjlogqj

be the Shannon entropies of Pand Q, respectively.

Theorem 1.10. The following uncertainty relation holds:

HP+HQ2logc,

where c=maxi,jϕiψj.

For details, see [15, 16].

Definition 6. The Fourier transformation of ψL2Ris defined as

ψ̂ω=ψte2πiωtdt.

We also define

QR=fL2Rt2ft2dt<.

Proposition 2. If ψL2R,ψ2=1satisfies ψ,ψ̂QR, then

Sψ+Sψ̂loge2,

where

Sψ=ψt2logψt2dt,Sψ̂=ψ̂t2logψ̂t2dt.

For details, see [17].

Theorem 1.11 ([18]). For any X,YMnC,A,BMn,+C, the following holds:

  1. IA,BgfXY+IA,BgfY12maxIA,BgfX+YIA,BgfXY.

  2. IA,BgfX+IA,BgfYmaxIA,BgfX+YIA,BgfXY.

  3. IA,BgfX+IA,BgfY2maxIA,BgfX+YIA.BgfXY.

Proof 1. The Hilbert-Schmidt norm satisfies

X2+Y2=12X+Y2+XY212maxX+Y2XY2.E13

Since IA,BgfXXis the second power of the Hilbert-Schmidt norm, X=IA,BgfX. We then obtain the result by substituting (13),

2. By the triangle inequality of a general norm, we apply the triangle inequality for X=IA,BgfX.

3. We prove the following norm inequality:

X+YX+Y+XY.E14

Since

X=12X+Y+12XY12X+Y+12XY

and

Y=12Y+X+12YX12Y+X+12YX,

we add two inequalities and obtain (14).

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Kenjiro Yanagi (April 28th 2020). Uncertainty Relations, Quantum Mechanics, Paul Bracken, IntechOpen, DOI: 10.5772/intechopen.92137. Available from:

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