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

## 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
• 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.

### 4.1 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).

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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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