Uncertainty relations for any multi observables

0.16 MB·

by  Jinchuan Hou

UNCERTAINTY RELATIONS FOR ANY MULTI OBSERVABLES
JINCHUAN HOU, KAN HE
6 Abstract. Uncertainty relations describe the lower bound of product of standard
1
0 deviations of observables. By revealing a connection between standard deviations
2
of quantum observables and numerical radius of operators, we establish a universal

n
a uncertainty relation for any k observables, of which the formulation depends on the
J
even or odd quality of k. This universal uncertainty relation is tight at least for the
4
2 cases k = 2n and k = 3. For two observables, the uncertainty relation is exactly a

] simpler reformulation of Schr¨odinger’s uncertainty principle.
h
p
-
t
n
a 1. Introduction
u
q
[ In the past ninety years, the theory of quantum mechanics was applied in lots of

1
other sciences, including Information Science, Chemistry and Biology (Ref. [2, 3, 9]).
v
8
The uncertainty principle, discovered first by Heisenberg in 1927 (Ref. [7]), is often
3
3
6 considered as one of the most important topics of quantum theory (Ref. [8, 13]) and
0
. can be linked to quantum entanglement and other important topics (Ref. [1, 10]).
1
0
Heisenberg’s uncertainty principle says that
6
1 ~
v: ∆q∆p , (1.1)
≥ 2
i
X
where ∆ and ∆ denote standard deviations of the position qˆ and momentum pˆ re-
r q p
a
spectively, ~ = 1 qˆpˆ pˆqˆ is the reduced Planck constant. Recalled that a quantum
2|h − i|
system can be simulated in a complex Hilbert space H with the inner product and
h·|·i
a pure state is described by a unit vector x . Quantum observables for a state x
| i | i
are self-adjoint operators on H with domain containing x (Ref. [13]). The value of
| i
observable A for the pure state x is A = x A x . In 1929, Robertson generalized
| i h i h | | i
Heisenberg’s uncertainty principle, which says that, for observables A,B and pure state
x ,
| i
1
∆ ∆ [A,B] . (1.2)
A B
≥ 2|h i|
PACS 2010. 03.65.Ta,03.65.Db, 02.30.Tb.
Key words and phrases. quantum states, uncertainty principle, observables, deviations.
1
2 JINCHUANHOU,KANHE
Where [A,B] = AB BA is the Lie product of A and B,
−
∆ = A2 A 2 and ∆ = B2 B 2
A B
h i−h i h i−h i
p p
are the standard deviations of A and B, respectively [11]. Schro¨dinger gave a uncer-
tainty principle, which is sharper than Robertson’s and asserts that
1 1
∆ ∆ [A,B] 2 + A,B A B 2, (1.3)
A B ≥ r4|h i| |2h{ }i−h ih i|
where A,B = AB + BA is the Jordan product of A and B [12]. Schro¨dinger’s
{ }
uncertainty principle holdsfor mixed state, too. Recall that a mixed stateρis apositive
operator on H with trace 1. Then the value of observable A for the state ρ is A =
h i
Tr(Aρ)andthestandarddeviationofAis∆ = A2 A 2 = Tr(A2ρ) Tr(Aρ)2.
A
h i−h i −
p p
Here we assume that both Tr(Aρ) and Tr(A2ρ) are finite.
What happens for multi observables?
There is a simple way to get certain uncertainty relation from the uncertainty prin-
ciples in (1.2) or (1.3). For example, let A,B,C be three observables, then by applying
(2) one gets
1
∆2∆2 ∆2 [A,B] [B,C] [A,C] . (1.4)
A B C ≥ 8|h ih ih i|
But (1.4) is not sharp enough.
~
Let A = qˆ,B = pˆand C = rˆ= pˆ qˆ; then [p,q] = [q,r] = [r,p] = , and thus (1.4),
− − i
together with (1.1), gives
~
∆2∆2∆2 ( )3.
q p r ≥ 2
However, in [15], a tight uncertainty relation is given that
~
∆2∆2∆2 (τ )3 (1.5)
q p r ≥ 2
with τ = 2 > 1.
√3
This also happens for Pauli matrices X,Y,Z. As [X,Y] = 2iZ, one has 1 [X,Y] =
2|h i|
Z . Similarly 1 [X,Z] = Y and 1 [Y,Z] = X . Thus by (1.4)
|h i| 2|h i| |h i| 2|h i| |h i|
∆2 ∆2 ∆2 X Y Z .
X Y Z ≥ |h ih ih i|
But it was announced by S.-M. Fei that
8
∆2 ∆2 ∆2 X Y Z . (1.6)
X Y Z ≥ 3√3|h ih ih i|
This inequality is also tight and achieves “=” at ρ = 1(I + 1 X + 1 Y + 1 Z).
2 √3 √3 √3
UNCERTAINTY RELATIONS 3
Therefore, toobtainuncertaintyrelationsformultiobservablesthataresharpenough,
one needs new approaches. Let A ,A ,...,A be any k observerbles of a quantum
1 2 k
system. The purpose of this paper is to establish a lower bound of ∆ ∆ ∆ in
A1 A2 ··· Ak
terms of A A , A A and A2 . For the case when k = 2, the uncertainty relation
h i ji h iih ji h ji
is equivalent to Schro¨dinger’s uncertainty principle that is tight and has a simpler
representation. For the case when k = 3, we show that the untainty relation is tight
by taking Pauli matrices as observables. All proofs of the main result and the lemmas
will be presented in the appendix section.
2. uncertainty relations for multi observables
Our main idea is based on the following observation, which establishes a formula to
connect the standard deviation of a quantum observable A of a state x to the norm
| i
as well as the numerical radius of [A, x x ], the Lie product of A and the rank one
| ih |
projection x x .
| ih |
Let T be a bounded linear operator acting on a complex Hilbert space H. The
numerical range of T is the set W(T) = x T x : x H, x = 1 , and the
{h | | i | i ∈ k| ik }
numerical radius of T is w(T) = sup λ : λ W(T) . The topic of numerical range
{| | ∈ }
and numerical radius plays an important role in mathematics and is applied into many
areas (Ref. [4, 5, 6]). Denote by T the operator norm of T.
k k
Lemma 2.1. Let x be a pure state and A an observable for it. Then
| i
∆ = [A, x x ] = w([A, x x ]).
A
k | ih | k | ih |
By Lemma 2.1, for any observables A ,A ,...,A for a pure state x ,
1 2 k
| i
Πk ∆ = Πk [A , x x ] Πk [A , x x ] w(Πk [A , x x ]). (2.1)
j=1 Aj j=1k j | ih | k ≥ k j=1 j | ih | k ≥ j=1 j | ih |
Note that, the value of Πk ∆ does not depend on the order arrange of observables
j=1 Aj
but w(Πk [A , x x ]) does. Therefore, the inequality (2.1) can be sharped to
j=1 j | ih |
Πk ∆ maxw(Πk [A , x x ]), (2.2)
j=1 Aj ≥ π j=1 π(j) | ih |
where the maximum is over all permutations π of (1,2,...,k). Thus the question
of establishing an uncertainty relation for k observales is reduced to the question of
4 JINCHUANHOU,KANHE
calculating the numerical radius of the operator
D(π) = Πk [A , x x ], (2.3)
k j=1 π(j) | ih |
which is an operator of rank 2.
≤
Theexactvalueofw(D(π))iscomputableandwecanestablishanuncertainty relation
k
for any multi observables by (2.2). For simplicity, and with no loss of generality, we
state our results only for π = id.
One may ask why do not work on the stronger inequality
Πk ∆ Πk [A , x x ] ?
j=1 Aj ≥ k j=1 j | ih | k
In fact, as we show in the Appendix section, this stronger inequality leads to weaker
uncertainty relations. So the numerical radius is the better choice.
Thefollowingisourmainresult, hereweagreeonΠ a = 1ifΛ = . Itissurprising
j Λ j
∈ ∅
that our uncertainty relation for any k observables has different formulation depending
on the even or odd quality of the integer k.
Theorem 2.2. Let A ,A ,...,A with k 2 be observables.
1 2 k
≥
(1) If k = 2n, then
Π2n ∆
j=1 Aj (2.4)
1(Πn 1 A A A A )( A A A A +∆ ∆ ).
≥ 2 j=−1|h 2j 2j+1i−h 2jih 2j+1i| |h 1 2ni−h 1ih 2ni| A1 A2n
(2) If k = 2n+1, then, identifying 2n+2 with [(2n+2) mod (2n+1)] = 1,
Π2n+1∆
j=1 Aj
1[2Π2n+1 A A A A
≥ 2 j=1 |h j j+1i−h jih j+1i| (2.5)
+∆2 Πn A A A A 2
A1 j=1|h 2j 2j+1i−h 2jih 2j+1i|
+∆2 Πn A A A A 2]1.
A2n+1 j=1|h 2j−1 2ji−h 2j−1ih 2ji| 2
Obviously, “=” holds if and only if
Πk [A , x x ] = Πk [A , x x ] = w(Πk [A , x x ]). (2.6)
j=1k j | ih | k k j=1 j | ih | k j=1 j | ih |
ThustheuncertaintyrelationistightifEq.(2.6)holdsforsomeobserverblesA ,A ,...,A
1 2 k
and some state. This is the case as will be illustrated in Section 4.
We remak that Theorem 2.2 holds for any state ρ with Tr(A ρ) < and Tr(A2ρ) <
| j | ∞ j
,j = 1,2,...,k. Toseethis, denoteby (H)betheHilbert-SchimitclassinH,which
2
∞ C
UNCERTAINTY RELATIONS 5
is a Hilbert space with inner product T,S = Tr(T S). Then, a positive operator ρ is
†
h i
a state if and only if √ρ is a unit vector in (H). For a self-adjoint operator A on H,
2
C
define a linear operator L on (H) by L T = AT if Tr(T A2T) < . It is clear that
A 2 A †
C ∞
LA is self-adjoint as L†A = LA† = LA. Note that
A = Tr(Aρ) = √ρ L √ρ = L
A A
h i h | | i h i
and thus
∆ = ∆ , L L = L = AB .
A LA h A Bi h ABi h i
Then, Theorem 2.2 is true by applying (2.1) to L ,L ,...,L and the pure state
A1 A2 Ak
√ρ .
| i
Before to see the uncertainty relations presented by theorem 2.2 is sharper than
thoseobtainedbyHeisenberg’suncertaintyprinciple(1.2)andSchro¨dinger’suncertainty
principle (1.3), we illustrate some application of Theorem 2.2 for the cases k = 2.
3. The case of k = 2: a reformulation of Schro¨dinger’s principle
Applying Theorem 2.2 (1) to the case when k = 2, the following result is immediate.
Theorem 3.1. Let A and B be observables for a state. Then
∆ ∆ AB A B , (3.1)
A B
≥ |h i−h ih i|
which is equivalent to Schr¨odinger’s uncertainty principle.
The expression of inequality is quite simpler than that of Schro¨dinger’s uncertainty
principle. We show that (3.1) is in fact equivalent to Schro¨dinger’s uncertainty principle
(1.3).
To check it, write A B = r and AB = s +it, where s,t,r R. Then BA =
h ih i h i ∈ h i
s it. A simple computation gives
−
AB A B = (s r)2 +t2,
|h i−h ih i| −
p
1 1
[A,B] 2+ A,B A B 2 = (s r)2 +t2
r4|h i| |2h{ }i−h ih i| −
p
and
1
[A,B] = t .
2|h i| | |
6 JINCHUANHOU,KANHE
So, we get
∆ ∆ A B AB
A B
≥ |h ih i−h i|
= 1 [A,B] 2+ 1 A,B A B 2 (3.2)
4|h i| |2h{ }i−h ih i|
q
1 [A,B] .
≥ 2|h i|
Now we are at a position to show that Theorem 2.2 is sharper than the uncertainty
relations obtained by the approach mentioned in the introduction section.
Let A ,A ,...,A be observables.
1 2 k
If k = 2n is even, by inequalities (3.2) one has
Πk ∆ = (Πn 1(∆ ∆ ))(∆ ∆ )
j=1 Aj j=−1 A2j A2j+1 A1 A2n
(Πn 1 A A A A ) A A A A (3.3)
≥ j=−1|h 2j 2j+1i−h 2jih 2j+1i| h 1 2ni−h 1ih 2ni|
1 (Πn 1 [A ,A ] ) [A ,A ] ,
≥ 2n j=−1|h 2j 2j+1 i| |h 1 2n i|
which is weaker than the inequality (2.4) since ∆ ∆ A A A A .
A1 A2n ≥ |h 1 2ni−h 1ih 2ni|
If k = 2n+1 is odd, by (3.2) again we have
Πk ∆2 = (Πn (∆ ∆ ))(Πn (∆ ∆ ))(∆ ∆ )
j=1 Aj j=1 A2j−1 A2j j=1 A2j A2j+1 A1 A2n+1
(Πn ( A A A A )( A A A A ) )
≥ j=1| h 2j−1 2ji−h 2j−1ih 2ji h 2j 2j+1i−h 2jih 2j+1i | · (3.4)
A A A A
1 2n+1 1 2n+1
·|h i−h ih i|
1 (Πn ( [A ,A ] [A ,A ] )) [A ,A ] ,
≥ 22(2n+1) j=1 |h 2j−1 2j ih 2j 2j+1 i| |h 1 2n+1 i|
which is clearly weaker than the inequality (2.5) as a2 + b2 2ab and ∆ ∆
≥ A1 A2n+1 ≥
A A A A .
1 2n+1 1 2n+1
|h i−h ih i|
4. Uncertainty relations for three or four observables
By Theorem 2.2 and a careful check of its proof, one gets a uncertainty relation for
any three observables like the following.
Theorem 4.1. Let A,B,C be three observables for a state ρ in a state space H, then
∆2∆2 ∆2 1(∆2 AB A B 2+∆2 BC B C 2)
A B C ≥ 4 C|h i−h ih i| A|h i−h ih i| (4.1)
+1 ( AB A B )( BC B C )( AC A C ) .
2| h i−h ih i h i−h ih i h i−h ih i |
Particularly, for the case when ∆ ∆ = AC A C or AB = A B or
A C
|h i − h ih i| h i h ih i
dimH = 2,
1
∆ ∆ ∆ (∆ BC B C +∆ AB A B ). (4.2)
A B C A C
≥ 2 |h i−h ih i| |h i−h ih i|
UNCERTAINTY RELATIONS 7
The inequalities (4.1) and (4.2) are tight as illustrated by applying to Pauli matrices.
Example 4.2. Uncertainty relations for Pauli matrices.
Let X,Y,Z be Pauli matrices, that is,
0 1 0 i 1 0
X =  , Y =  − , Z =  .
1 0 i 0 0 1
     − 
Recall that, for any dense matrix ρ M (C), ρ has a representation
2
∈
1
ρ = (I +r X +r Y +r Z)
2 1 2 3
2
with Bloch vector (r ,r ,r )t R3 and r2 + r2 + r2 1; and ρ is pure if and only
1 2 3 ∈ 1 2 3 ≤
if r2 + r2 + r2 = 1. Recall also that XY = iZ, YZ = iX and ∆2 = 1 A 2 for
1 2 3 A − h i
A X,Y,Z , X2 = Y2 = Z2 = 1 and ( X , Y , Z ) = (r ,r ,r ).
1 2 3
∈ { } h i h i h i h i h i h i
Applying the inequality (4.2) of Theorem 4.1 to X,Y,Z we get
∆ ∆ ∆
X Y Z
1(∆ i X Y Z +∆ i Z X Y ) (4.3)
≥ 2 X| h i−h ih i| Z| h i−h ih i|
= 1( (1 X 2)( X 2+ Y 2 Z 2)+ (1 Z 2)( Z 2 + X 2 Y 2) ).
2 −h i h i h i h i −h i h i h i h i
p p
Obviously, the inequality (4.3) is tight and “=” holds if the Bloch vector satisfies
r = r = 1 and r = 0.
| 1| | 3| √2 2
This illustrates that Theorem 2.2 is tight for three observables. Since Schr¨odinger’s
uncertaintyprinciple(1.3)is tightandouruncertainty relationis equivalentto Schr¨odinger’s
uncertainty principle by (3.2), Theorem 2.2 is also tight for two obserables.
Moreover, by Theorem 3.1,
(1 Z 2)(1 X 2) Y 2 + X 2 Z 2,
−h i −h i ≥ h i h i h i
hence we have
∆2 ∆2 ∆2
X Y Z
1[(1 X 2)( X 2 + Y 2 Z 2)+(1 Z 2)( Z 2 + X 2 Y 2)]
≥ 4 −h i h i h i h i −h i h i h i h i
+1 (1 Z 2)(1 X 2)( X 2 + Y 2 Z 2)( Z 2 + X 2 Y 2)
2 −h i −h i h i h i h i h i h i h i
p
1[(1 X 2)( X 2 + Y 2 Z 2)+(1 Z 2)( Z 2 + X 2 Y 2)] (4.4)
≥ 4 −h i h i h i h i −h i h i h i h i
+1 ( Y 2 + X 2 Z 2)( X 2 + Y 2 Z 2)( Z 2 + X 2 Y 2)
2 h i h i h i h i h i h i h i h i h i
p
( Y 2 + X 2 Z 2)( X 2 + Y 2 Z 2)( Z 2+ X 2 Y 2)
≥ h i h i h i h i h i h i h i h i h i
p2√2 X Y Z 3.
2
≥ |h ih ih i|
8 JINCHUANHOU,KANHE
Particularly, one has
∆2X∆2Y∆2Z ≥ 2√2|hXihYihZi|23. (4.5)
Observe that we always have
8
1 ∆2 ∆2 ∆2 = (1 r2)(1 r2)(1 r2)
≥ X Y Z − 1 − 2 − 3 ≥ 27
since the function (1 r2)(1 r2)(1 r2) has its minimum value 8 at r = r = r =
− 1 − 2 − 3 27 | 1| | 2| | 3|
1 and the maximum value 1 at ρ = 1I . Moreover, r r r achieves simultaneously
√3 2 2 | 1 2 3|
its maximum value 1 at r = r = r = 1 . Thus the inequality (4.5) can be
3√3 | 1| | 2| | 3| √3
sharped to
8√4 3
∆2X∆2Y∆2Z ≥ 3 |hXihYihZi|23. (4.6)
The inequality (4.6) is tight in the sense that “=” holds if r = r = r = 1 .
| 1| | 2| 3| √3
Compare (4.3) with (1.6) and (4.6). Although these inequalities are all tight, one of
the remarkable advantage of (4.3) is that, even if some of X , Y , Z are zero, we still
h i h i h i
may get a positive lower bound of ∆ ∆ ∆ . For instance, saying Y = 0, we have
X Y Z
h i
1
∆ ∆ ∆ (1 X 2) X 2 + (1 Z 2) Z 2 ;
X Y Z
≥ 2 −h i h i −h i h i
(cid:16)p p (cid:17)
saying Y = Z = 0, we have
h i h i
1
∆ ∆ ∆ (1 X 2) X 2,
X Y Z
≥ 2 −h i h i
p
while we cannot get any information from (1.6) and (4.6).
Before conclusion we state the uncertainty relation from Theorem 2.2 for four obser-
vations, which has a relatively simple expression.
Theorem 4.3. Let A ,A ,A ,A be observables. Then
1 2 3 4
∆ ∆ ∆ ∆
A1 A2 A3 A4 (4.7)
1 A A A A ( A A A A +∆ ∆ ).
≥ 2|h 2 3i−h 2ih 3i| |h 1 4i−h 1ih 4i| A1 A4
The inequality (4.7) is tight. For example, Consider bipartite continuous-variable
system. Let (A ,A ,A ,A ) = (qˆ ,pˆ ,qˆ ,pˆ ), where qˆ,pˆ are the position and momen-
1 4 2 3 1 1 2 2 i i
tum in the ith mode satisfying the canonical commutation relation. As Heisenberg’s
UNCERTAINTY RELATIONS 9
uncertainty principle (1.1) is tight, we say that
∆ ∆ ∆ ∆
q1 p1 q2 p2 (4.8)
1 qˆ pˆ qˆ pˆ ( qˆ pˆ qˆ pˆ +∆ ∆ ).
≥ 2|h 2 2i−h 2ih 2i| |h 1 1i−h 1ih 1i| q1 p4
is tight, the “=” is attained at ρ = e.
Similarly, considering the positions and momentums (qˆ ,pˆ ,qˆ ,pˆ ,...,qˆ ,pˆ ) in a
1 1 2 2 n n
n-partite continuous-variable system, one sees that the uncertainty relation (2.4) in
Theorem 2.2 is tight.
However we do not know whether the uncertainty relation (2.5) is tight for odd
k = 2n+1 5.
≥
5. Conclusion
Uncertainty relations discover lower bounds of the product of standard deviations
of several observables. Larger the lower bound is, more powerful the corresponding
uncertaintyrelationis. Therearenoknownuncertaintyrelationsthatvalidforarbitrary
multi observavles. By finding the equality of deviation and the norm of the Lie product
of the observable and the pure state, we reduce the question of establishing uncertainty
relation of multi observerbles to the question of computing the numerical radius of an
operator of rank 2. This enable us establish a universal uncertainty relation for any k
≤
observables, of which, the formulation depends on the even or odd quality of k. For two
observables, our uncertainty relation is exactly a simpler reformulation of Schro¨dinger’s
uncertainty principle. The uncertainty relation provided in this paper is tight, at least
for the cases of two and three observables, as illustrated by examples.
References
[1] M. Berta, M. Christandl, R. Colbeck, J. M. Renes, R. Renner, The uncertainty principle in the
presence of quantum memory, Nature Physics 6, 659-662(2010) doi:10.1038/nphys1734
[2] C. H. Bennett, D. P. DiVincenzo, Quantum information and computation, Nature 404, 247-255
(2000) doi:10.1038/35005001
[3] G. S. Engel, T. R. Calhoun, E. L. Read, T. K. Ahn, T. Mancal, et al., Evidence for wavelike
energy transfer through quantum coherence in photosynthetic systems, Nature 446, 782 (2007).
[4] K. E. Gustafson, D. K. M. Rao, Numerical Range, Springer-Verlag, New York, Inc., 1997.
[5] P. R. Halmos, A Hilbert space problem book, Springer-Verlag,1982.
10 JINCHUANHOU,KANHE
[6] K. He, J.-C. Hou, X.-L. Zhang, Maps preserving numerical radius or cross norms of products of
self-adjoint operators, Acta Mathematica Sinica-English Series 26 (2010), 1071-1086.
[7] W.Heisenberg,U¨berdenanschaulichenInhaltderquantentheoretischenKinematikundMechanik.
Z. Phys. 43, 172-198 (1927).
[8] W. Heisenberg, The physical principles of quantum theory, (University of Chicago Press, 1930).
[9] B.P.Lanyon,J.D.Whitfield,G.G.Gillett,M.E.Goggin,M.P.Almeida,etal.,Towardsquantum
chemistry on a quantum computer, Nature Chemistry, (2010) 2, 106-111doi:10.1038/nchem.483
[10] R. Prevedel, D. R. Hamel, R. Colbeck, K. Fisher , K. J. Resch, Experimental investigation of
the uncertainty principle in the presence of quantum memory and its application to witnessing
entanglement, Nature Physics 7 757-761(2011) doi:10.1038/nphys2048
[11] H. P. Robertson, The Uncertainty Principle, Phys. Rev. 34, 163-Published 1 July 1929
[12] E.Schr¨odinger,ZumHeisenbergschenUnscha¨rfeprinzip.Berl.Ber.296-303(1930).[Englishtrans-
lation: A.Angelow,M.C.Batoni,AboutHeisenberguncertaintyrelation.Bulg.J.Phys.26(5/6),
193-203 (1999). arXiv:quant-ph/9903100
[13] J.vonNeumann,MathematicalFoundationsofQuantumMechanics,(PrincetonUniversityPress,
1932).
[14] D. S. Keeler,L. Rodman, I. M. Spitkovsky,The numricalrange of3 3 matrices, Linear Algebra
×
Appl., 252 (1997), 115-139.
[15] S.Kechrimparis,S.Weigert,Heisenberguncertaintyrelationforthreecanonicalobservables,Phys.
Rev. A 90, 062118(2014)
[16] L. Maccone, A. K. Pati, Stronger Uncertainty Relations for All Incompatible Observables, Phys.
Rev. Lett. 113, 260401(2014)
6. Appendix
In the appendix, we give the proofs of theorems 2.1 and 2.2.
Proof of Theorem 2.1. Let H be the associated Hilbert space for the pure state x and
| i
the observable A. Write A x in the form A x = α x + β y , where normalized y
| i | i | i | i | i
is orthogonal to x . Since A is self-adjoint we have α = x A x R. Moreover,
| i h | | i ∈
by self-adjointness of A, the Lie product of A and the rank one projection x x is
| ih |
represented by the following matrix relative to decomposition H = [x] [y] x,y, ,
⊥
⊕ ⊕{ }
here [x] = span x . Then
{ }
0 β¯
[A, x x ] =  −  0.
| ih | β 0 ⊕
 