Corrigendum: Spectral thresholding quantum tomography for low rank states (2015 New J Phys 17 113050)

CORRIGENDUM • OPEN ACCESS

Corrigendum: Spectral thresholding quantum

tomography for low rank states (2015 New J.

Phys. 17 113050)

To cite this article: Cristina Butucea et al 2016 New J. Phys. 18 069501

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CORRIGENDUM

Corrigendum: Spectral thresholding quantum tomography for low

rank states (2015

New J. Phys.

17

113050

)

Cristina Butucea1_{, Mădălin Guţă}2

and Theodore Kypraios2

1 _{Université Paris-Est Marne-la-Vallée, LAMA(UMR 8050), UPEMLV F-77454, Marne-la-Vallée, France} 2 _{University of Nottingham, School of Mathematical Sciences, University Park, Nottingham NG7 2RD, UK}

In this corrigendum to the paper Butuceaet al(2015New J. Phys.17113050)we point out an error in one of the theoretical results describing the upper bound to the operator norm error of the least squares estimator. We provide a corrected version of the upper bound with a new convergence rate, and discuss the implications for other results which rely on the above upper bound.

Proposition 1 as stated in the paper is incorrect, in particular the dependence of the upper boundn( ) 2_on

the number of atomskis not valid. The error lies in the evaluation of the upper boundWof the variance term in the concentration bound. Below we provide a new version of proposition 1 with a corrected ratenc( ) 2replacing

the raten( ) 2stated in the paper. Ignoring the logarithmic factors, the new upper bound scales as3k N compared to erroneous rate2k N_{, wherekis the number of atoms andN=n3}k_{is the total number of}

measurements. We note that although the corrected bound is weaker that the one claimed in the paper, it is still an improvement compared to the previously known bound[2]which scaled as4k N_.

We will now discuss the implication of the correction to subsequent results in the paper. Proposition 2, theorem 1, corollary 1, and theorem 2 establish error rates for estimators obtained by normalising, penalising or thresholding the least square estimator. The proofs of these results use the operator norm error raten( ) 2_{as a}

generic expression, and are therefore not affected by its concrete dependence on the number of atomsk. Therefore proposition 2, theorem 1, corollary 1, and theorem 2 hold true when the operator norm rate is taken to have expressionnc( ) 2in proposition 1 below. In particular, the upper bounds on the Frobenius square norm

error in corollary 1 and theorem 2, will scale asr·n ec( )2=r3k Nrather thanrd N=r2k N. The remaining results including the lower bound in theorem 3 and the simulation results are independent of proposition 1 and do not require any correction.

Proposition 1.Letr_n( )ls be the linear estimator ofr. Then, for anye>0, the following operator norm inequality holds, fornlarge enough, with probability larger than1-eunderr

( )

( )

r_nls -rn ec ,

where

( ) ·

n e

e

= ⎛ +

⎝ ⎜ ⎞_⎠⎟

N

4 3 log 2 c

k k

2 1

withN≔n·3k_{the total number of measurements. The same bound holds whenk=k n( )as long asn e( ) 0.}

Proof of proposition 1.Note that the empirical frequencies can written as f( ∣ )o s = å I X( =o) n i si 1

, , where the

random variablesXs,iare independent for all settingssand allifrom 1 ton. To estimate the risk of the linear

estimator we write

( ( ∣ )) ( ∣ )) ( ∣ )

( ( ) ( ∣ )) ( ∣ )

≔

( )

( )

( )



å å å

å å å å

å å

r r s

s

- =

-= =

-f p A

n I X p

A

W

o s o s o s

o o s o s

2 3 1

2 3 .

n ls

k d

i

i _{k d}

i i b o s

b b b

b o s

s b _b b

s s

,

,

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whereWs,iare independent and centered Hermitian random matrices. We will apply the following extension of

the Bernstein matrix inequality[1]due to Tropp, see also[4,6].

Proposition 2(Bernstein inequality, Tropp).LetY1,¼,Ynbe independent, centered,m´mHermitian random matrices. Suppose that, for some constantsV W, >0we have Yj V, for alljfrom 1 ton, and that

( )

å_j Y_j2 W. Then, for allt 0,

(  )

 + +  

-+ ⎛ ⎝

⎜ ⎞

⎠ ⎟

Y Y t m t

W tV

... 2 exp 2

3 . n

1

2

In our setup,Ws,iplay the role ofYj. We boundWs,i Vfor allsandiandås,i(W Ws*,i s,i)W,

whereV W, are evaluated below. We have

ℓ

( ∣ )

· ∣ ( ) ( ∣ )∣ ·

∣ ( ) ( ∣ )∣

· ≔

ℓ ℓ ℓ ℓ ℓ

( )

( )

∣ ∣

    





å å

å



å

å å

å

s

d

=

-=

-= = + =

Î

= = =

⎜ ⎟ ⎜ ⎟

⎛ ⎝

⎞ ⎠

⎛

⎝ ⎞⎠

W n

A

I X p

n I X p

n n

k

n n V

o s

o o s

o o s

1

2 3

1 1

2 3

2 2

1 3

2 2

1 3

2 2 1

1

3 2

2

3 .

i _{k d} i

k d j E

b s i

k k

E E k

k

k

k k

k s

b o b

b s b

b b o

s

, ,

, ,

0 : 0

j j b

Now, denote by_B( ∣ )_{o s = å 2 3}- -k d( )_A ( ∣ )_{o s s}

b b b bso thatWs,i= åoB( ∣ )( (o s I Xs,i=o)-p( ∣ ))o s . Then

( )

( ∣ ) · ( ( ) ( )) · ( ∣ )

( ∣ ) · ( ( ) ( )) · ( ∣ )

( ∣ ) ( ∣ ) · ( ∣ ) ≔   ( )

*

*

*

*





å å

å å å

å å

å å

= = = ¢ ¢

= = = ¢ ¢

¢

¢ W W

n B I X I X B

n B I X I X B

n p B B n T

o s o o o s

o s o o o s

o s o s o s

1

Cov ,

1

Cov ,

1 1

. 1

i

i i

i

i i

s

s s

s o o

s s

s o o

s s

s o

, ,

2

,

, ,

,

,1 ,1

In the last inequality we used that

( )s ¢≔ ( (I X =o) I X( = ¢ =o)) p( ∣ )o s d ¢-p( ∣ ) ·o s p( ∣ )o s¢

Cov o o, Cov s,1 , s,1 o o,

which implies the following inequality between2k´2k_{matrices:Cov( )s p( )s wherep( )s is the diagonal}

matrix with elementsp( )so o, ¢=p( ∣ )o sdo o, ¢.

By expressingB( ∣ )o s in terms ofsbas above, we get

( ∣ ) ( ∣ )_{( )} ( ∣ )_{( )}

 =

å å

å

s s

¢

¢

¢ ¢

n T n p

A A

o s o s o s

1 1

2 3k d 2 3k d . s o b b

b b

b

b b b

,

Before giving the upper bound we introduce some combinatorial notations which will be used below. Let

{ }

Î x y z I

b , , , k_{and recall thatE ≔ {i b: =I}Ì{1 ,...,k}} i

b . We say thatbagrees with a settingsifbj=sjfor

alljÎE_bc. In this casebis completely determined by the setEb, for afixeds. This fact will be used to replace the

sums overbandb¢with those overEbandEb¢inT. Indeed sinceAb( ∣ )o s is proportional toj EÏ b db sj j, , the sums

overbandb¢inTare restricted to sequences which agree withs. We denote by

≔ ( ⧹ ) ( ⧹ )

Ç

¢ ¢

È

¢

Eb Eb Eb Eb Eb Eb andEb

Ç

Eb¢the symmetric difference and respectively the intersection ofEb

andEb¢. With these notations we have

( ∣ ) ¢( ∣ )s s¢=



·s

Î D ¢

A o s A o s o

j E E j

b b b b g

b b

whereg=g s( ,EbDEb¢)is the sequence withEg=(EbDEb¢)c, and it agrees withs. With these notations we have

2

( ∣ ) ( ∣ ) ( ∣ ) · ( ) ∣ ∣ ∣ ∣ ( ) ∣ ∣ ∣ ∣ ( ) ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ( )

å å

å



å å å



å

å å



s s s = = = Î D + D Î D + D - D + Î D

- D D

¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ T p o p o p o o s o s o s 1 2 3 1 2 3 1 2 3

3 2 3 2

k

E E

j E E j

E E E E

k E E

j E E j

E E E E

k E E

k E E

E E

j E E j

k k E E E E

s o g s s o g s s o g s 2 , , 2 , , , , b b b b

b b b b

b b

b b

b b b b

b b

b b

b b

b b

b b b b

In the last expression we rewrite the sum over settingssas a double sum overs˜and˜sc_{wheres˜is the restriction of} stoEbDEb¢ands˜cis the restriction to(EbDEb¢)c. Note thatg s( ,EbDEb¢)depends onsonly through the component˜s. Then

( ∣ ) · ( ∣ ) · ( ) ∣ ∣ ( ) ˜ ˜ ∣ ∣ (˜ ) ˜ (˜ ) (˜ ) ( )

å å



å å å



å

å

s s

r s rs s

=

= =

Î D

- D D

Î D

- D D

D D = D ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ p o p o

o s o s

2 3 2 3

1

2 Tr

j E E j

k k E E E E

j E E j

k k E E E E

E E E E _k

E E E s o

g s

s s o

g s

s

g s g s

g g g , , , , : c c b b

b b b b

b b

b b b b

b b b b

g b b

In the second equality we have used formulas(2.3)and(2.6)in the paper[3], to evaluate the interior sum as a Fourier coefficient ofρ. In the third equality we replaced the sum overs˜with an equivalent sum over sequences gsuch thatEg=(EbDEb¢)c.

Note that any pair(E E, ¢)is uniquely determined by three disjoint subsets,( ⧹E E E E E¢, ¢⧹ ,

Ç

E¢), or equivalently by the symmetric differenceD≔E ED ¢together withF≔ ⧹E E¢ ÌDandM≔E

Ç

E¢. The sum overE E, ¢in(2)is computed by summing over all triplesD F M, , satisfying the above conditions:

( ) ( ) ( ) ∣ ∣ ∣ ∣ ∣ ∣ ( ) ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

å

å

å å

å

å

å

å

å

rs s rs s rs s = = = Ç Ç ¢ - D ¢

+ ¢

= D ¢

Ì =Æ -+ = =Æ -+ = T 1 2 3 3 Tr 1 2 3 3 Tr 1 2 2 3 3 Tr k E E

k E E

E E

E E E

k

D F D M M D

k D D M E D k D D

M M D

k D D M E D g g g g g g g g g 2 , : 2

: 2 :

2

: 2 :

c c c g g g

Indeed, the sum overFÌDgives a factor2D _{since the summands do not depend onF. Next, the sum overMis}

performed by summing over the sizem=∣ ∣M and the binomial coefficient represents the number of setsMof a given size. ∣ ∣ ( ) ( ) ( ) ( ) ( ) ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 

å

å

å

å

å

å

å

rs s rs s rs s r = -= = = = - -+ = -= = Ä ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞⎠ ⎛⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠

T k D

m 1 2 2 3 3 Tr 3 4 2 9 10 9 Tr 5 6 1 5 Tr 5

6 2 . 3

k D

D

m

k D k D

D m

E D k

D

D k D

E D k D D E D k k k g g g g g g g g g 2

0 2 :

: : c c c g g g

Thefinal sum goes over subsetsDand over sequencesgsuch thatE =Dc

g , and is similar to the Fourier

decomposition ofρexcept that each terms is weighted by the factor5-D_{. In fact, a closer inspection shows that}

the weighted sum is nothing but the output state of a product of depolarising channels acting in parallel in the stateρ, where an individual depolarising channel is defined by

( ) 

 + s + s + s ⎜⎛ + s + s + s⎟

⎝ ⎞⎠

I r r r I r r r

: 1

2 :

1

2 5 5 5 .

x x y y z z x x

y

y z z

For an arbitrary quantum channel, letnp( ) ≔T sup Trt ( ( ) )T t p1 p be itsp-norm, where the supremum is

taken over all input statesτ; in particular forp ¥this becomes the¥-normn¥( ) ≔T sup_tT( )t . For the

depolarising channeldefined above, the¥-norm can be computed easily by applying the channel to an arbitrary pure state and is equal ton¥( ) =3 5. Moreover, it is known[5]that the depolarising channel has

multiplicativep-norm, i.e.n (Ä)=n ( ) _,

p k p k which implies thatÄk( )r(3 5)k. Together with(3)this

· ·

· ( )

  ⎜ ⎟⎛ ⎜ ⎟ = =

⎝ ⎞⎠ ⎛⎝ ⎞⎠

n T n n N

1 1 5

6 2

3 5

3 3

3

. 4

k

k k k

k k

Putting together(1)and(4)we obtain

( )

· ≕



_{å å}

 W W* 

n W

3

3 .

i

i i

k

k s

s, s,

We apply now the matrix Bernstein inequality in proposition2to get, for anyt>0:

( ) ·

· ( )

( )

 

 r -r  

-+

r + ₊

⎛ ⎝

⎜ ⎞

⎠ ⎟

t n t

t

2 exp 2

1 2 3 .

n

ls k

k

1 2

1

We chooset >0such that

·

· ( ) e

-+

+

+

⎛ ⎝

⎜ n t ⎞_⎠⎟

t

2 exp 2

1 2 3 ,

k

k

1 2

1

which leads, fornlarge enough, tot =n ec( )such that

( ) ·

n e

e

= ⎛ +

⎝

⎜ ⎞

⎠ ⎟

N

4 3

log 2 . c

k k

2 1

,

Acknowledgments

MGʼs work was supported by the EPSRC Grant No. EP/J009776/1.

References

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[3]Butucea C, Guta M and Kypraios T 2015 Spectral thresholding quantum tomography for low rank statesNew J. Phys.17113050

[4]Gross D 2011 Recovering low-rank matrices from few coefficients in any basisIEEE Transactions on Information Theory571548–66

[5]King C 2002 A remark on low rank matrix recovery and noncommutative Bernstein type inequalitiesJ. Math. Phys.434641

[6]Koltchinskii V 2013 A remark on low rank matrix recovery and noncommutative bersntein type inequalitiesIMS Collections9213–26

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