Priority choice two-qubit tomographyzon8.physd.amu.edu.pl/~bartkiewicz/presentation_CEWQO2015.pdf · Prioritychoicetwo-qubittomography KarolBartkiewicz [email protected] FacultyofPhysics,AdamMickiewiczUniversityinPoznań,Poland

Priority choice two-qubit tomographyKarol Bartkiewicz

[email protected] of Physics, AdamMickiewicz University in Pozna, Poland

July 8, 2015

bg=whiteIn collaborationwith

Antonn CernochKarel LemrAdamMiranowicz

ARTICLE K. Bartkiewicz, A. Cernoch, K. Lemr,A.Miranowicz, arXiv:1506.01317

Karol Bartkiewicz |Priority choice two-qubit tomography 2/42

bg=whiteOutline of the presentation

1 IntroductionQuantum tomographyTomographic stability

2 Error analysisMaximum errorStandard error

3 Optimal tomographyTheoryExperiment

4 Conclusions

bg=whiteQuantum state tomography

Platos Allegory of the Cave by Jan Saenredam,according to Cornelis van Haarlem, 1604, Albertina, Vienna

bg=whiteQuantum state tomography (QST)

ApplicationsFull characterization of physical realityquantum states (full information)quantum processesquantum detectors

Negativity for mixed two-qubit statesFull QST: 16measurementsInvariants: 9 fundamental or 6 composite

ARTICLE K. Bartkiewicz, J. Beran, K. Lemr,M. Norek, A.Miranowicz,Phys. Rev. A 91, 022323 (2015).

ARTICLE K. Bartkiewicz, P. Horodecki, K. Lemr, A.Miranowicz, K. yczkowski,Phys. Rev. A 91, 032315 (2015).

bg=whiteQuantum state tomography (QST)

ApplicationsFull characterization of physical realityquantum states (full information)quantum processesquantum detectors

Negativity for mixed two-qubit statesFull QST: 16measurementsInvariants: 9 fundamental or 6 composite

ARTICLE K. Bartkiewicz, J. Beran, K. Lemr,M. Norek, A.Miranowicz,Phys. Rev. A 91, 022323 (2015).

ARTICLE K. Bartkiewicz, P. Horodecki, K. Lemr, A.Miranowicz, K. yczkowski,Phys. Rev. A 91, 032315 (2015).

Approacheslinear inversionleast-squares inversionBayesianmean estimationlinear regression estimationThere exists a number of QST protocols and choosing the best oneappeared to be not a simple task.

ARTICLE A.Miranowicz, K. Bartkiewicz, J. Perina Jr., M. Koashi, N. Imoto,F. Nori, Phys. Rev. A 90, 062123 (2014).

Approacheslinear inversionleast-squares inversionBayesianmean estimationlinear regression estimationThere exists a number of QST protocols and choosing the best oneappeared to be not a simple task.

ARTICLE A.Miranowicz, K. Bartkiewicz, J. Perina Jr., M. Koashi, N. Imoto,F. Nori, Phys. Rev. A 90, 062123 (2014).

Two-qubit density matrix is an array of real numbers x1, ..., x16 with its elements givenas follows

(x) =

x1 x2 + ix3 x4 + ix5 x6 + ix7

x2 ix3 x8 x9 + ix10 x11 + ix12x4 ix5 x9 ix10 x13 x14 + ix15x6 ix7 x11 ix12 x14 ix15 x16

Our approach can be generalized to d-dimensional states.

Two-qubit density matrix is an array of real numbers x1, ..., x16 with its elements givenas follows

(x) =

x1 x2 + ix3 x4 + ix5 x6 + ix7

x2 ix3 x8 x9 + ix10 x11 + ix12x4 ix5 x9 ix10 x13 x14 + ix15x6 ix7 x11 ix12 x14 ix15 x16

Our approach can be generalized to d-dimensional states.

Linear inversion

Ax = b x = A1b

A is the coefficient matrixb is the observation vectorx = vec() = [11,Re12, Im12,Re13, Im13, ..., 44]

T

is a real vector describing an unknown state

bg=whiteTomographic stability

An examplex = A1b

A =

[6 75 6

] A1 =

[6 75 6

]b = [0.7, 0.6]T x = [0, 0.1]T

b = [0.71, 0.59]T x = [0.13,0.01]T

IfA is almost singular, then small deviations in observations b causelarge deviations in reconstructed values x!

An examplex = A1b

A =

[6 75 6

] A1 =

[6 75 6

]b = [0.7, 0.6]T x = [0, 0.1]T

b = [0.71, 0.59]T x = [0.13,0.01]T

An examplex = A1b

A =

[6 75 6

] A1 =

[6 75 6

]b = [0.7, 0.6]T x = [0, 0.1]T

b = [0.71, 0.59]T x = [0.13,0.01]T

An examplex = A1b

A =

[6 75 6

] A1 =

[6 75 6

]b = [0.7, 0.6]T x = [0, 0.1]T

b = [0.71, 0.59]T x = [0.13,0.01]T

Gastinel-Kahan theoremdist,(A) := min

{A P,A,

: P is singular

},

dist,(A) =1

cond,(A).

ARTICLE W.Kahan,Numerical linear algebra,Canad. Math. Bull. 9, 757(1966).

Yu. I. Bogdanov, G. Brida, M. Genovese, S. P. Kulik, E. V. Moreva, andA. P. Shurupov, Phys. Rev. Lett. 105, 010404 (2010).

Gastinel-Kahan theoremdist,(A) := min

{A P,A,

: P is singular

},

dist,(A) =1

cond,(A).

ARTICLE W.Kahan,Numerical linear algebra,Canad. Math. Bull. 9, 757(1966).

Yu. I. Bogdanov, G. Brida, M. Genovese, S. P. Kulik, E. V. Moreva, andA. P. Shurupov, Phys. Rev. Lett. 105, 010404 (2010).

Condition numbercond,(A) = A, A1,

A, = maxx 6=0 Axxcond,(A) = + for a singular matrixAcond,(A) 1

ARTICLE A.M. Turing, Rounding-off errors in matrix processes,Quart. J. Mech. Appl. Math. 1, 287 (1948).

Spectral normA2 = max[svd(A)] max(A)

where the function svd(A) returns the singular values ofA.Singular value decomposition

x = A1b = (V D1UT )b =ni=1

uTi b

ivi

U = [u1, ..., un] and V = [v1, ..., vn] are the left- andright-hand singular vectors forAD = diag([1, ..., n]) is a diagonal matrix of the singularvalues i forA

Spectral normA2 = max[svd(A)] max(A)

where the function svd(A) returns the singular values ofA.Singular value decomposition

x = A1b = (V D1UT )b =ni=1

uTi b

ivi

U = [u1, ..., un] and V = [v1, ..., vn] are the left- andright-hand singular vectors forAD = diag([1, ..., n]) is a diagonal matrix of the singularvalues i forA

Condition number (A) cond2(A) =

max[svd(A)]

min[svd(A)]

svd(A) returns the singular values ofA(A) = 1 for the optimal tomography

Condition number

(A) cond2(A) =

max[eig(AA)]

min[eig(AA)]

eig(AA) returns the eigenvalues ofAA

Atkinsons inequality1

(A)

bb xx

(A)bb

ARTICLE K. E. Atkinson, An Introduction to Numerical Analysis(Wiley, New York, 1989).

bg=whiteError analysis

Modified Atkinsons inequality1

(A)

b||b + b||

xx + x

(A) bb + b

.

This follows directly from linearity of the inversion problem.ARTICLE K. Bartkiewicz, A. Cernoch, K. Lemr, A.Miranowicz,

arXiv:1506.01317

Modified Atkinsons inequality1

(A)

b||b + b||

xx + x

(A) bb + b

.

This follows directly from linearity of the inversion problem.ARTICLE K. Bartkiewicz, A. Cernoch, K. Lemr, A.Miranowicz,

arXiv:1506.01317

Implications of Atkinsons inequalitykR

4d2(A)

E kR22

State dimension d = 4ErrorE T [(x), (x+ x)] = 1

2Tr

()2

0 k := b/(b) 2

2 for the Poissonian statisticsThe distance between a physical state and its tomographicestimate + is limited!

Implications of Atkinsons inequalitykR

4d2(A)

E kR22

State dimension d = 4ErrorE T [(x), (x+ x)] = 1

2Tr

()2

0 k := b/(b) 2

2 for the Poissonian statisticsThe distance between a physical state and its tomographicestimate + is limited!

Maximum error

Emax R 2d (A)

(b)x + xb + b

know your (A) and state dimension d = 4measure b+ b and estimate (b)calculate x+ x = A1(b+ b)

The limit onmaximum error is independent!

Maximum error

Emax R 2d (A)

(b)x + xb + b

know your (A) and state dimension d = 4measure b+ b and estimate (b)calculate x+ x = A1(b+ b)

The limit onmaximum error is independent!

S0 R0 S R

Uncertainty disk = +

R0 =R

2d2(A)

R = 2d (A) (b)x+xb+b

The physical state is located on a disk of radius betweenR0 andR.The size of uncertainty disc is limited byR.

S0 R0 S R

Uncertainty disk = +

R0 =R

2d2(A)

The physical state is located on a disk of radius betweenR0 andR.The size of uncertainty disc is limited byR.

Standard error

Estandard d

2(A)

(b)x + xb + b

=R

22

if k = 1 the deviations b (b)

The limit on standard error is independent!

Themost probable range of errorskR

4d2(A)

E kR22

Chebyshevs inequality: 1 1/k2 Pr(E kR/22)Let S = R/2, then

Pr (0 E S) 12

The Chebyshevs inequality holds if R/22 is a good estimate ofEstandard, but also ifEstandard R/22 (it is the case!).

S0 R0 S R

Standard error = +

S =d (A) (b)x+xb+b

S0 =S

2d2(A)

With probability > 1/2 the physical state is located on a disk ofradius between S0 and S.

S0 R0 S R

Standard error = +

S =d (A) (b)x+xb+b

S0 =S

2d2(A)

With probability > 1/2 the physical state is located on a disk ofradius between S0 and S.

S0 R0 S R

Error estimation = +

R0 =R

2d2(A)

S = R/2

S0 = R0/2

R or S (A) appear to be a good indicators of error robustness.

S0 R0 S R

Error estimation = +

R0 =R

2d2(A)

S = R/2

S0 = R0/2

R or S (A) appear to be a good indicators of error robustness.

bg=whiteOptimal tomography

Overview of tomographies(A) = 1 Optimal tomography: 16measurements(A) =

2 Pauli matrices: 16measurements

(A) =

5 MUB tomography: 20measurements(A) =

9 Standard tomography: 36measurements

(A) =

60.1 [James et al., 2001]: 16measurementsARTICLE A.Miranowicz, K. Bartkiewicz, J. Perina Jr., M. Koashi, N. Imoto,

F. Nori, Phys. Rev. A 90, 062123 (2014).ARTICLE K. Bartkiewicz, A. Cernoch, K. Lemr, A.Miranowicz,

arXiv:1506.01317R (A)

(A) =

arXiv:1506.01317R (A)

(A) =

arXiv:1506.01317R (A)

(A) =

60.1 [James et al., 2001]: 16measurementsARTICLE R. B. A. Adamson and A.M. Steinberg, Phys. Rev. Lett. 105,

030406 (2010).R (A)

(A) =

60.1 [James et al., 2001]: 16measurementsARTICLE J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, Advances in Atomic,

Molecular, andOptical Physics 52, 105 (2005).R (A)

(A) =

60.1 [James et al., 2001]: 16measurementsARTICLE D. F. V. James, P. G. Kwiat,W. J.Munro, and A. G.White, Phys. Rev.

A 64, 052312 (2001).R (A)

MatrixA

AT =

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 1 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

Separable (local) observablesb1

= |HHHH|, b2 = |HV HV |,b3

= |V HV H|, b4 = |V V V V |,

b5=

1

2|HH| 1, b6 =

1

2|HH| 2,

b7=

1

21 |HH|, b8 =

1

22 |HH|,

b9=

1

2|V V | 1, b10 =

1

2|V V | 2,

b11=

1

21 |V V |, b12 =

1

22 |V V |...

Pauli matrices: 1 = |DD| |AA|, 2 = |LL| |RR|

Separable (local) observablesb1

= |HHHH|, b2 = |HV HV |,b3

= |V HV H|, b4 = |V V V V |,

b5=

1

2|HH| 1, b6 =

1

2|HH| 2,

b7=

1

21 |HH|, b8 =

1

22 |HH|,

b9=

1

2|V V | 1, b10 =

1

2|V V | 2,

b11=

1

21 |V V |, b12 =

1

22 |V V |...

Pauli matrices: 1 = |DD| |AA|, 2 = |LL| |RR|

Global observablesb13

= 12(|++| ||),

b14= 12(|

++| ||),b15

= 12(|++| ||),

b16= 12(|

++| ||).

Entangled states | = (|HH |V V )/2, | = (|HV |V H)/

2, | = (|HH i|V V )/2, | = (|HV

i|V H)/

2 can be transformed into each other by local transforma-tions. It is enough to have a singlet | detector!

For composite measurements b is given by the Skellam distribution.

= 12(|++| ||),

b14= 12(|

++| ||),b15

= 12(|++| ||),

b16= 12(|

++| ||).

2, | = (|HH i|V V )/2, | = (|HV

i|V H)/

= 12(|++| ||),

b14= 12(|

++| ||),b15

= 12(|++| ||),

b16= 12(|

++| ||).

2, | = (|HH i|V V )/2, | = (|HV

i|V H)/

bg=whiteExperimental two-qubit tomography

Experimental setup

MT

INPU

T

STATEPREP.

STATETOMOGRAPHY

QWP HWP

QWP HWPQWP

HWP

QWPHWP

HWP

BS

POL

DETE

CTION

Detectiondata collection: 5slocal: BS outnonlocal: BS in

Photon pairs were generated by SPDC occurring in a pair of BBOcrystals (2.103 cc. per second for 200mWpumping@ 355nm)

ARTICLE K. Bartkiewicz, A. Cernoch, K. Lemr, A.Miranowicz,arXiv:1506.01317

Experimental setup

MT

INPU

T

STATEPREP.

STATETOMOGRAPHY

QWP HWP

QWP HWPQWP

HWP

QWPHWP

HWP

BS

POL

DETE

CTION

Experimental setup

MT

INPU

T

STATEPREP.

STATETOMOGRAPHY

QWP HWP

QWP HWPQWP

HWP

QWPHWP

HWP

BS

POL

DETE

CTION

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

0.02 0.12 0.22 0.32 0.42 0.52 0.62 0.72 0.82

E

Standard error

S0 R0 S R

With probability 1/2 the physical state is closer to than the upper limit of a bar.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

0.02 0.12 0.22 0.32 0.42 0.52 0.62 0.72 0.82

E

Standard error

S0 R0 S R

With probability 1/2 the physical state is closer to than the upper limit of a bar.

Relatively placed uncertainty discsO optimal,M MUB, S standardQSTcentresmeasured states relative positions given by trace distanceintersections contain the physical statesintersections are closest toO

Representative statesThe reconstructed states can be approximatedwith n = |nn|:|4 = (|DR i|AL)/

2

|7 = |HV |9 = (|HV |V H)/

2

|14 = |e1ae1bwhere |e1a = (0.6556 + 0.6248i)|H+ 0.4241|V and |e1b = (0.1415 0.7165i)|H+ 0.6831|V .

moremethodsmore dimensions for graphical comparison

bg=whiteConclusions

Various QSTs implementedwith the same setupQST errors estimatedwithout knowing the physical stateBest error estimates for the optimal QSTOptimal tomography is themost error robust

bg=whiteSupported by

Quantum engineering and quantumphase transitions in optical and

low-dimensional condensedmattersystems

Registry No.:DEC-2011/03/B/ST2/01903

bg=whiteSupported by

Nonlinear properties of lowdimesional quantum states: directmeasurement and applications inquantum information processing

Registry No.:DEC-2013/11/D/ST2/02638

bg=white

Thank you for yourattention!

ARTICLE K. Bartkiewicz, A. Cernoch, K. Lemr,A.Miranowicz, arXiv:1506.01317

ARTICLE

A.Miranowicz, K. Bartkiewicz, J. Perina Jr.,M. Koashi, N. Imoto, F. Nori,Phys. Rev. A 90, 062123 (2014).

bg=whiteUncertainty discs

bg=whiteMeasured states

Wehave prepared 17 different states of high purity, which approximately correspond to:

|1 = (|HH |V V )/2, |2 = (|HH+ |V V )/

2,

|3 = (|HH i|V V )/2, |4 = (|DR i|AL)/

2,

|5 = (|HV + i|V H)/2, |6 = (|HV + |V H)/

2,

|7 = |HV , |8 = (|HH+ i|V V )/2,

|9 = (|HV |V H)/2, |10 = (|HV i|V H)/

2,

|11 = (|DL+ i|AR)/2, |12 = (|DL i|AR)/

2,

|13 = |e1ae1b, |14 = |e2ae2b,|15 = 0.79|HV 0.61|V H, |16 = 0.50|HV 0.87|V H,

|17 = 0.35|HV 0.94|V H;

where |e1a = (0.6556 + 0.6248i)|H+ 0.4241|V ,|e1b = (0.1415 0.7165i)|H+ 0.6831|V ,|e2a = (0.9608 + 0.2091i)|H+ 0.1822|V ,and |e2b = (0.2613 + 0.7338i)|H+ 0.6271|V .

bg=whiteMaximum likelihood estimation

IntroductionQuantum tomographyTomographic stability

Error analysisMaximum errorStandard error

Optimal tomographyTheoryExperiment

Conclusions

fd@rm@0: fd@rm@1: