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DEPARTMENT OF PHYSICS UNIVERSITY OF TORONTO, 60 ST. GEORGE STREET, TORONTO, ONTARIO, CANADA M5S 1A7

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Daniel F. V. JAMESDepartment of Physics &Department of Physics &

Center for Quantum Information and Quantum ControlCenter for Quantum Information and Quantum ControlUniversity of TorontoUniversity of Toronto

QELS ’10, San Jose CAQFF-Quantum State Reconstruction/QFF-1

21 May 2010

Measuring and Characterizing Quantum States and Processes




Outline

1. Single qubit tomography = Polarimetry

3. Positivity: The first big problem, and its fixes.

2. Two (and more) photons (and other qubits).

4. Characterizing quantum states.

5. Quantum Processes.

6. Scalability: the second big problem.

7. Conclusions and resources.




§1: State of a Single Qubit

€

ψ =α H + βV

• Photon polarization based qubits

• Measure multiple (assumed identical) copies: frequency of “clicks” gives estimate of ||2

€

V V• Measure Single Copy by projecting on to : get answer “click” or “no click”– One bit of information about α and

: you know one of them is non-zero

polarizerVdetector




• Find relative phase of α and by performing a unitary operation before beam splitter:

e.g.:

• Frequency of “clicks” now gives an estimate of • Systematic way of getting all the data needed….

€

ψ → ′ ψ = 12

α + β( ) H + 12

α − β( )V

polarizerVdetectorQWP+HWP(unitary

transformation)

€

12 α − β 2




Stokes Parametersfilterdetector

G. G. Stokes, Trans Cambridge Philos Soc 9 399 (1852)

Measure intensity with four different filters:

€

n3 = N Tr R R ρ{ } = N Tr Π 3ρ{ }(iv) RCP€

n0 = N2 Tr ρ{ } = N

2 Tr Π 0ρ{ }(i) 50% intensity

€

n2 = N Tr D D ρ{ } ≡ N Tr Π 2ρ{ }(iii) 45o polarizer

€

n1 = N Tr H H ρ{ } ≡ N Tr Π1ρ{ }(ii) H-polarizer




Stokes Parameters

€

S0 =2n0 =N R ρ R + L ρ L( )

€

S1 =2 n1−n0( ) =N R ρ L + L ρ R( )

€

S2 =2 n2 −n0( ) =iN R ρ L − L ρ R( )

€

S3 =2 n3 −n0( ) =N R ρ R − L ρ L( )

• These 4 parameters completely specify polarization of beam• Beam is an ensemble of photons…

€

ρ =12

SiS0

σii=0

4

∑Pauli matrices




Outline






ψ =aHH +bHV +cVH +dVV

Density Matrix ρ =

a2

a* b a* c a* d

b* a b2

b* c b* d

c* a c* b c2

c* d

d* a d* b d* c d2

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

• Pure states– Ideal case

• Mixed states– Quantum state is random: need averages and

correlations of coefficients

§2: Two Qubit Quantum States




State Creation by OPDC

€

cosθ H +e iφ sinθV →

cosθVV +e iφ sinθ HH

€

Schmidt decomposition: ψ =aφϕ +b φ⊥ϕ⊥

Arbitrary state: change basis…

€

H → VV




€

na,b = N Tr Πa ⊗Πb( )ρ{ }

Coincidence Rate measurements for two photons

Two-Qubit Quantum State Tomography

Polarizing beamsplitter (PBS)

Half-wave plate(HWP)

Beam stop

QWP

HWP

PBS

Ar+

laser

Two-crystalentangler

Quarter-waveplate (QWP)

Decoherers

HWP

Stateselector Tomographic analyzer Detectors

Filter Lens

ApertureSingle-photondetectorsource measurement




• Doesn’t give the right answer….

Linear combination of na,b yields the two-photon Stokes parameters:

€

Sa,b = N Tr σa ⊗σb( )ρ{ }

€

ρ =Sa,b

S0,0a,b=0

3

∑ σa ⊗σb( )

From the two-photon Stokes parameters, we can get an estimate of the density matrix:




Outline







§3: The First Problem…• Properties of Density matrices:

- Hermitian:- unit trace:- non-negative definite all eigenvalues are non-negative:€

ρ =ρ †

€

Tr ρ{ } =1

€

ψ ρ ψ ≥0 for all ψ

€

ρ = Sa ,bS 0,0

a ,b=0

3

∑ σ a ⊗σ b( )What about: ?

2/3 - must try

harder!

“happy beach” of positive states

“sea” of negative matrices

experimental data (with error bars)

Q: how to find the “best”

positive ρ from noisy data?




Goodness?• Data is random, with (say) a Gaussian distribution, with mean given by the expectation values determined by the density matrix:

€

P na ,b{ },ρ( ) =1

2πna ,b

Exp −na ,b −NTr Πa ⊗Πb( )ρ{ }( )

2

2na ,b

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥all a ,b{ }

∏

• find the “best” ρ by ensuring this probability is a maximum




Enforcing Positivity• We need a ρ such that

€

ψ ρ ψ ≥0 for all ψ

• Incorporate constraints in numerical search so that the eigenvalues are positive

-tedious mucking about finding the eigenvalues each step

-a bunch of Lagrange multipliers to find• “Cholesky decomposition”- André-Louis Cholesky (French Army Officer, 1875-

1918)- non-negative matrices can be written as follows:

€

ρ =T †T

€

ψ ρ ψ = ψT †T ψ

φ{ = φ φ ≥ 0




• Maximum Likelihood fit to "physical" density matrix– Density matrix must be Hermitian, normalized, non-

negative– Numerically Minimize the function:

€

f t1,t2 ,...t16( ) = Tr ρ t1,t2 ,...t16( ) Πa ⊗Πb( ){ } −na ,b( )2na ,b

a ,b=0

3

∑

Maximum Likelihood Tomography*

– where: ρ = TT†/Tr{TT†} and

€

T t1,t2 ,...t16( ) =

t1 0 0 0t5 + it6 t2 0 0

t11 + it12 t7 + it8 t3 0t15 + it16 t13 + it14 t9 + it10 t4

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

* D. F. V. James, et al., Phys Rev A 64, 052312 (2001).




Example: Measured Density Matrix(a) Real—Theoretical (b) Imaginary—Theoretical

(c) Real—Measured (d) Imaginary—Measured

0.4

0.2

0

0.4

0.2

0

0.4

0.2

0

0.4

0.2

0

HHHV

VHVV

VV

VH

HV

HH

VV

VH

HV

HH

HHHV

VHVV

VV

VH

HV

HH

HHHV

VHVV

HHHV

VHVV

VV

VH

HV

HH




Quantum State Tomography I•Sublevels of Hydrogen (partial) (Ashburn et al, 1990)•Optical mode (Raymer et al., 1993)•Molecular vibrations (Walmsley et al, 1995)•Motion of trapped ion (Wineland et al., 1996)•Motion of trapped atom (Mlynek et al., 1997)•Liquid state NMR (Chaung et al, 1998)•Entangled Photons (Kwiat et al, 1999)•Entangled ions (Blatt et al., 2002; 8 ions: 2005)•Superconducting qubits (Martinis et al., 2006)




Outline








§4: Characterizing the StatePurity

€

Entropy S =−Tr ρ lnρ{ } =− λi lnλii∑

€

"Linear Entropy" 43 1−Tr ρ2{ }( )

€

F ρ1,ρ2( ) = Tr ρ1ρ2 ρ1{ }[ ]2

Fidelity: how close are two states?

€

F ψ1,ψ2( ) = ψ1 ψ22

Pure states:

Mixed states: doesn’t work:

€

Tr ρ1ρ2{ }

€

Tr ρ2{ } ≠ 1

€

"Purity" Tr ρ2{ }




• Pure states

• How much entanglement is in this state?

Measures of Entanglement

ψ =α HH +βHV +γVH +δVV

€

C =2αδ −γ–Concurrence:

E =h 1− 1−C 2( ) 2[ ] , where h x[ ] =−x ln x( ) − 1−x( )ln1−x( )

–Concurrence is equivalent to Entanglement :

–C=0 implies separable state

–C=1 implies maximally entangled state (e.g. Bell states)

€

E = −Tr ρA ln ρA( ){ } where ρA =TrB ψ ψ{ }–Entropy of reduced density matrix of one photon




• Mixed states can be de-composed into incoherent sums of pure (non-orthogonal) states:

Entanglement in Mixed States

ρ = pi ψ i ψ ii∑

C = piC ψ i( )i∑

•“Average” Concurrence: dependent on decomposition

C min =minψ i{ }

piC ψ i( )i∑

•“Minimized Average Concurrence”:

–Independent of decomposition

–C=0 implies separable state

–C=1 implies maximally entangled state (e.g. Bell states)

–Analytic expression (Wootters, ‘98) makes things very convenient!




€

R =ρΣρTΣ

€

Σ =

0 0 0 −10 0 1 00 1 0 0−1 0 0 0

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

Two Qubit Mixed State Concurrence

Transpose (in computational basis)

“spin flip matrix”

€

C =Max λ1 − λ2 − λ3 − λ4 ,0[ ]

Eigenvalues of R(in decreasing order)

W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998)




“Map” of Hilbert Space*1.0

0.8

0.6

0.4

0.2

0.20 0.4 0.5 0.8 1.0

Linear entropy

Classical states:along T = 0 axis

Maximallymixed state

Forbidden region

Maximally entangled(Bell states)

* D.F.V. James and P.G .Kwiat, Los Alamos Science, 2002

MEMS states

*W. J. Munro et al., Phys Rev A 64, 030302-1 (2001)

€

ρMEMS C( ) =

g C( ) 0 0 C / 20 1−2g C( ) 0 00 0 0 0

C / 2 0 0 g C( )

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟

€

g C( ) =C / 2 if C ≥2 / 31/ 3 if C < 2 / 3

⎧ ⎨ ⎩




Outline









§5: Process Tomography• Trace Preserving Completely Positive Maps: Every thing that could possibly happen to a quantum state

“operator-sum formalism”

“Kraus operators”

€

′ ρ = E iρE i†

i

∑ ; E i†E i

i

∑ = I

€

Γν =12σm(ν ) ⊗σn(ν )

set of basis matrices, e.g.:

€

Γ1 =12σ0 ⊗σ1, Γ2 =1

2σ0 ⊗σ2 , Γ4 =1

2σ1 ⊗σ0 ,etc,etc.

€

Tr ΓμΓν{ } =δν,μTrace orthogonality:




€

Ei = εi,νΓνν∑Decompose the Kraus operators:

€

εi,νε *i,μi∑ =χνμ

χ is a Hermitian, positive 16x16 matrix(“error correlation matrix”), with the constraints-

€

χμνTr ΓμΓλΓν{ }μ,ν

∑ =Tr Γλ{ }

where-

€

′ ρ = χμνΓμρΓνμ,ν∑

then-

χ is almost like ρ“Choi-Jamiolkowski isomorphism”




• Estimate probability from counts 16x16 = 256 data:

• Recover χμν by linear inversion- problematic in constraining positivity- close analogy with state tomography…

Process Tomography

€

ψa{ }16 Input states

€

φb{ }16 Projection states

€

χμν

€

pab = χμν φb Γμ ψa

μν

∑ ψa Γν φb

*I. L. Chuang and M. A. Nielsen, J. Mod Op. 44, 2455 (1997)

*




• Numerically optimize

where: (256 free parameters)

• Constraints on χμν :- positive- Hermitian- additional constraint for physically allowed

process:

€

f t1,...t256( ) = χ μν t1,...t256( ) φb Γμ ψaμν

∑ ψa Γν ψb −pab ⎛

⎝ ⎜

⎞

⎠ ⎟ ⎟

2

paba ,b=1

16

∑

€

χμν =T †T

€

χμνTr ΓμΓλΓν{ }μ,ν

∑ =Tr Γλ{ }

Maximum Likelihood Process Tomography




Process Tomography of UQ Optical CNOT*

Actual CNOT Most Likely χμν matrix

*J. L. O’Brien et al., “Quantum process tomography of a controlled-NOT gate,” Phys Rev Lett, 93, 080502 (2004); quant-ph/0402166.




Outline












§6: Scalability?• record: 8 qubit W-state (Blatt et al., 2006)

• Why not more? N qubit state tomography requires 4N-1 measurements (& numerical optimization in a 4N-1 dimensional space)




Fixes?• Measurements: you can get a good guess at the density matrix with fewer measurements (it still requires exponential searching) (Aaronson, 2006)• Direct Characterization: In some cases you can get the information you need more directly, without the tedious mucking around with the density matrix (e.g. entanglement witnesses; noise characterization)• Push the envelope: How far can we go using smart computer science before we hit the wall?

- convex optimization- improved data handling and

processing-other approaches to ‘optimziation’




- ‘quick and dirty’: zero out the negative eigenvalues rather than perform an exhaustive optimization.- ‘forced purity’: we are trying to make specifc states, so why not use that fact?

Are we being too pedantic in looking for the optimal density matrix to fit a given data set, when a simpler numerical technique produces a good estimation (i.e. within the error bars)?

Is “the best” the enemy of “good enough”

Alternatives:

Both give positive matrices quickly: but are they the actual states in question?•M. Kaznady and D. F. V. James, Phys Rev A 79, 022109 (2009);

arXiv:0809.2376.




- Fidelities for 2-qubit states:

Numerical experiments- choose a state- simulate measurement data with a Poisson RNG- estimate state using code-compare estimated and actual state

MLE

Q&D

FP




Bayesian Approach*





*Robin Blume-Kohout, “Optimal,reliable estimation of quantum states,” New Journal of Physics 12 043034 (2010)




Polynomial Time Tomography?*

*S. T. Flammia et al., “Heralded Polynomial-Time Quantum State Tomography” arXiv:1002:3839; see also M. Cramer and M. Plenio, arXiv:1002.3780

1. Choose a few-parameter set of states suitable for what you are trying to do.

2. Find a protocol to find the best set of parameters to fit your data.

3. Check that the state thus recovered is a good approximation to the actual data




Outline







7. Conclusions and resources.




Conclusions

• Maybe these techniques can give reasonably good characterization of a dozen or so qubits….• Beyond that, how an we know quantum computers is doing what it’s meant to?

- well characterized components.- error correction: you can’t know if it’s bust or not, so you’d best fix it anyway.- answers are easy to check.




Resources

D.F.V. James and P.G .Kwiat, “Quantum State Entanglement: Creation, characterization, and application,” Los Alamos Science, 2002

• A gentle introduction:

• More details (books, compilation volumes etc.):U. Leonhardt, Measuring the Quantum State of Light (Cambridge, 1997) [tomography of harmonic oscillator modes via inverse Radon transforms]

Quantum State Estimation, Lecture Notes in Physics, Vol. 649, M. G. A. Paris and J. Řeháček, eds. (Springer, Heidelberg, 2004)

Asymptotic Theory of Quantum Statistical Inference: Selected Papers, edited by M. Hayashi (World Scientific, Singapore, 2005)










Thanks to...

Dr. René StockAsma Al-QasimiOmal GamelMax Kaznady

• My Group: QuickTime™ and a decompressor


• Funding Agencies:

Ardavan DarabiFaiyaz HasanTimur RvachovBassam Helou

• Collaborators/helpers:

Paul Kwait

Andrew White

Bill Munro

Christian Roos

HartmutHäffner

Steve Flammia

Robin Blume-Kohout

department of physics university of toronto, 60 st. george street, toronto, ontario, canada m5s 1a7...

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