Linear-optical quantum information processing:

A few experiments

LinearLinear--optical quantum optical quantum information processing: information processing:

A few experimentsA few experimentsMiloslav Dušek

Lucie Čelechovská, Karel Lemr, Michal Mičuda, Antonín Černoch, Jaromír Fiurášek, Miroslav Ježek,

Jan Soubusta, Radim Filip, Konrad Kieling, Jens Eisert, Helena Fikerová, Martina Miková

Miloslav DuMiloslav Duššek ek Lucie Lucie ČČelechovskelechovskáá, , KarelKarel LemrLemr, , MichalMichal MiMiččuda, uda,

AntonAntoníínn ČČernoch, ernoch, JaromJaromíírr FiurFiurášášekek, , Miroslav JeJežžekek, , JanJan SoubustaSoubusta,, RadimRadim FilipFilip, , Konrad Kieling, Konrad Kieling,

Jens EisertJens Eisert, Helena , Helena FFikerovikerováá, Martina Mikov, Martina Mikováá

Department of Optics, Palacký University, Olomouc,

Joint Laboratory of Optics of Palacký University and Institute

of Physics of Academy of Sciences of the Czech Republic,Institute of Physics and Astronomy, University of Potsdam,

Institute for Advanced Study, Berlin.

Department of Optics, Department of Optics, PalackPalackýý UUniversityniversity, , OlomoucOlomouc, ,

Joint Laboratory of Optics of PalJoint Laboratory of Optics of Palackýacký University and Institute University and Institute

of Physics of Academy of Sciences of the Czech Republic,of Physics of Academy of Sciences of the Czech Republic,

Institute of Physics and Astronomy, University of Potsdam,Institute of Physics and Astronomy, University of Potsdam,

Institute for Advanced Study, Berlin.Institute for Advanced Study, Berlin.

� Information has a physical character – what we can do with it depends on the

physical system which carries it.

� Quantum systems behave more strangely than classical ones (superpositions

of states, entanglement, “non-locality”, intrinsic randomness and non-linearity

of quantum measurements).

� Quantum effects offer the solution of some tasks which cannot be solved in

“classical” information theory or whose classical solution is unknown (secure

distribution of cryptographic key, factorization of large numbers in polynomial time and other “hard” computational problems).

� Classical bit: 0 or 1

Quantum bit:

� Quantum register – superposition of states

of the whole register, e.g.

(entangled state)

� Information has a physical character – what we can do with it depends on the

physical system which carries it.

� Quantum systems behave more strangely than classical ones (superpositions

of states, entanglement, “non-locality”, intrinsic randomness and non-linearity

of quantum measurements).

� Quantum effects offer the solution of some tasks which cannot be solved in

“classical” information theory or whose classical solution is unknown (secure

distribution of cryptographic key, factorization of large numbers in polynomial time and other “hard” computational problems).

� Classical bit: 0 or 1

Quantum bit:

� Quantum register – superposition of states

of the whole register, e.g.

(entangled state)

Quantum information processingQuantum information processingQuantum information processing

0 1α β+0 1α β+

( )1

200 11+( )1

200 11+

� Quantum cryptography

� Secret communication: security is guarantied by the laws of physics

(secure key distribution, eavesdropping can be detected)

� Can profitably use: quantum repeaters, distillation of entanglement,

quantum memories, etc.

� Quantum computation

� Efficient (polynomial) algorithms for factorization (using quantum Fourier

transform), discrete logarithm, database search, etc.

� Can exploit: quantum correction codes, preparation of complex entangled

states, quantum teleportation, etc.

� To built quantum circuits we need quantum gates (analogous to classical

logical gates)

� Qubits in quantum gates must interact with each other in a controlled way but

must not interact with the environment (in order to keep superpositions).

This is a difficult task.

� Many platforms are considered: trapped ions, atoms in a cavity, electrons in a

quantum dot, light etc.

� Quantum cryptography

� Secret communication: security is guarantied by the laws of physics

(secure key distribution, eavesdropping can be detected)

� Can profitably use: quantum repeaters, distillation of entanglement,

quantum memories, etc.

� Quantum computation

� Efficient (polynomial) algorithms for factorization (using quantum Fourier

transform), discrete logarithm, database search, etc.

� Can exploit: quantum correction codes, preparation of complex entangled

states, quantum teleportation, etc.

� To built quantum circuits we need quantum gates (analogous to classical

logical gates)

� Qubits in quantum gates must interact with each other in a controlled way but

must not interact with the environment (in order to keep superpositions).

This is a difficult task.

� Many platforms are considered: trapped ions, atoms in a cavity, electrons in a

quantum dot, light etc.

Quantum information processingQuantum information processingQuantum information processing

� Photons are good carriers of quantum informationbut they do not interact with each other well.

� Quantum gates need controlled interaction between qubits.Interaction of qubits requires non-linearity.

� Quantum measurement is nonlinear (breaks superpositions)⇨ it can emulate nonlinearity in linear optical quantum gates (after the

measurement on an auxiliary system the state of the whole system collapses).

� But quantum measurement is also probabilistic (gives random results)⇨ linear optical quantum gates are probabilistic (sometimes fail).

� Fortunately, for many small-scale quantum computing tasks this is not a key problem. Especially for quantum information processing immediately after the quantum transmission the probability of success lower than one is not essential because the losses on the quantum channel are usually a few orders of magnitude higher.

� Linear-optical gates are experimentally feasible.

� They works directly with photons without the necessity to transfer the quantum state of a photonic qubit into another quantum system like an ion etc. (photons are useful for communication purposes).

� Photons are good carriers of quantum informationbut they do not interact with each other well.

� Quantum gates need controlled interaction between qubits.Interaction of qubits requires non-linearity.

� Quantum measurement is nonlinear (breaks superpositions)⇨ it can emulate nonlinearity in linear optical quantum gates (after the

measurement on an auxiliary system the state of the whole system collapses).

� But quantum measurement is also probabilistic (gives random results)⇨ linear optical quantum gates are probabilistic (sometimes fail).

� Fortunately, for many small-scale quantum computing tasks this is not a key problem. Especially for quantum information processing immediately after the quantum transmission the probability of success lower than one is not essential because the losses on the quantum channel are usually a few orders of magnitude higher.

� Linear-optical gates are experimentally feasible.

� They works directly with photons without the necessity to transfer the quantum state of a photonic qubit into another quantum system like an ion etc. (photons are useful for communication purposes).

Linear-optical quantum information processing

LinearLinear--optical quantum information optical quantum information

processingprocessing

Programmable unambiguous discriminator of weak coherent states

L. Bartůšková, A. Černoch, J. Soubusta, M. Dušek, Phys. Rev. A 77, 034306 (2008)

Programmable unambiguous Programmable unambiguous

discriminator of weak coherent statesdiscriminator of weak coherent statesL. BartL. Bartůůšškovkováá, A. , A. ČČernoch, J. ernoch, J. SoubustaSoubusta, M. Du, M. Duššek, ek, PhysPhys. . RevRev. A . A 7777, 034306 (2008), 034306 (2008)

An unknown state can equal to either one of the two, in general non-orthogonal,

program states.

[M. Sedlák et al., Phys. Rev. A 76, 022326 (2007)]

An unknown state can equal to either one of the two, in general non-orthogonal,

program states.

[M. [M. SedlSedláákk et al., Phys. Rev. A 76, 022326 (2007)]et al., Phys. Rev. A 76, 022326 (2007)]

1 ? 2 2 ? 1 clicks: , clicks: D Dα α α α= =1 ? 2 2 ? 1 clicks: , clicks: D Dα α α α= =

0 1 2

1 2 1, ,

2 3 3T T T= = =0 1 2

1 2 1, ,

2 3 3T T T= = =

826 nm, 4 nsActive stabilization

826 nm, 4 ns826 nm, 4 ns

Active stabilizationActive stabilization

Programmable unambiguous discriminator of weak coherent states

Programmable unambiguous Programmable unambiguous

discriminator of weak coherent statesdiscriminator of weak coherent states

2

suc 1 21 exp , is detector efficiency3

pη

α α η

= − − −

2

suc 1 21 exp , is detector efficiency3

pη

α α η

= − − −

The number of program states can be increased

⇨ quantum database search [M. Sedlák, et al., Phys. Rev. A 76, 022326 (2007)]

The number of program states can be increased The number of program states can be increased

⇨⇨ quantum database search quantum database search [M. [M. SedlSedláákk, et al., Phys. Rev. A 76, 022326 (2007)], et al., Phys. Rev. A 76, 022326 (2007)]

Encoding two qubits into a single qutritL. Bartůšková, A. Černoch, R. Filip, J. Fiurášek, J. Soubusta, M. Dušek, Phys. Rev.

A 74, 022325 (2006)

Encoding two Encoding two qubitsqubits into a single into a single qutritqutritL. BartL. Bartůůšškovkováá, A. , A. ČČernoch, R. Filip, J. ernoch, R. Filip, J. FiurFiurášášekek, J. , J. SoubustaSoubusta, M. Du, M. Duššek, ek, PhysPhys. . RevRev..

A A 7474, 022325 (2006), 022325 (2006)

( )1 2 1 2 1 2

10 1 2

Nαα α β β βΦ = + +( )1 2 1 2 1 2

10 1 2

Nαα α β β βΦ = + +

1 20 0

1 20 0

1 20 1

1 20 1

1 21 1

1 21 1

⇓⇓ ⇓⇓ ⇓⇓

}}1 1 11 1

2 2 22 2

0 1

0 1

α β

α β

Ψ = +

Ψ = +

1 1 11 1

2 2 22 2

0 1

0 1

α β

α β

Ψ = +

Ψ = +1 2

Ψ ⊗ Ψ1 2Ψ ⊗ Ψ

[A.Grudka and A.Wojcik, Phys.Lett. A 314, 350 (2003)]

[[A.Grudka and A.Wojcik, Phys.Lett. A 314, A.Grudka and A.Wojcik, Phys.Lett. A 314,

350 (2003)350 (2003)]]

Alice encodes an arbitrary pure product state of two qubits into a state of one qutrit. Bob can then restore either of the two encoded qubit states but not both of them simultaneously. Both the encoding and decoding

are probabilistic but error free.

Alice encodes an arbitrary pure product state of two Alice encodes an arbitrary pure product state of two qubitsqubits into a state of into a state of

one one qutritqutrit. Bob can then restore either of the two encoded . Bob can then restore either of the two encoded qubitqubit states states

but not both of them simultaneously. Both the encoding and decodbut not both of them simultaneously. Both the encoding and decoding ing

are probabilistic but are probabilistic but error freeerror free..

Encoding two qubits into a single qutritEncoding two Encoding two qubitsqubits into a single into a single qutritqutrit

( )1 2 4 1 2 4 1 2 4

1 2 1 2 1 2

1 3 1001 010 100 , , ,

4 4 3f f f f f f f f fT R T R R Tα α α β η β β η+ − + = = =( )

1 2 4 1 2 4 1 2 41 2 1 2 1 2

1 3 1001 010 100 , , ,

4 4 3f f f f f f f f fT R T R R Tα α α β η β β η+ − + = = =

Photon pairs at 826 nm:

SPDC in LiIO3, cw pump

Photon pairs at 826 nm: Photon pairs at 826 nm:

SPDC in LiIOSPDC in LiIO33, , cwcw pumppump

If D3 clicks there is at most

one photon in f1, f2, f4.

These three modes

constitute a qutrit.

If D3 clicks there is at most If D3 clicks there is at most

one photon in f1, f2, f4. one photon in f1, f2, f4.

These three modes These three modes

constitute a constitute a qutritqutrit..

Programmable gate for an arbitrary rotationof a single qubit along the z axis

M. Mičuda, M. Ježek, M. Dušek, J. Fiurášek, Phys. Rev. A 78, 062311 (2008)

Programmable gate for an arbitrary rotationProgrammable gate for an arbitrary rotation

of a single of a single qubitqubit along the along the zz axisaxisM. MiM. Miččuda, M. Jeuda, M. Ježžek, M. Duek, M. Duššek, J. ek, J. FiurFiurášášekek, , PhysPhys. . RevRev. A . A 7878, 062311 (2008), 062311 (2008)

It applies a unitary phase shift to a data qubit.

The value of the phase shift is determined by the state of a program qubit.[G. Vidal et al., Phys. Rev. Lett. 88, 047905 (2002)]

It applies a unitary phase shift to a data It applies a unitary phase shift to a data qubitqubit. .

The value of the phase shift is determined by The value of the phase shift is determined by

the state of a program the state of a program qubitqubit..[G. Vidal et al., Phys. Rev. [G. Vidal et al., Phys. Rev. LettLett. 88, 047905 (2002)]. 88, 047905 (2002)]

Polarization encoding is usedPolarization encoding is usedPolarization encoding is usedPhoton pairs at 814 nm: SPDC

in LiIO3, cw pump (CUBE)

Photon pairs at 814 nm: SPDC Photon pairs at 814 nm: SPDC

in LiIOin LiIO33, , cwcw pump (CUBE)pump (CUBE)

Programmable gate for an arbitrary rotationof a single qubit along the z axis

Programmable gate for an arbitrary rotationProgrammable gate for an arbitrary rotation

of a single of a single qubitqubit along the along the zz axisaxis

� An exact specification of the phase shift

would require infinitely many classical bits. Here the information is encoded into a single

qubit.

� Using a different set of program states the

device can operate as a programmable partial polarization filter.

� An exact specification of the phase shift

would require infinitely many classical bits. Here the information is encoded into a single

qubit.

� Using a different set of program states the

device can operate as a programmable partial polarization filter.

( )( )

1

2

1

2

out

Input: ,

If the program qubit is found in state

Then the data qubit is in state

i

D P

P

i

D

H V H e V

H V

H e V

φ

φ

ψ α β ψ

ψ α β

= + = +

± = ±

= ±

( )( )

1

2

1

2

out

Input: ,

If the program qubit is found in state

Then the data qubit is in state

i

D P

P

i

D

H V H e V

H V

H e V

φ

φ

ψ α β ψ

ψ α β

= + = +

± = ±

= ±

Reconstructed process matrices(real part – left, imaginary part – right).

Process fidelity exceeds 97%.

Reconstructed process matrices(real part – left, imaginary part – right).

Process fidelity exceeds 97%.

Partial-SWAP gates includingentangling square-root of SWAP

A. Černoch, J. Soubusta, L. Bartůšková, M. Dušek, J. Fiurášek, Phys. Rev. Lett. 100,180501 (2008)

PartialPartial--SWAP gates includingSWAP gates including

entangling squareentangling square--root of SWAProot of SWAPA. A. ČČernoch, J. ernoch, J. SoubustaSoubusta, L. Bart, L. Bartůůšškovkováá, M. Du, M. Duššek, J. ek, J. FiurFiurášášekek, , PhysPhys. . RevRev. . LettLett. . 100100,,

180501180501 (2008)(2008)

Photon pairs at 826 nm:

SPDC in LiIO3, cw pump

Photon pairs at 826 nm: Photon pairs at 826 nm:

SPDC in LiIOSPDC in LiIO33, , cwcw pumppump

, where ,iU e Iφφ

− −

+ − − + −Ψ Ψ= Π + Π Π = Π = − Π, where ,iU e Iφφ

− −

+ − − + −Ψ Ψ= Π + Π Π = Π = − Π

The gate is successful only if we detect a single photon in each output port.The gate is successful only if we detect a single photon in eachThe gate is successful only if we detect a single photon in each output port.output port.

(to implement Uϕ three CNOT gates are required in general)

Our scheme can reliably implement a whole class of inequivalentoperations by changing the length of one interferometer arm.

(to implement Uϕϕ three CNOT gates are required in general)

Our scheme can reliably implement a whole class of inequivalentoperations by changing the length of one interferometer arm.

0 ... identity

... SWAP2

... SWAP

φ

πφ

φ π

=

=

=

0 ... identity

... SWAP2

... SWAP

φ

πφ

φ π

=

=

=

Partial-SWAP gates includingentangling square-root of SWAP

PartialPartial--SWAP gates includingSWAP gates including

entangling squareentangling square--root of SWAProot of SWAP

( )1

2V H i H V← −( )1

2V H i H V← −

V HV HOutput 2-qubit state tomographyInput state:

Output 2Output 2--qubit state tomographyqubit state tomography

Input state: Input state:

Quantum process

tomographyfor ϕ =π/2

(matrix 16x16)

Quantum processQuantum process

tomographytomography

for for ϕϕ ==ππ/2/2

(matrix 16x16) (matrix 16x16)

Partial symmetrization and anti-symmetrization of two-qubit states

A. Černoch, J. Soubusta, L. Bartůšková, M. Dušek, J. Fiurášek, New J. Phys. 11,023005 (2009).

Partial Partial symmetrizationsymmetrization and antiand anti--

symmetrizationsymmetrization ofof twotwo--qubitqubit statesstatesA. Černoch, J. Soubusta, L. Bartůšková, M. Dušek, J. Fiurášek, New J. Phys. 11,

023005 (2009).

The filter enables an arbitrary attenuation of either the symmetric or anti-symmetric part of the input two-qubit state.

The filter enables an arbitrary attenuation of either the symmetric or anti-symmetric part of the input two-qubit state.

, where ,i

S AV T e T I

φ − −

+ − − + −Ψ Ψ= Π + Π Π = Π = − Π, where ,i

S AV T e T I

φ − −

+ − − + −Ψ Ψ= Π + Π Π = Π = − Π

Partial symmetrization and anti-symmetrization of two-qubit states

Partial Partial symmetrizationsymmetrization and antiand anti--symmetrizationsymmetrization ofof twotwo--qubitqubit statesstates

36 input states, 9 measurement bases, ϕ = 0

Tomography of output states, Process tomography

36 input states, 9 measurement bases, 36 input states, 9 measurement bases, ϕϕ = 0= 0

Tomography of output states, Process tomographyTomography of output states, Process tomography

Partial symmetrization, TS=1Partial Partial symmetrizationsymmetrization, , TTSS=1=1Partial anti-symmetrization, TA=1Partial antiPartial anti--symmetrizationsymmetrization, , TTAA=1=1

Utilization for optimal universal asymmetric quantum cloning

A. Černoch, J. Soubusta, L. Čelechovská, M. Dušek, J. Fiurášek, Phys. Rev. A 80,062306 (2009),

Utilization for optimal universal asymmetric quantum cloningUtilization for optimal universal asymmetric quantum cloning

A. Černoch, J. Soubusta, L. Čelechovská, M. Dušek, J. Fiurášek, Phys. Rev. A 80,062306 (2009),

� The phase shift can be tuned to any given value.

� For each phase shift the gate operates at the maximum success probability achievable within the framework of linear optics.

� The phase shift can be tuned to any given value.

� For each phase shift the gate operates at the maximum success probability achievable within the framework of linear optics.

Optimal controlled phase gate with an arbitrary phase shift

K. Lemr, A. Černoch, J. Soubusta, K. Kieling, J. Eisert, M. Dušek, submitted to Phys.

Rev. Lett.

Optimal controlled phase gate with Optimal controlled phase gate with

an arbitrary phase shiftan arbitrary phase shiftK. Lemr, A. Černoch, J. Soubusta, K. Kieling, J. Eisert, M. Dušek, submitted to Phys. Phys.

Rev. Lett. Rev. Lett.

Bulk optical elements

Polarization encoding of qubit states

Bulk optical elements

Polarization encoding of qubit states

[K. Kieling, J.L. O'Brien, J. Eisert, New J. Phys. 12, 13003 (2010)][K. Kieling, J.L. O'Brien, J. Eisert, New J. Phys. 12, 13003 (2010)]

, ,

, ,

, ,

, ,ie ϕ

→

→

→

→

0 0 0 0

0 1 0 1

1 0 1 0

1 1 1 1

, ,

, ,

, ,

, ,ie ϕ

→

→

→

→

0 0 0 0

0 1 0 1

1 0 1 0

1 1 1 1

Optimal controlled phase gate with an arbitrary phase shift

Optimal controlled phase gate with Optimal controlled phase gate with

an arbitrary phase shiftan arbitrary phase shift

Choi matrix for φ =πReconstructed: Ideal:

ChoiChoi matrix for matrix for φφ ==ππReconstructed: Ideal:Reconstructed: Ideal:

Choi matrix for φ =π /2

Reconstructed: Ideal:

ChoiChoi matrix for matrix for φφ ==ππ /2/2

Reconstructed: Ideal:Reconstructed: Ideal:

ReReRe

ReReRe

ImImIm

The optimum success probability

is not monotonous in the phase.

The optimum success probability

is not monotonous in the phase.

Programmable phase gate – increasing probability of success

H. Fikerová, M. Miková, M. Dušek

Programmable Programmable phase phase gate gate –– increasing increasing

probability of successprobability of successH. H. FikerovFikerováá, , MM. . MikovMikováá,, M. M. DuDuššekek

It applies a unitary phase shift to a data qubit.

The value of the phase shift is determined by the state of a program qubit.

It applies a unitary phase shift to a data It applies a unitary phase shift to a data qubitqubit. .

The value of the phase shift is determined by The value of the phase shift is determined by

the state of a program the state of a program qubitqubit..

Theoretical limit of the success probability: 50%

Success probab. of our orig. implementation: 25%

Theoretical limit of the success probability:Theoretical limit of the success probability: 50%50%

Success Success probabprobab. of our orig. implementation: . of our orig. implementation: 25%25%

( )1

20 1

0 1ie

φα β

−

−

− +→

If the program qubit is found in state

Then the data qubit is in state

We ignored these results, but we can change

( )1

20 1

0 1ie

φα β

−

−

− +→

If the program qubit is found in state

Then the data qubit is in state

We ignored these results, but we can change

Programmable phase gate – increasing probability of success

Programmable Programmable phase phase gate gate –– increasing increasing

probability of successprobability of success

Fiber-optics implementation - spatial-mode encoding is usedFiberFiber--optics implementation optics implementation -- spspatialatial--modemode encoding is usedencoding is used

Photon pairs at 814 nm: SPDC in LiIO3, cw pumpPhoton pairs at 814 nm: SPDC in LiIOPhoton pairs at 814 nm: SPDC in LiIO33, , cwcw pumppump

Success probability: 50%Success probability: Success probability: 50%50%

Electrronic feed-forward

The output quantum state is changed according to the measurement result (useful also for other experiments)

ElectrronicElectrronic feedfeed--forwardforward

The output quantum state is changed according to the The output quantum state is changed according to the

measurement result (useful also for other experiments)measurement result (useful also for other experiments)

det / 25 V, 1.5 VU Uλ≈ ≈det / 25 V, 1.5 VU Uλ≈ ≈

Other experimentsOther experimentsOther experiments

� Optimal symmetric and asymmetric phase-covariant

quantum cloning� J. Soubusta, L. Bartůšková, A. Černoch, M. Dušek, J. Fiurášek, Phys. Rev.

A 78, 052323 (2008)

� J. Soubusta, L. Bartůšková, A. Černoch, J. Fiurášek, M. Dušek, Phys. Rev. A 76, 042318 (2007)

� Bartůšková, M. Dušek, A. Černoch, J. Soubusta, J. Fiurášek, Phys. Rev. Lett. 99, 120505 (2007)

� A. Černoch, L. Bartůšková, J. Soubusta, M. Ježek, J. Fiurášek, M. Dušek, Phys. Rev. A 74, 042327 (2006)

� Programmable quantum-state discriminator and

Phase-covariant quantum multimeter� J. Soubusta, A. Černoch, J. Fiurášek, M. Dušek, Phys. Rev. A 69, 052321

(2004)

�� Optimal symmetric and asymmetric phaseOptimal symmetric and asymmetric phase--covariantcovariant

quantum cloningquantum cloning�� J. J. SoubustaSoubusta, L. Bart, L. Bartůůšškovkováá, A. , A. ČČernoch, M. Duernoch, M. Duššek, J. ek, J. FiurFiurášášekek, , PhysPhys. . RevRev. .

A A 7878,, 052323 (2008) 052323 (2008)

�� J. J. SoubustaSoubusta, L. Bart, L. Bartůůšškovkováá, A. , A. ČČernoch, J. ernoch, J. FiurFiurášášekek, M. Du, M. Duššek, ek, PhysPhys. . RevRev. .

A A 7676, 042318 (2007) , 042318 (2007)

�� BartBartůůšškovkováá, M. Du, M. Duššek, A. ek, A. ČČernoch, J. ernoch, J. SoubustaSoubusta, J. , J. FiurFiurášášekek, , PhysPhys. . RevRev. .

LettLett. . 9999, 120505 (2007) , 120505 (2007)

�� A. A. ČČernoch, L. Barternoch, L. Bartůůšškovkováá, J. , J. SoubustaSoubusta, M. Je, M. Ježžek, J. ek, J. FiurFiurášášekek, M. Du, M. Duššek, ek,

PhysPhys. . RevRev. A . A 7474, 042327 (2006), 042327 (2006)

�� PProgrammablerogrammable quantumquantum--statestate discriminatordiscriminator andand

PPhasehase--covariantcovariant quantumquantum multimetermultimeter�� J. J. SoubustaSoubusta, A. , A. ČČernoch, J. ernoch, J. FiurFiurášášekek, M. Du, M. Duššekek,, PhysPhys. . RevRev. A . A 6969, 052321 , 052321

(2004)(2004)

Thank you for your attentionThank you for your attention