Demonstration of entanglement-by-measurement of solid-state … · 2017-04-12 · Maximilian Holst,...

||

Maximilian Holst, Joshua Maas

"Demonstration of entanglement-by-measurementof solid-state qubits” Pfaff et al., Nature Physics (2013)

|| 2

Entanglement by measurement

Maximilian Holst, Joshua Maas 10.04.2017

|| 2


LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS2444

PL (arb. units)

Qubit 1

Ancilla e13C

14N

Qubit 2 Electron

56

+2,800

64

MW

frq. (MH

z)

72+13C +14N

|CN⟩ =a b c

d e f

0 25RF pulse length (µs)

50 0 80RF pulse length (µs)

160

1

P C (0

)

0

1

P N (0

)

00 1

N/π2

0

1

Ram

sey

sign

al (

a.u.

) 2

5 msπ/2 π/2 ( N)

φ

φ

|11⟩

|10⟩

|01⟩

|00⟩

|C⟩ = |1⟩

|C⟩ = |0⟩

ms = ¬1

ms = 0

|1⟩a

|0⟩a

RFC RFNRFN

Figure 1 | Entanglement by measurement and qubits in diamond. a, Schematic illustration of entanglement by measurement. Two qubits are made tointeract with an ancillary system, for instance an ancilla qubit to which both qubits couple. Subsequent readout of the ancilla can project the qubits in anentangled state without requiring a direct interaction between the qubits. b, The nitrogen-vacancy centre in diamond. The spins of a close-by 13C nucleusand the 14N nucleus of the nitrogen-vacancy centre serve as qubits. The nitrogen-vacancy electron spin is used as an ancilla. c, Energy level spectrum forthe mS = 0 to mS = �1 electron spin transition. The data show photoluminescence (PL) against the applied microwave frequency. The transition splits intosix well-resolved resonances owing to the hyperfine interactions with the 13C (hyperfine constant 12.796 MHz) and the 14N (hyperfine constant2.184 MHz), enabling conditional operations on the electron (arrows). Our definitions of the qubit and ancilla states are indicated. The vertical arrowsindicate transitions between the electron mS = 0 to mS = �1 transitions for the four different two-qubit states (dashed for the even states). d,e, Coherentsingle-qubit control of the 13C and 14N spins by radiofrequency (RF) pulses. PC(0) (PN(0)) is the probability to find the 13C (14N) spin in state |0i. Solidlines are sinusoidal fits. The error bars are smaller than the symbols. f, Ramsey-type experiment on the 14N with a 5 ms delay between the two ⇡/2-pulses.The phase �N of the second ⇡/2-pulse is swept. From the phase difference between the curves for the 13C spin prepared in |0i and in |1i we estimate adirect interaction strength between the nuclear spins of (30±13) Hz. Solid lines are sinusoidal fits. All error bars are one statistical s.d. Sample size is 1,000for d,e, and 100 for f.

|C⟩

|N⟩P̂

P̂

R π2

R ' π2

a

c d e

b

0001

1011

0001

1011

0001

1011

00

01

10

11

¬1

0

1

¬0.5

0.0

0.5

¬0.5

1

0

0.0

0.5

φ

φ

|1⟩a

|C⟩ = |0⟩

|N⟩ = |0⟩

00

01

10

11

00

01

10

11

Figure 2 | Projection into a Bell state by a non-destructive qubit parity measurement. a, Circuit diagram of the parity measurement. We condition onoutcome 0 for the ancillary electron readout. For outcome 1, the measurement is aborted. b, Circuit diagram of the protocol to create entanglement bymeasurement. We first initialize the qubits into |00i by measurement. After creating a maximal superposition state, the parity measurement projects thequbits into a Bell state. c–e, Real part of the measured density matrix after initialization (c), for the maximal superposition state (d) and for the Bell state|8+i (e). For numbers, imaginary parts and errors, see Supplementary Information.

onto eigenstates17,19,20. Possible sources of nuclear qubit dephasingduring prolonged optical readout are uncontrolled flips of theelectron spin in the excited state20 and differences in hyperfine

strength between the electronic ground and excited state21. To avoidsuch dephasing, we use a short ancilla readout time. By conditioningon detection of at least one photon (outcome |0ia), we obtain

30 NATURE PHYSICS | VOL 9 | JANUARY 2013 | www.nature.com/naturephysics

Pfaff et al., Nature Physics 9, 29–33 (2013)


|| 2



PL (arb. units)

Qubit 1

Ancilla e13C

14N

Qubit 2 Electron

56

+2,800

64

MW

frq. (MH

z)

72+13C +14N

|CN⟩ =a b c

d e f



160

1

P C (0

)

0

1

P N (0

)

00 1

N/π2

0

1

Ram

sey

sign

al (

a.u.

) 2

5 msπ/2 π/2 ( N)

φ

φ

|11⟩

|10⟩

|01⟩

|00⟩

|C⟩ = |1⟩

|C⟩ = |0⟩

ms = ¬1

ms = 0

|1⟩a

|0⟩a

RFC RFNRFN


|C⟩

|N⟩P̂

P̂

R π2

R ' π2

a

c d e

b

0001

1011

0001

1011

0001

1011

00

01

10

11

¬1

0

1

¬0.5

0.0

0.5

¬0.5

1

0

0.0

0.5

φ

φ

|1⟩a

|C⟩ = |0⟩

|N⟩ = |0⟩

00

01

10

11

00

01

10

11





| i = |00i+ |01i+ |10i+ |11i


Separable state:


|| 2



PL (arb. units)

Qubit 1

Ancilla e13C

14N

Qubit 2 Electron

56

+2,800

64

MW

frq. (MH

z)

72+13C +14N

|CN⟩ =a b c

d e f



160

1

P C (0

)

0

1

P N (0

)

00 1

N/π2

0

1

Ram

sey

sign

al (

a.u.

) 2

5 msπ/2 π/2 ( N)

φ

φ

|11⟩

|10⟩

|01⟩

|00⟩

|C⟩ = |1⟩

|C⟩ = |0⟩

ms = ¬1

ms = 0

|1⟩a

|0⟩a

RFC RFNRFN


|C⟩

|N⟩P̂

P̂

R π2

R ' π2

a

c d e

b

0001

1011

0001

1011

0001

1011

00

01

10

11

¬1

0

1

¬0.5

0.0

0.5

¬0.5

1

0

0.0

0.5

φ

φ

|1⟩a

|C⟩ = |0⟩

|N⟩ = |0⟩

00

01

10

11

00

01

10

11





| i = |00i+ |01i+ |10i+ |11i

P̂ = |00i h00|+ |11i h11|


Separable state:

Measurement:


|| 2



PL (arb. units)

Qubit 1

Ancilla e13C

14N

Qubit 2 Electron

56

+2,800

64

MW

frq. (MH

z)

72+13C +14N

|CN⟩ =a b c

d e f



160

1

P C (0

)

0

1

P N (0

)

00 1

N/π2

0

1

Ram

sey

sign

al (

a.u.

) 2

5 msπ/2 π/2 ( N)

φ

φ

|11⟩

|10⟩

|01⟩

|00⟩

|C⟩ = |1⟩

|C⟩ = |0⟩

ms = ¬1

ms = 0

|1⟩a

|0⟩a

RFC RFNRFN


|C⟩

|N⟩P̂

P̂

R π2

R ' π2

a

c d e

b

0001

1011

0001

1011

0001

1011

00

01

10

11

¬1

0

1

¬0.5

0.0

0.5

¬0.5

1

0

0.0

0.5

φ

φ

|1⟩a

|C⟩ = |0⟩

|N⟩ = |0⟩

00

01

10

11

00

01

10

11





| i = |00i+ |01i+ |10i+ |11i

P̂ = |00i h00|+ |11i h11|


P̂ | i = |00i+ |11i =��+

↵

Separable state:

Measurement:

Maximally entangled state:


|| 3

Overview

▪ Part 1 - Physical system

▪ Part 2 - Entanglement by measurement

▪ Part 3 - Results


|| 4

Overview





|| 5

NV centre


|| 5

NV centre

▪ NV = nitrogen vacancy


|| 5

NV centre


Doherty et al., Measuring the defect structure orientation of a single NV centre in diamond, New Journal of Physics, Vol. 16, 063067

centreʼs [111] major symmetry axis has four possible alignments (see figure 2(b)). The centreʼsminor symmetry axis is defined as being orthogonal to its major symmetry axis and alsocontained within one of the centreʼs three reflection planes (e.g. ¯[112]), which corresponds tothe direction joining a point on the centreʼs major symmetry axis and one of the vacancyʼsnearest neighbour carbon atoms. Considering an isolated single −NV centre, if the alignment ofits major symmetry axis is known (i.e. via magnetic field alignment), then even with knowledgeof the crystal orientation, the orientation of the centreʼs defect structure is not fully determined.

As depicted in figure 3(a), the one-electron orbital level structure of the −NV centrecontains three defect orbital levels (a1, ex and ey). EPR observations and ab initio calculationsindicate that these defect orbitals are highly localized to the centre [29–32]. Figure 3(b) showsthe centreʼs many-electron electronic structure generated by the occupation of the three defectorbitals by four electrons [33, 34], including the zero phonon line energies of the optical(1.945 eV/637 nm) [35] and infrared (1.190 eV/1042 nm) [36–38] transitions. The energyseparations of the spin triplet and singlet levels ( ↔A E2

3 1 and ↔A E11 3 ) are unknown.

As depicted in the inset of figure 3(b), the ground A23 level exhibits a zero field fine

structure splitting between the =m 0s and ±1 spin sub-levels of ∼D 2.87 GHz, which isprincipally due to first-order electron spin–spin interaction [39]. The spin quantization axis isthus defined by the trigonal unpaired electron spin density distribution of the ground A2

3 level tobe parallel to the centreʼs major symmetry axis. Spin–orbit and spin–spin mixing of the A2

3 andE3 levels makes the A2

3fine structure susceptible to electric fields, yet does not significantly

perturb the g-factor of the spin magnetic interaction from its free electron value [40].A detailed derivation of the spin-Hamiltonian that describes the A2

3fine structure in the

presence of electric and magnetic fields has been previously reported [40]. The spin-Hamiltonian derived in [40] is

γ= + − + ⃗ · ⃗ − − + +( ) ( )( )( )H D k E S S B k E S S k E S S S S2 3 , (1)z z z e x x x y y y x y y x2 2 2

Figure 2. (a) A diamond unit cell depicting the four possible alignments of the NVdefect structureʼs major symmetry axis. (b) The defect structure of an NV centre,including its major trigonal, symmetry axis z, its reflection planes and one (of the threepossible) definitions of its minor symmetry axis x.

4

New J. Phys. 16 (2014) 063067 M W Doherty et al


|| 5

NV centre


▪ Point defect in diamond ▪ One is missing ▪ One is replaced by

Doherty et al., Measuring the defect structure orientation of a single NV centre in diamond, New Journal of Physics, Vol. 16, 063067

13C

13C

13C

14N

centreʼs [111] major symmetry axis has four possible alignments (see figure 2(b)). The centreʼsminor symmetry axis is defined as being orthogonal to its major symmetry axis and alsocontained within one of the centreʼs three reflection planes (e.g. ¯[112]), which corresponds tothe direction joining a point on the centreʼs major symmetry axis and one of the vacancyʼsnearest neighbour carbon atoms. Considering an isolated single −NV centre, if the alignment ofits major symmetry axis is known (i.e. via magnetic field alignment), then even with knowledgeof the crystal orientation, the orientation of the centreʼs defect structure is not fully determined.

As depicted in figure 3(a), the one-electron orbital level structure of the −NV centrecontains three defect orbital levels (a1, ex and ey). EPR observations and ab initio calculationsindicate that these defect orbitals are highly localized to the centre [29–32]. Figure 3(b) showsthe centreʼs many-electron electronic structure generated by the occupation of the three defectorbitals by four electrons [33, 34], including the zero phonon line energies of the optical(1.945 eV/637 nm) [35] and infrared (1.190 eV/1042 nm) [36–38] transitions. The energyseparations of the spin triplet and singlet levels ( ↔A E2

3 1 and ↔A E11 3 ) are unknown.

As depicted in the inset of figure 3(b), the ground A23 level exhibits a zero field fine

structure splitting between the =m 0s and ±1 spin sub-levels of ∼D 2.87 GHz, which isprincipally due to first-order electron spin–spin interaction [39]. The spin quantization axis isthus defined by the trigonal unpaired electron spin density distribution of the ground A2

3 level tobe parallel to the centreʼs major symmetry axis. Spin–orbit and spin–spin mixing of the A2

3 andE3 levels makes the A2

3fine structure susceptible to electric fields, yet does not significantly

perturb the g-factor of the spin magnetic interaction from its free electron value [40].A detailed derivation of the spin-Hamiltonian that describes the A2

3fine structure in the

presence of electric and magnetic fields has been previously reported [40]. The spin-Hamiltonian derived in [40] is

γ= + − + ⃗ · ⃗ − − + +( ) ( )( )( )H D k E S S B k E S S k E S S S S2 3 , (1)z z z e x x x y y y x y y x2 2 2

Figure 2. (a) A diamond unit cell depicting the four possible alignments of the NVdefect structureʼs major symmetry axis. (b) The defect structure of an NV centre,including its major trigonal, symmetry axis z, its reflection planes and one (of the threepossible) definitions of its minor symmetry axis x.

4

New J. Phys. 16 (2014) 063067 M W Doherty et al


|| 6

Electronic ground state configuration


|| 6


▪ 6 electrons in the NV centre


|| 6



▪ 4 dangling bonds result in the orbitals�1,�2,�3,�N a01, a1


|| 6



▪ 4 dangling bonds result in the orbitals

▪ 2 electrons occupy the orbitals and form a spin triplet state ▪ interpret them as 1 effective “ancilla” electron with spin

�1,�2,�3,�N a01, a1

ex

, ey

S = 1


|| 7

Energy level splitting


|| 7


▪ Zero-field splitting


|| 7



▪ Zeeman splitting ▪ Magnetic field: B = 5 · 10�4 T


|| 7



▪ Zeeman splitting ▪ Magnetic field:

▪ Hyperfine splitting ▪ Nuclear spin of carbon: ▪ Nuclear spin of nitrogen:

B = 5 · 10�4 T

IC = 1/2IN = 1


|| 8

Qubits


|| 8

QubitsLETTERS NATURE PHYSICS DOI: 10.1038/NPHYS2444

PL (arb. units)

Qubit 1

Ancilla e13C

14N

Qubit 2 Electron

56

+2,800

64

MW

frq. (MH

z)

72+13C +14N

|CN⟩ =a b c

d e f



160

1

P C (0

)

0

1

P N (0

)

00 1

N/π2

0

1

Ram

sey

sign

al (

a.u.

) 2

5 msπ/2 π/2 ( N)

φ

φ

|11⟩

|10⟩

|01⟩

|00⟩

|C⟩ = |1⟩

|C⟩ = |0⟩

ms = ¬1

ms = 0

|1⟩a

|0⟩a

RFC RFNRFN


|C⟩

|N⟩P̂

P̂

R π2

R ' π2

a

c d e

b

0001

1011

0001

1011

0001

1011

00

01

10

11

¬1

0

1

¬0.5

0.0

0.5

¬0.5

1

0

0.0

0.5

φ

φ

|1⟩a

|C⟩ = |0⟩

|N⟩ = |0⟩

00

01

10

11

00

01

10

11







|| 8

Qubits

▪ Ancilla electron: |Aia = (ms)


PL (arb. units)

Qubit 1

Ancilla e13C

14N

Qubit 2 Electron

56

+2,800

64

MW

frq. (MH

z)

72+13C +14N

|CN⟩ =a b c

d e f



160

1

P C (0

)

0

1

P N (0

)

00 1

N/π2

0

1

Ram

sey

sign

al (

a.u.

) 2

5 msπ/2 π/2 ( N)

φ

φ

|11⟩

|10⟩

|01⟩

|00⟩

|C⟩ = |1⟩

|C⟩ = |0⟩

ms = ¬1

ms = 0

|1⟩a

|0⟩a

RFC RFNRFN


|C⟩

|N⟩P̂

P̂

R π2

R ' π2

a

c d e

b

0001

1011

0001

1011

0001

1011

00

01

10

11

¬1

0

1

¬0.5

0.0

0.5

¬0.5

1

0

0.0

0.5

φ

φ

|1⟩a

|C⟩ = |0⟩

|N⟩ = |0⟩

00

01

10

11

00

01

10

11







|| 8

Qubits

▪ Ancilla electron: |Aia = (ms)

|1ia = (�1)

|0ia = (0)


PL (arb. units)

Qubit 1

Ancilla e13C

14N

Qubit 2 Electron

56

+2,800

64

MW

frq. (MH

z)

72+13C +14N

|CN⟩ =a b c

d e f



160

1

P C (0

)

0

1

P N (0

)

00 1

N/π2

0

1

Ram

sey

sign

al (

a.u.

) 2

5 msπ/2 π/2 ( N)

φ

φ

|11⟩

|10⟩

|01⟩

|00⟩

|C⟩ = |1⟩

|C⟩ = |0⟩

ms = ¬1

ms = 0

|1⟩a

|0⟩a

RFC RFNRFN


|C⟩

|N⟩P̂

P̂

R π2

R ' π2

a

c d e

b

0001

1011

0001

1011

0001

1011

00

01

10

11

¬1

0

1

¬0.5

0.0

0.5

¬0.5

1

0

0.0

0.5

φ

φ

|1⟩a

|C⟩ = |0⟩

|N⟩ = |0⟩

00

01

10

11

00

01

10

11







|| 8

Qubits

▪ Ancilla electron:

▪ Qubits:

|Aia = (ms)

|CNi = (mIC ,mIN )

|1ia = (�1)

|0ia = (0)


PL (arb. units)

Qubit 1

Ancilla e13C

14N

Qubit 2 Electron

56

+2,800

64

MW

frq. (MH

z)

72+13C +14N

|CN⟩ =a b c

d e f



160

1

P C (0

)

0

1

P N (0

)

00 1

N/π2

0

1

Ram

sey

sign

al (

a.u.

) 2

5 msπ/2 π/2 ( N)

φ

φ

|11⟩

|10⟩

|01⟩

|00⟩

|C⟩ = |1⟩

|C⟩ = |0⟩

ms = ¬1

ms = 0

|1⟩a

|0⟩a

RFC RFNRFN


|C⟩

|N⟩P̂

P̂

R π2

R ' π2

a

c d e

b

0001

1011

0001

1011

0001

1011

00

01

10

11

¬1

0

1

¬0.5

0.0

0.5

¬0.5

1

0

0.0

0.5

φ

φ

|1⟩a

|C⟩ = |0⟩

|N⟩ = |0⟩

00

01

10

11

00

01

10

11







|| 8

Qubits

▪ Ancilla electron:

▪ Qubits:

|Aia = (ms)

|CNi = (mIC ,mIN )

|1ia = (�1)

|0ia = (0)

|11i = (�1/2, 1)

|10i = (�1/2, 0)

|01i = (1/2, 1)

|00i = (1/2, 0)


PL (arb. units)

Qubit 1

Ancilla e13C

14N

Qubit 2 Electron

56

+2,800

64

MW

frq. (MH

z)

72+13C +14N

|CN⟩ =a b c

d e f



160

1

P C (0

)

0

1

P N (0

)

00 1

N/π2

0

1

Ram

sey

sign

al (

a.u.

) 2

5 msπ/2 π/2 ( N)

φ

φ

|11⟩

|10⟩

|01⟩

|00⟩

|C⟩ = |1⟩

|C⟩ = |0⟩

ms = ¬1

ms = 0

|1⟩a

|0⟩a

RFC RFNRFN


|C⟩

|N⟩P̂

P̂

R π2

R ' π2

a

c d e

b

0001

1011

0001

1011

0001

1011

00

01

10

11

¬1

0

1

¬0.5

0.0

0.5

¬0.5

1

0

0.0

0.5

φ

φ

|1⟩a

|C⟩ = |0⟩

|N⟩ = |0⟩

00

01

10

11

00

01

10

11







|| 9

Overview





||Maximilian Holst, Joshua Maas 10.04.2017 10




mixed state

��+↵CN

= |00iCN + |11iCN



mixed stateinitialization

singel qubit rotations

��+↵CN

= |00iCN + |11iCN

||Maximilian Holst, Joshua Maas

three qubit gates

parity measurement

10.04.2017 10


mixed stateinitialization


��+↵CN

= |00iCN + |11iCN

||Maximilian Holst, Joshua Maas

three qubit gates

parity measurement

10.04.2017 10


mixed state

ancilla readout

initialization


��+↵CN

= |00iCN + |11iCN


Ancilla readout


Ancilla readout


NV-levelscheme

—— radiative, spin-conserving transition


Ancilla readout


NV-levelscheme

measure flourescence: Observed a photon?!

yes no

—— radiative, spin-conserving transition


Ancilla readout


NV-levelscheme


yes no

~~~ non-radiative transition—— radiative, spin-conserving transition


Ancilla readout


NV-levelscheme


yes no

Non-destructive readout sequence

Pfaff et.al., Nature Physics 9, 29–33 (2013)

~~~ non-radiative transition—— radiative, spin-conserving transition


Initialization


Parity measurement

Pfaff et.al., Nature Physics 9, 29–33 (2013)

|eveniCN


Bell state preparation

|| 15

Overview





|| 16

Part 3 - Results


|| 16

Part 3 - Results▪ State tomography

F =⌦�+

��⇢m��+

↵= (90± 3)%


|| 16


F =⌦�+

��⇢m��+

↵= (90± 3)%


PL (arb. units)

Qubit 1

Ancilla e13C

14N

Qubit 2 Electron

56

+2,800

64

MW

frq. (MH

z)

72+13C +14N

|CN⟩ =a b c

d e f



160

1

P C (0

)

0

1

P N (0

)

00 1

N/π2

0

1

Ram

sey

sign

al (

a.u.

) 2

5 msπ/2 π/2 ( N)

φ

φ

|11⟩

|10⟩

|01⟩

|00⟩

|C⟩ = |1⟩

|C⟩ = |0⟩

ms = ¬1

ms = 0

|1⟩a

|0⟩a

RFC RFNRFN


|C⟩

|N⟩P̂

P̂

R π2

R ' π2

a

c d e

b

0001

1011

0001

1011

0001

1011

00

01

10

11

¬1

0

1

¬0.5

0.0

0.5

¬0.5

1

0

0.0

0.5

φ

φ

|1⟩a

|C⟩ = |0⟩

|N⟩ = |0⟩

00

01

10

11

00

01

10

11







|| 16


▪ CHSH inequality

F =⌦�+

��⇢m��+

↵= (90± 3)%

|S| = |E(�1, ✓1)� E(�1, ✓2)� E(�2, ✓1)� E(�2, ✓2)| 2


PL (arb. units)

Qubit 1

Ancilla e13C

14N

Qubit 2 Electron

56

+2,800

64

MW

frq. (MH

z)

72+13C +14N

|CN⟩ =a b c

d e f



160

1

P C (0

)

0

1

P N (0

)

00 1

N/π2

0

1

Ram

sey

sign

al (

a.u.

) 2

5 msπ/2 π/2 ( N)

φ

φ

|11⟩

|10⟩

|01⟩

|00⟩

|C⟩ = |1⟩

|C⟩ = |0⟩

ms = ¬1

ms = 0

|1⟩a

|0⟩a

RFC RFNRFN


|C⟩

|N⟩P̂

P̂

R π2

R ' π2

a

c d e

b

0001

1011

0001

1011

0001

1011

00

01

10

11

¬1

0

1

¬0.5

0.0

0.5

¬0.5

1

0

0.0

0.5

φ

φ

|1⟩a

|C⟩ = |0⟩

|N⟩ = |0⟩

00

01

10

11

00

01

10

11







|| 16


▪ CHSH inequality

F =⌦�+

��⇢m��+

↵= (90± 3)%

|S| = |E(�1, ✓1)� E(�1, ✓2)� E(�2, ✓1)� E(�2, ✓2)| 2


b d

¬X

+Z

¬Y

a

00

0

1

1

0

01

10

11

Result

0001

1011

State prepared

Fraction of events

00 01 10State

11 00 01 10State

11 00 01 10State

11 00 01 10State

11

Ψ+ Ψ¬Φ+ Φ¬

State

1

2

⎥S⎥

c

Frac

tion

of e

vent

sFr

actio

n of

eve

nts

|Φ¬⟩

|Ψ+⟩

E = 0.71 ± 0.02 E = ¬0.51 ± 0.02 E = ¬0.44 ± 0.03 E = ¬0.62 ± 0.02

E = 0.53 ± 0.02 E = 0.68 ± 0.02 E = 0.58 ± 0.02 E = 0.54 ± 0.02

C1, N1 C2, N1 C1, N2 C2, N2

Figure 4 | Bell’s inequality violation. a, Characterization of the single-shot readout without post-selection. The readout fidelities for the four eigenstatesstates |00i, |01i, |10i and |11i are (90.0±0.9)%, (92.3±0.9)%, (92.6±0.8)% and (95.2±0.7)%, respectively. b, For all four Bell states, we measure Susing a set of bases for which a maximum of |S| is expected. The bases correspond to rotating the 13C qubit by {⇡/4, 3⇡/4} and the 14N qubit by {0,⇡/2},all around �Y. c, Histograms of the four single-shot measurements for |8�i (300 repetitions per measurement) and |9+i (500 repetitions). d, CHSHparameters. The resulting values for |S| are 2.43±0.06, 2.28±0.05, 2.33±0.04 and 2.17±0.04 (from left to right). The dashed lines mark the classical(bottom) and quantum (top) limits. Error bars are 1 s.d.

�Y axis by angles � and ✓ , respectively (Fig. 4b). Figure 4c showsthe resulting data for |8�i and |9+i. We determine S=E(�1,✓1)�E(�1,✓2)�E(�2,✓1)�E(�2,✓2) and observe a violation of theCHSH(Clauser–Horne–Shimony–Holt) inequality, |S|2, bymore than 4s.d. for each of the four Bell states, with amean of h|S|i=2.30±0.05(Fig. 4d). The obtained values for S are lower than the theoreticalmaximum of 2

p2 owing to finite fidelities of the prepared Bell state

and the single-shot readout. The main errors arise from imperfectmicrowave⇡-pulses. On the basis of the separate characterization ofa created Bell state (Fig. 2e) and of the readout (Fig. 4a) we expectS = 2.31± 0.09, in agreement with the experimental result. For aperfect readout, a value of S=2.5±0.1would be obtained.

This violation of Bell’s inequalitywithout assuming fair samplingdemonstrates that our heralded parity measurement creates high-purity entangled states that can be used as input for deterministicquantum protocols such as deterministic teleportation. In contrast,early pioneering experiments with solid-state nuclear spins con-sidered a subset of the full state and generated pseudo-pure statesthat contain no entanglement16,26. Therefore, our work constitutesthe first unambiguous demonstration of entanglement betweennuclear spins in a solid.

We have generated entanglement between two nuclear spinsin diamond through a qubit parity measurement. Our schemedoes not require a direct interaction between qubits and uses thefast but more fragile electron spin exclusively as an ancilla forthe measurement. The protocol can be directly applied to otherhybrid electron–nuclear systems such as phosphorous donors insilicon27,28. The generation of entanglement within a local registercan be supplemented with remote entanglement through opticalchannels7,29–31 to enable scalable quantum networks. Moreover,the presented parity measurements are a primary building block

for deterministic measurement-based controlled NOT gates8 andquantum error correction14. Therefore, our results mark an impor-tant step towards quantum computation in the solid state based onentangling,manipulating and protecting qubits bymeasurement.

MethodsSample and set-up. We use a naturally occurring nitrogen-vacancy centre inhigh-purity type IIa chemical-vapour-deposition-grown diamond with a h111icrystal orientation. Details of the experimental set-up are given in ref. 20. Allexperiments are performed at temperatures between 8.7 and 8.85 K and with anapplied magnetic field of ⇠5G. We determine the following dephasing times (T ⇤

2 )by Ramsey-type measurements: (1.1±0.02) µs for the electron spin, (3.0±0.2)msfor the 13C nuclear spin and (11.0±0.7)ms for the 14N nuclear spin. The spectrumof the m

s

= 0 to ms = �1 transitions in Fig. 1c is measured by electron spinresonance with off-resonant optical excitation.

Nuclear spin initialization and readout. The nuclear spins are initialized bymeasurement. First the ancilla electron is initialized by depleting |0ia by opticalpumping. Second the ancilla is flipped conditionally on state |00i and read out.Successful initialization in |0ia|00i is heralded by a measurement outcome |0ia.The two-qubit state is read out in two steps. First we initialize the ancilla in |1ia.Second we sequentially probe each two-qubit eigenstate by flipping the ancillaconditional on the state probed and then reading out the ancilla. The resultof the two-qubit readout is given by the first eigenstate for which the ancillareadout outcome is |0ia. In the CHSH experiments we obtain single-shot readoutby repeating the probing sequence until ancilla outcome |0ia is obtained forone of the probed states.

For further details on the methods, characterization and error analysis see theSupplementary Information.

Received 7 June 2012; accepted 10 September 2012;published online 14 October 2012

References1. Raussendorf, R. & Briegel, H. J. A one-way quantum computer. Phys. Rev. Lett.

86, 5188–5191 (2001).



PL (arb. units)

Qubit 1

Ancilla e13C

14N

Qubit 2 Electron

56

+2,800

64

MW

frq. (MH

z)

72+13C +14N

|CN⟩ =a b c

d e f



160

1

P C (0

)

0

1

P N (0

)

00 1

N/π2

0

1

Ram

sey

sign

al (

a.u.

) 2

5 msπ/2 π/2 ( N)

φ

φ

|11⟩

|10⟩

|01⟩

|00⟩

|C⟩ = |1⟩

|C⟩ = |0⟩

ms = ¬1

ms = 0

|1⟩a

|0⟩a

RFC RFNRFN


|C⟩

|N⟩P̂

P̂

R π2

R ' π2

a

c d e

b

0001

1011

0001

1011

0001

1011

00

01

10

11

¬1

0

1

¬0.5

0.0

0.5

¬0.5

1

0

0.0

0.5

φ

φ

|1⟩a

|C⟩ = |0⟩

|N⟩ = |0⟩

00

01

10

11

00

01

10

11








|| 16


▪ CHSH inequality

▪ Correlation measurement

F =⌦�+

��⇢m��+

↵= (90± 3)%

|S| = |E(�1, ✓1)� E(�1, ✓2)� E(�2, ✓1)� E(�2, ✓2)| 2


b d

¬X

+Z

¬Y

a

00

0

1

1

0

01

10

11

Result

0001

1011

State prepared

Fraction of events

00 01 10State

11 00 01 10State

11 00 01 10State

11 00 01 10State

11

Ψ+ Ψ¬Φ+ Φ¬

State

1

2

⎥S⎥

c

Frac

tion

of e

vent

sFr

actio

n of

eve

nts

|Φ¬⟩

|Ψ+⟩

E = 0.71 ± 0.02 E = ¬0.51 ± 0.02 E = ¬0.44 ± 0.03 E = ¬0.62 ± 0.02

E = 0.53 ± 0.02 E = 0.68 ± 0.02 E = 0.58 ± 0.02 E = 0.54 ± 0.02

C1, N1 C2, N1 C1, N2 C2, N2










s






86, 5188–5191 (2001).



PL (arb. units)

Qubit 1

Ancilla e13C

14N

Qubit 2 Electron

56

+2,800

64

MW

frq. (MH

z)

72+13C +14N

|CN⟩ =a b c

d e f



160

1

P C (0

)

0

1

P N (0

)

00 1

N/π2

0

1

Ram

sey

sign

al (

a.u.

) 2

5 msπ/2 π/2 ( N)

φ

φ

|11⟩

|10⟩

|01⟩

|00⟩

|C⟩ = |1⟩

|C⟩ = |0⟩

ms = ¬1

ms = 0

|1⟩a

|0⟩a

RFC RFNRFN


|C⟩

|N⟩P̂

P̂

R π2

R ' π2

a

c d e

b

0001

1011

0001

1011

0001

1011

00

01

10

11

¬1

0

1

¬0.5

0.0

0.5

¬0.5

1

0

0.0

0.5

φ

φ

|1⟩a

|C⟩ = |0⟩

|N⟩ = |0⟩

00

01

10

11

00

01

10

11








|| 16


▪ CHSH inequality

▪ Correlation measurement

F =⌦�+

��⇢m��+

↵= (90± 3)%

|S| = |E(�1, ✓1)� E(�1, ✓2)� E(�2, ✓1)� E(�2, ✓2)| 2


b d

¬X

+Z

¬Y

a

00

0

1

1

0

01

10

11

Result

0001

1011

State prepared

Fraction of events

00 01 10State

11 00 01 10State

11 00 01 10State

11 00 01 10State

11

Ψ+ Ψ¬Φ+ Φ¬

State

1

2

⎥S⎥

c

Frac

tion

of e

vent

sFr

actio

n of

eve

nts

|Φ¬⟩

|Ψ+⟩

E = 0.71 ± 0.02 E = ¬0.51 ± 0.02 E = ¬0.44 ± 0.03 E = ¬0.62 ± 0.02

E = 0.53 ± 0.02 E = 0.68 ± 0.02 E = 0.58 ± 0.02 E = 0.54 ± 0.02

C1, N1 C2, N1 C1, N2 C2, N2










s






86, 5188–5191 (2001).


NATURE PHYSICS DOI: 10.1038/NPHYS2444 LETTERS

+Z

N¬X

NY

+Y ¬X

Cα

α

a

d e

c

0

1

P (e

ven)

b

0Z-basis

1

N¬X

P (e

ven)

PC (0)NY

0 1/2 1 3/2 2α/π

0 1/2 1 3/2 2α/π

0

Frac

tion

of e

vent

s

1

00 01 10State

11 00 01 10State

11

|1⟩a

|N⟩

|C⟩

|Ψ¬⟩ |Ψ+⟩

|Ψ+⟩

|Ψ¬⟩

P C (

0)

P (e

ven)

Figure 3 | Bell state analysis. a, Readout in different single-qubit bases. The lines represent different measurement bases, achieved by single-qubitrotations before readout along Z. b, Measurement of |8+i in different bases, as defined in a. Probability for even correlations P (even) (outcome |00i or|11i) versus the angle ↵ between �X and the measurement basis for the 13C spin, C↵ , for two different measurement bases for the 14N spin, N�X and NY

(filled dots). Solid lines are fits to the expected sinusoidal transformation behaviour. The probability for the single-qubit result 0 for the 13C spin, PC(0),shows no dependence on the measurement basis (open dots). Results in the Z-basis (no rotations applied after projection) are shown on the right-handside. c, Adapted Toffoli gates for projection into the odd two-qubit subspace. d, Histograms of readout results (240 repetitions) for |9+i and |9�i in theZ-basis of both qubits. e, Measurement for the Bell states |9±i in bases C↵ and N�X . Solid lines are sinusoidal fits. All error bars are 1 s.d. The sample sizeis 240 for each data point.

a parity measurement that is highly non-destructive (measuredancilla post-readout state fidelity of (99±1)%) at the cost of a lowersuccess probability (⇠3%).

Our parity measurement is ideally suited to generate high-fidelity entangled states that are heralded by the measurementoutcome. To also enable deterministic quantum gates, the mea-surement must yield a high-fidelity single-shot outcome for bothmeasurement results |0ia and |1ia. This can be achieved by furtherimprovement of the electron readout, for instance by increasingthe photon collection efficiency22 or by reducing electron spin flipsthrough proper tuning of electronic levels23.

We apply the parity measurement to project the two nuclearspin qubits into a Bell state. Figure 2b shows the circuit diagram ofthe protocol. We track the two-qubit state evolution by performingquantum state tomography at three stages. First, the qubits areinitialized by projectivemeasurement into |00i (Fig. 2c). The ancillais reset to |1ia to be re-used in the subsequent parity measurementand final readout. Then, we create a maximal superposition byapplying a ⇡/2-pulse to each of the qubits, so that the two-qubitstate is |CNi = (|00i + |01i + |10i + |11i)/2 (Fig. 2d). This statecontains no two-qubit correlations. Finally, the paritymeasurementprojects the nuclear spins into the even subspace, and thus into theBell state |8+i= (|00i+|11i)/

p2. We find a fidelity with the ideal

state of F = h8+|⇢m|8+i = (90± 3)%, where ⇢m is the measureddensity matrix (Fig. 2e). The deviation from a perfect Bell statecan be fully explained by imperfect microwave ⇡-pulses that resetthe ancilla to |1ia after the projection steps. The high fidelity ofthe output state is consistent with the non-destructive nature ofthe parity measurement.

For amaximally entangled state, a measurement of a single qubityields a random result, whereas two-qubit correlations aremaximal.We access these correlations through two-qubit measurementsin different bases (Fig. 3a). The ⇡/2 pulses implementing thebasis rotations effectively transform the original state into otherBell states, which results in oscillations in the parity (Fig. 3b).

In contrast, the single-qubit outcomes are found to be random,independent of themeasurement basis (Fig. 3b).

The parity measurement can also project the qubits directlyinto each of the other Bell states. We create the states |9+i =(|01i+|10i)/

p2 and |9�i = (|01i� |10i)/

p2 by projecting into

the odd subspace (Fig. 3c). The phase of the resulting state ispre-set deterministically by adjusting the phase of the pulsesthat create the initial superposition. We characterize the states|9+i and |9�i by correlation measurements in different bases(Fig. 3d,e). The visibility yields a lower bound for the statefidelity of (91± 1)% and (90± 1)%, respectively (SupplementaryMethods). These results are consistent with the value obtained fromquantum state tomography (Fig. 2e) and confirm the universalnature of our scheme. Furthermore, our results indicate howthe parity measurement could be used to implement a non-destructive Bell-state analyser8,13. Although |9+i and |9�i showidentical odd-parity correlations in the Z -basis (Fig. 3d), theycan be distinguished by a second parity measurement after abasis rotation (Fig. 3e).

Finally, we use our measurement-based scheme to observe aviolation of Bell’s inequality with spins in a solid. This experimentplaces high demands on both the fidelity of the entangled stateand on its readout24, and therefore provides a pertinent benchmarkfor quantum computing implementations. We adapt the readoutprotocol to obtain a measurement of the complete two-qubit statein a single shot (Fig. 4a) and therefore do not rely on a fair-samplingassumption24,25. To fully eliminate the need for post-selection, weconfirm before each experimental run that the nitrogen-vacancycentre is in its negative charge state19 and that the optical transitionsare resonant with the readout and pump laser20.

We project into each of the four Bell states |8±i and |9±i andmeasure the correlation function E(�,✓) = P�,✓ (00)+ P�,✓ (11)�P�,✓ (01)� P�,✓ (10) for all combinations of the Bell angles �1,2 ={⇡/4, 3⇡/4} and ✓1,2 = {0, ⇡/2}. P�,✓ (X) is the probability tomeasure stateX after a rotation of the 13C and 14N qubits around the

NATURE PHYSICS | VOL 9 | JANUARY 2013 | www.nature.com/naturephysics 31


PL (arb. units)

Qubit 1

Ancilla e13C

14N

Qubit 2 Electron

56

+2,800

64

MW

frq. (MH

z)

72+13C +14N

|CN⟩ =a b c

d e f



160

1

P C (0

)

0

1

P N (0

)

00 1

N/π2

0

1

Ram

sey

sign

al (

a.u.

) 2

5 msπ/2 π/2 ( N)

φ

φ

|11⟩

|10⟩

|01⟩

|00⟩

|C⟩ = |1⟩

|C⟩ = |0⟩

ms = ¬1

ms = 0

|1⟩a

|0⟩a

RFC RFNRFN


|C⟩

|N⟩P̂

P̂

R π2

R ' π2

a

c d e

b

0001

1011

0001

1011

0001

1011

00

01

10

11

¬1

0

1

¬0.5

0.0

0.5

¬0.5

1

0

0.0

0.5

φ

φ

|1⟩a

|C⟩ = |0⟩

|N⟩ = |0⟩

00

01

10

11

00

01

10

11









|| 17

Correlation measurement


|| 17




+Z

N¬X

NY

+Y ¬X

Cα

α

a

d e

c

0

1

P (e

ven)

b

0Z-basis

1

N¬XP

(eve

n)

PC (0)NY

0 1/2 1 3/2 2α/π

0 1/2 1 3/2 2α/π

0

Frac

tion

of e

vent

s

1

00 01 10State

11 00 01 10State

11

|1⟩a

|N⟩

|C⟩

|Ψ¬⟩ |Ψ+⟩

|Ψ+⟩

|Ψ¬⟩

P C (

0)

P (e

ven)











p2 and |9�i = (|01i� |10i)/







|| 17


Pfaff et al., Nature Physics 9, 29–33 (2013)Pfaff et al., Nature Physics 9, 29–33 (2013)


+Z

N¬X

NY

+Y ¬X

Cα

α

a

d e

c

0

1

P (e

ven)

b

0Z-basis

1

N¬XP

(eve

n)

PC (0)NY

0 1/2 1 3/2 2α/π

0 1/2 1 3/2 2α/π

0

Frac

tion

of e

vent

s

1

00 01 10State

11 00 01 10State

11

|1⟩a

|N⟩

|C⟩

|Ψ¬⟩ |Ψ+⟩

|Ψ+⟩

|Ψ¬⟩

P C (

0)

P (e

ven)











p2 and |9�i = (|01i� |10i)/







+Z

N¬X

NY

+Y ¬X

Cα

α

a

d e

c

0

1

P (e

ven)

b

0Z-basis

1

N¬XP

(eve

n)

PC (0)NY

0 1/2 1 3/2 2α/π

0 1/2 1 3/2 2α/π

0

Frac

tion

of e

vent

s

1

00 01 10State

11 00 01 10State

11

|1⟩a

|N⟩

|C⟩

|Ψ¬⟩ |Ψ+⟩

|Ψ+⟩

|Ψ¬⟩

P C (

0)

P (e

ven)











p2 and |9�i = (|01i� |10i)/







��+↵= |00i+ |11i

|| 18

Two-qubit readout


|| 18

Two-qubit readout



6

10µs

πa

10µs

πb

10µs

πc

10µs

πd

Ex

MW

Ex

Condition

A2

MW π00 π

200ns200µs6µs

10µs150µs 0 ≥1 0

0: continue1: abort

optional

10µs

π11

10µs

π’11ms=+1ms=-1

2x

3x

2xrepeat for 10, 01, and 00

Ex

MW

π00 π11 π

Ex

A2

MW

Condition ≥1

0: continue1: abort

200ns6µs

π/2(φ)π/2(φ′)

RFC

RFN

N(θ)C(θ′)

RFC

RFNReadout

100µs100µs

10µs

≥30False

True

Ex

A2

Green

Condition

Verify/resetcharge &

resonance

FailFail Fail

Init

Init

π/2(φ)π/2(φ′)

RFC

RFN

N(θ)C(θ′)

RFC

RFNParity

Parity

ReadoutVerify/resetcharge &

resonanceFailFail Fail

next run

next run

A

B

FIG. S7. Register readout. We read out the two-qubit eigenstates (denoted a, b, c, d) in succession by mapping them sequentiallyonto the electron by applying a conditional ⇡-pulse, and then reading out the electron. The order in which we probe dependson the type of readout, see text.

correct.

A B

00 01 10 11

1

0

0001

1011

state prepared

result

00 01 10 11

1

0

0001

1011

state prepared

result 0

1fraction of events

FIG. S8. Characterization of the two-qubit readout. (A) Raw characteristics for the readout using permutations, and (B),after calibration. We first initialize into an eigenstate and subsequently probe all two-qubit states.

Readout for the CHSH measurements In order to obtain a single-shot readout of the two-qubit register, we modifythe readout protocol as shown in figure S9. We do not use permutations of the readout order, so the exact samereadout is used throughout the experiment.Note that we not only probe the computational basis states |xyi, but also the corresponding states |xy0i in the

m

s

= +1 manifold. During optical excitation, the electron spin can flip from m

s

= 0 to either ms

= �1 or +1. Thus,probing only the �1 states can lead to the state being trapped in m

s

= +1.The result of the readout is obtained as follows. We look for the first probe where the obtained number of photons

exceeds a threshold (initially two), which determines the readout result. If no attempt surpasses the threshold, its valueis decremented by one, and the procedure is repeated. Only in case where no photon at all has been obtained duringthe whole sequence, the measurement result remains unknown. This happens in < 0.2% of all cases for the presenteddata. When determining the CHSH parameter S, we always assume that unknown outcomes lower its absolute value(worst case scenario).

Charge and resonance condition

As shown in figure S10, we verify that the NV center is in its negative charge state and that the lasers are onresonance with the E

x

and A2

transitions. If required, we perform a reset of the electronic state by applying a greenlaser pulse. In the CHSH measurements this is done before creating the entanglement, in all other experiments at theend of the experimental sequence.

|| 19

Summary


|| 19

Summary


▪ NV centre level scheme

|| 19

Summary




PL (arb. units)

Qubit 1

Ancilla e13C

14N

Qubit 2 Electron

56

+2,800

64

MW

frq. (MH

z)

72+13C +14N

|CN⟩ =a b c

d e f



160

1

P C (0

)

0

1

P N (0

)

00 1

N/π2

0

1

Ram

sey

sign

al (

a.u.

) 2

5 msπ/2 π/2 ( N)

φ

φ

|11⟩

|10⟩

|01⟩

|00⟩

|C⟩ = |1⟩

|C⟩ = |0⟩

ms = ¬1

ms = 0

|1⟩a

|0⟩a

RFC RFNRFN


|C⟩

|N⟩P̂

P̂

R π2

R ' π2

a

c d e

b

0001

1011

0001

1011

0001

1011

00

01

10

11

¬1

0

1

¬0.5

0.0

0.5

¬0.5

1

0

0.0

0.5

φ

φ

|1⟩a

|C⟩ = |0⟩

|N⟩ = |0⟩

00

01

10

11

00

01

10

11






|| 19

Summary



▪ Single-/Multi-qubit gates


PL (arb. units)

Qubit 1

Ancilla e13C

14N

Qubit 2 Electron

56

+2,800

64

MW

frq. (MH

z)

72+13C +14N

|CN⟩ =a b c

d e f



160

1

P C (0

)

0

1

P N (0

)

00 1

N/π2

0

1

Ram

sey

sign

al (

a.u.

) 2

5 msπ/2 π/2 ( N)

φ

φ

|11⟩

|10⟩

|01⟩

|00⟩

|C⟩ = |1⟩

|C⟩ = |0⟩

ms = ¬1

ms = 0

|1⟩a

|0⟩a

RFC RFNRFN


|C⟩

|N⟩P̂

P̂

R π2

R ' π2

a

c d e

b

0001

1011

0001

1011

0001

1011

00

01

10

11

¬1

0

1

¬0.5

0.0

0.5

¬0.5

1

0

0.0

0.5

φ

φ

|1⟩a

|C⟩ = |0⟩

|N⟩ = |0⟩

00

01

10

11

00

01

10

11






|| 19

Summary




▪ Ancilla readout


PL (arb. units)

Qubit 1

Ancilla e13C

14N

Qubit 2 Electron

56

+2,800

64

MW

frq. (MH

z)

72+13C +14N

|CN⟩ =a b c

d e f



160

1

P C (0

)

0

1

P N (0

)

00 1

N/π2

0

1

Ram

sey

sign

al (

a.u.

) 2

5 msπ/2 π/2 ( N)

φ

φ

|11⟩

|10⟩

|01⟩

|00⟩

|C⟩ = |1⟩

|C⟩ = |0⟩

ms = ¬1

ms = 0

|1⟩a

|0⟩a

RFC RFNRFN


|C⟩

|N⟩P̂

P̂

R π2

R ' π2

a

c d e

b

0001

1011

0001

1011

0001

1011

00

01

10

11

¬1

0

1

¬0.5

0.0

0.5

¬0.5

1

0

0.0

0.5

φ

φ

|1⟩a

|C⟩ = |0⟩

|N⟩ = |0⟩

00

01

10

11

00

01

10

11






|| 19

Summary




▪ Ancilla readout

Entanglement of nuclear spins in NV centres)


PL (arb. units)

Qubit 1

Ancilla e13C

14N

Qubit 2 Electron

56

+2,800

64

MW

frq. (MH

z)

72+13C +14N

|CN⟩ =a b c

d e f



160

1

P C (0

)

0

1

P N (0

)

00 1

N/π2

0

1

Ram

sey

sign

al (

a.u.

) 2

5 msπ/2 π/2 ( N)

φ

φ

|11⟩

|10⟩

|01⟩

|00⟩

|C⟩ = |1⟩

|C⟩ = |0⟩

ms = ¬1

ms = 0

|1⟩a

|0⟩a

RFC RFNRFN


|C⟩

|N⟩P̂

P̂

R π2

R ' π2

a

c d e

b

0001

1011

0001

1011

0001

1011

00

01

10

11

¬1

0

1

¬0.5

0.0

0.5

¬0.5

1

0

0.0

0.5

φ

φ

|1⟩a

|C⟩ = |0⟩

|N⟩ = |0⟩

00

01

10

11

00

01

10

11







Demonstration of entanglement-by-measurement of solid-state … · 2017-04-12 · Maximilian Holst,...

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Transcript of Demonstration of entanglement-by-measurement of solid-state … · 2017-04-12 · Maximilian Holst,...