(Continuous Variables) Quantum Optics, Quantum Information

the Orthodox Academy of Crete - (Continuous Variables) Quantum Optics, Quantum Information and Relativistic Quantum Information. 1/24

(Continuous Variables) Quantum Optics, QuantumInformation and Relativistic Quantum Information

David Edward Bruschi

York Centre of Quantum TechnologiesDepartment of Physics

University of Yorkthe (now Brexited) United Kingdom

XVII August MMXVII

Partially based on: Introductory Quantum Optics, C. C. Gerry and P. L. Knight.


1 IntroductionHigh energy physicsLow energy physics

2 Quantum OpticsQuantum Optics premisesQuantum Optics implementations

3 Convariance matrix formalismGaussian statesEntanglement of bipartite Gaussian statesExamples with Gaussian states

4 (Quantum) Information TheoryQuantum Information theory: teleportationConclusions


Introduction

High energy physics

Quantum Electro Dynamics (QED)

Relativistic and quantum Lagrangian LQED for matter interacting with light

LQED = ψ (i γµDµ −m) ψ − 1

4Fµν Fµν .

- γµ are Dirac matrices;

- ψ a bispinor field of spin-1/2 particles (e.g. electron-positron field);

- ψ ≡ ψ†γ0, called ”psi-bar”, also referred to as the Dirac adjoint;

- Dµ ≡ ∂µ + i e Aµ + i e Bµ is the gauge covariant derivative;

- e is the coupling constant, i.e., the electric charge of the bispinor field;

- m is the mass of the electron or positron;

- Aµ is the covariant four-potential of the electromagnetic fieldgenerated by the electron itself;

- Bµ is the external field imposed by external source;

- Fµν = ∂µAν − ∂νAµ is the electromagnetic field tensor.


Introduction

High energy physics

Quantum Electrodynamics

High energy physics - Interacting theory

- Construct vertex from the interaction coupled-term LI ∼ e ψ γµ ψ Aµ;

- Coupling strength λ ∼ e;

- Energy, momentum, charge must be conserved in physical processes;


Introduction

Low energy physics

Semiclassical theory

Low energy physics of light and matter

In the low energy physics regime we can safely ignore details of light-matterinteraction contained in LQED. We can employ a semiclassical theory.

Classical and quantum fields (the latter with [aks , a†k′s′ ] = δ3(k − k ′) δss′)

A(x , t) =∑s

∫d3k ek,s

[Ak,s e

i (k·x−ωk t) + A∗k,s e−i (k·x−ωk t)

],

A(x , t) =∑s

∫d3k ek,s

[ak,s e

i (k·x−ωk t) + a†k,s e−i (k·x−ωk t)

].

Classical field

Classical Hamiltonian of e.m. field:

H0 =∑s

∫d3x ω(k)A∗k,s Ak,s .

Quantized field

Quantum Hamiltonian of e.m. field

H0 =∑s

∫d3k ω(k) a†k,s ak,s + E0.


Introduction

Low energy physics





A(x , t) =∑s

∫d3k ek,s

[Ak,s e


],

A(x , t) =∑s

∫d3k ek,s

[ak,s e


].

Classical field


H0 =∑s


Quantized field


H0 =∑s



Introduction

Low energy physics





A(x , t) =∑s

∫d3k ek,s

[Ak,s e


],

A(x , t) =∑s

∫d3k ek,s

[ak,s e


].

Classical field


H0 =∑s


Quantized field


H0 =∑s



Quantum Optics

Quantum Optics premises

Moving to Quantum Optics

Quantized electric and magnetic fields

E (x , t) =i∑s

∫d3k ek,s

[ak,s e


]B(x , t) =

i

c

∑s

∫d3k

(k|k |× ek,s

)[ak,s e


].

We now make important considerations:

i) Most quantum optical situations, coupling of field to matter is throughelectric field interacting with a dipole moment or through somenonlinear type of interaction involving powers of the electric field;

ii) Focus on the electric field E (x , t);

iii) Magnetic field is “weaker” than the electric field by a factor of 1c ;

iv) Field couples to the spin magnetic moment of the electrons;

v) This interaction is negligible for essentially all the aspects of QuantumOptics that we are concerned with;

vi) Negligible spatial variation of field over dimensions of atomic system.


Quantum Optics


From the electromagnetic field to modes of light

Dipole approximation:

E (x , t) ∼ ex

[a e−i ωk t + a† e i ωk t

], [a, a†] = 1.

For our purposes: replace field operator E (x , t) by single bosonic mode a.

Significant states of one-mode a

Vacuum state |0〉: a |0〉 = 0.Number state |n〉: a† a |n〉 = n |n〉,

|n〉 =

(a†)n

√n!|0〉.

Coherent state |α〉:

|α〉 = exp[α a− α∗ a†] |0〉.

Single-mode squeezed state |s〉:

|s〉 = exp[s a† 2 − s∗ a2] |0〉.

Significant states of two-modes a,c

Vacuum state |0〉: a |0〉 = c |0〉 = 0Number state |n,m〉:[a† a + b† b] |n,m〉 = (n + m) |n,m〉,

|n,m〉 =

(a†)n

√n!

(b†)m

√m!|0〉.

Two-mode squeezed state |r〉:

|r〉 = exp[r a† b† − r∗ a b] |0〉.

We have [a, b] = [a, b†] = 0.


Quantum Optics


From the electromagnetic field to modes of light

Dipole approximation:

E (x , t) ∼ ex

[a e−i ωk t + a† e i ωk t

], [a, a†] = 1.

For our purposes: replace field operator E (x , t) by single bosonic mode a.

Significant states of one-mode a

Vacuum state |0〉: a |0〉 = 0.Number state |n〉: a† a |n〉 = n |n〉,

|n〉 =

(a†)n

√n!|0〉.

Coherent state |α〉:

|α〉 = exp[α a− α∗ a†] |0〉.

Single-mode squeezed state |s〉:

|s〉 = exp[s a† 2 − s∗ a2] |0〉.

Significant states of two-modes a,c

Vacuum state |0〉: a |0〉 = c |0〉 = 0Number state |n,m〉:[a† a + b† b] |n,m〉 = (n + m) |n,m〉,

|n,m〉 =

(a†)n

√n!

(b†)m

√m!|0〉.

Two-mode squeezed state |r〉:

|r〉 = exp[r a† b† − r∗ a b] |0〉.

We have [a, b] = [a, b†] = 0.


Quantum Optics


Phase space representation of interesting states

|α〉 = exp[α a− α∗ a†] |0〉 = e−|α|22∑

nαn√n!|n〉 .

ρ(T ) = 1cosh2 r

∑n tanh2 n r |n〉〈n| , tanh r = e

− ~ωkB T .

|s〉 = exp[s a† 2 − s∗ a2] |0〉 = 1cosh s

∑n tanhn s |2 n〉 .


Quantum Optics

Quantum Optics implementations

Operations in Quantum Optics

Time evolution / transformation of states ρ: ρ(λ) = U†(λ) ρ(0)U(λ),

Linear unitary operations: effectively reduced to

Free evolution/phase shifting:U(t) = exp[−i ωa t(= φ)a† a].

Beam splitting:U(θ) = exp[−i θ (a† b + a b†)].

Single mode squeezing:U(s) = exp[(s a†,2 − s∗a2)].

Two mode squeezing:U(r) = exp[(r a† b† − r∗a b)].

φ

r

a ' = e−iφaa

a

b

s a a ' = cosh(s)a+ sinh(s)a†

a ' = cosh(r)a+ sinh(r)b†

b ' = cosh(r)b+ sinh(r)a†

a

b

b ' = cos(θ )b− sin(θ )a

a ' = cos(θ )a+ sin(θ )b

Unitary Evolution / Transformation

Non-linear unitary operations: effectively reduced to

PDC: U(ξ) = exp[ξ a† b† c − ξ∗ a b c†]. ξ c

a ' = cosh(ξc )a+ sinh(ξc )b†

b ' = cosh(ξc )b+ sinh(ξc )a†


Quantum Optics









φ

r

a ' = e−iφaa

a

b




a

b









Quantum Optics









φ

r

a ' = e−iφaa

a

b




a

b









Quantum Optics


Quantum Optics laboratory

Now use BCH: eA eB = eA+B+ 12 [A,B]+... .

Figure: Figure from phys.org


Convariance matrix formalism

Gaussian states

Gaussian states of light

Picking experimentally realisable states of light

i) Very few states can be realised in optics laboratories (i.e., not |981〉);

ii) Employable states are prepared using linear optics;

iii) These state can be manipulated with linear optics.

We choose to work with Gaussian states

Gaussian states

i) Have Gaussian Wigner function;

ii) Wigner function is positive;

iii) Are defined by finite d.o.f.;

iv) Produced in every Q.O. lab.

W (ξ) =1

π2

∫R2 N

d2NX χs(κ) e i X Ω ξ



Gaussian states







Gaussian states





W (ξ) =1

π2

∫R2 N




Gaussian states







Gaussian states





W (ξ) =1

π2

∫R2 N




Gaussian states

Covariance matrix formalism

Gaussian states

- N modes of light a1, ..., aN ;

- Introduce X := (a1, ..., aN , a†1, ..., a

†N)T ;

- Commutation relations: [an, a†m] = i Ωnm.

- Symplectic form Ω: Ωnm = diag(−i ,−i , ..., i , i , ...).

- First moments d : d := 〈X〉ρ.

- Second moments σ: σnm := 〈Xn, X†m〉ρ − 2 〈Xn〉ρ〈X†m〉ρ.

Linear transformations

- Quadratic in an, a†m;

- N-mode linear represented by 2N × 2N symplectic matrix S .

- Symplectic: S†ΩS = Ω.

- Unitary U = exp[−i H t]: represented by S = exp[−i F (t)ΩH].Here H = 1

2 XT H X.



Gaussian states


Gaussian states



†N)T ;










2 XT H X.



Gaussian states


Gaussian states



†N)T ;










2 XT H X.



Gaussian states


Gaussian states



†N)T ;










2 XT H X.



Gaussian states


Gaussian states



†N)T ;










2 XT H X.



Gaussian states


Gaussian states



†N)T ;










2 XT H X.



Gaussian states


Gaussian states



†N)T ;










2 XT H X.



Gaussian states


Gaussian states



†N)T ;










2 XT H X.



Gaussian states


Gaussian states



†N)T ;










2 XT H X.



Gaussian states


Gaussian states



†N)T ;










2 XT H X.



Gaussian states


Gaussian states

Gaussian states are defined by only first and second moments: they aredefined univocally by the covariance matrix σ and the first moments d .

Williamson Theorem

σ = S† ν⊕ S

Here ν⊕ = diag(ν1, ν2, ..., ν1, ν2, ...) is the Williamson form of σ andνk ≥ 1 are the symplectic eigenvalues of σ.

Purity

N.B. The state σ is pure iff νk = 1 for all k. The symplectic eigenvaluesare νk = coth( ~ωk

Kb Tk).



Gaussian states


Gaussian states


Williamson Theorem

σ = S† ν⊕ S


Purity

N.B. The state σ is pure iff νk = 1 for all k. The symplectic eigenvaluesare νk = coth( ~ωk

Kb Tk).



Gaussian states


Gaussian states


Williamson Theorem

σ = S† ν⊕ S


Purity

N.B. The state σ is pure iff νk = 1 for all k . The symplectic eigenvaluesare νk = coth( ~ωk

Kb Tk).



Gaussian states

Quantum optics with covariance matrix formalism

Gaussian state evolution/transformations

σ(λ) = S†(λ)σ(0) S(λ) .

Gaussian vs. non-Gaussian states

Non-Gaussian / Hilbert

i) N modes;

ii) Infinite d.o.f;

iii) Tensor product Hilbert space;

iv) Unitary linear operators;

v) Trace operation: infinite sums.

vi) Entanglement: very difficultmeasures to compute.

Gaussian / CM formalism

i) N modes;

ii) Finite d.o.f.;

iii) Direct sum CM construction;

iv) 2N × 2N symplectic matrices;

v) Trace operation: deleterows/columns.

vi) Entanglement: see next.



Gaussian states

Quantum optics with covariance matrix formalism

Gaussian state evolution/transformations

σ(λ) = S†(λ)σ(0) S(λ) .

Gaussian vs. non-Gaussian states

Non-Gaussian / Hilbert

i) N modes;

ii) Infinite d.o.f;

iii) Tensor product Hilbert space;

iv) Unitary linear operators;

v) Trace operation: infinite sums.

vi) Entanglement: very difficultmeasures to compute.

Gaussian / CM formalism

i) N modes;

ii) Finite d.o.f.;

iii) Direct sum CM construction;

iv) 2N × 2N symplectic matrices;

v) Trace operation: deleterows/columns.

vi) Entanglement: see next.



Entanglement of bipartite Gaussian states

Entanglement of Gaussian states

Separability and entanglement

The state ρAB separable if exists ρAB = ρA ⊗ ρB . If not, it is entangled.EPR state: |Ψ〉 = 1√

2[|01〉+ |10〉].

Symplectic eigenvalues νk ≥ 1: spectrum of i Ωσ.

Symplectic spectrum of the partial transpose

Compute spectrum νk of i ΩP† σP . Here P implements partialtransposition of one mode.

Of these eigenvalues νk take the smallest (in absolute value), called ν−.







2[|01〉+ |10〉].











2[|01〉+ |10〉].







Examples with Gaussian states

An example: using Hilbert space formalism

Start from the vacuum state |0〉. Use linear optics and two-mode squeezethe vacuum with U(r) := exp[−r (a† b† − a b)]. (Lots of) algebra:

|Ψ(r)〉 = U(r) |0〉 =∑n

tanhn r

cosh r|n, n〉.

The state ρ(r) is:

ρ(r) = |Ψ(r)〉〈Ψ(r)| =∑n,m

tanhn+m r

cosh2 r|n, n〉〈m,m|.

Partial transpose mode b is

ρ(r) = |Ψ(r)〉〈Ψ(r)| =∑n,m

tanhn+m r

cosh2 r|n,m〉〈m, n|.

Choose measure: Negativity N :=Tr(√ρ† ρ)−12

More algebra: N = e2 r−12 .




An example: using Covariance Matrix formalism

Start from the vacuum state σ0. Use linear optics and two-mode squeezethe vacuum with S(r). (Little) algebra:

σ0 =

1 0 0 00 1 0 00 0 1 00 0 0 1

, S(r) =

cosh r 0 0 sinh r

0 cosh r sinh r 00 sinh r cosh r 0

sinh r 0 0 cosh r

.

The state σ(r) := S†(r)σ0 S(r) is:

σ(r) =

cosh 2r 0 0 sinh 2r

0 cosh 2r sinh 2r 00 sinh 2r cosh 2r 0

sinh 2r 0 0 cosh 2r

.




An example: using Covariance Matrix formalism

Partial transpose mode b is implemented by P and obtain P† σP

P =

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

, P† σP =

cosh 2r sinh 2r 0 0sinh 2r cosh 2r 0 0

0 0 cosh 2r sinh 2r0 0 sinh 2r cosh 2r

.

Compute the spectrum of i ΩP† σP, which is

e2 r , ν− = e−2 r ,−e2 r ,−e−2 r , .

Choose measure: Negativity N := 1−ν−2 ν−

.

Simply obtain: N = e2 r−12 .


(Quantum) Information Theory

Quantum Information

Information theory aims at understanding how to...

Store

Transmit

Decode Quantify

Employ

Secure

...information

Classical IT Discrete QI CV QI

Systems Bits (“0” and “1”) Qbits (|0〉 and |1〉) Modes of lightDimensions 2N 2N Infinite

Gates Classical Boolean Unitaries Linear unitariesMeasurements Magnetic readout Projective Projective



Quantum Information

Information theory aims at understanding how to...

Store

Transmit

Decode Quantify

Employ

Secure

...information

Classical IT Discrete QI CV QI

Systems Bits (“0” and “1”) Qbits (|0〉 and |1〉) Modes of lightDimensions 2N 2N Infinite

Gates Classical Boolean Unitaries Linear unitariesMeasurements Magnetic readout Projective Projective



Quantum Information theory: teleportation

Discrete Variables

i) Generate EPR. One qubit to A, other to B;

ii) Bell measurement at A of EPR pair qubitand the qubit (|φ〉) to be teleported. Yieldone of four measurement outcomes,encoded in two classical bits;

iii) Using the classical channel, the two bitsare sent from A to B. (Speed less than c);

iv) Result of measurement at A, EPR pairqubit at B in one of four possible states.Of these, one identical to the originalquantum state |φ〉, other three are closelyrelated. Which of these four possibilitiesactually obtains is encoded in the twoclassical bits. Knowing this, the qubit atlocation B is modified to result in a qubitidentical to |φ〉.

Figure: Figure fromWikipedia




Continuous Variables

i) Generate EPR-like state: two-modesqueezed state;

ii) CV version of Bell measurement.Beam split modes a and “in”.Homodyne detect quadratures x−and p+;

iii) Classical channel, same as before;

iv) Similar as before. Use classicalinformation to perform extradisplacement. Obtain initial stateρin.

Figure: Protocol and figure based onSection IV.B in Laser Physics 16,1418 (2006)




Relativistic Quantum Teleportation

i) Generate usual squeezed state;

ii.i) Homodyne measurement at A of localmode and the mode to be teleported;

ii.ii) Rob moves! Field inside Rob’s cavity isaffected by motion. Mode from squeezedstate is mixed to other modes;

iii) Using the classical channel, the two bitsare sent from A to R. (Speed less than c);

iv) Result like before but depends also onmotion. Figure: PRL 110, 113602

(2013)



Conclusions

Conclusions

Information Theory Thermodynamics Quantum Mechanics

Relativity

Quantum Information

Black Hole Thermodyn. QFT in curved spacetime

Relativistic Quantum Information



Conclusions

Acknowledgments

Thank you

Documents

(Continuous Variables) Quantum Optics, Quantum Information