continuous variable quantum informationhhlee/slides-adesso.pdf · 2018. 10. 26. · multimode...

CONTINUOUS VARIABLE

QUANTUM INFORMATION

Gerardo Adesso

School of Mathematical Sciences

University of Nottingham (UK)

[email protected]

https://quantingham.wordpress.com

mailto:[email protected]

Plan

LECTURE I. (Wednesday 24)

• Quantum phase space methods and the symplectic group

LECTURE II. (Thursday 25)

• Gaussian states: informational properties and correlations

LECTURE III. (Friday 26)

• Gaussian channels: description and classification

LECTURE IV. (Saturday 27)

• Gaussian quantum technologies: teleportation and beyond

CONTINUOUS VARIABLE

QUANTUM INFORMATION

LECTURE I.

Qubits

*are not continuous variables systems!

Spin, polarisation, etc. are discrete

variables: upon measuring in a

given basis, you can obtain a finite,

discrete set of possible results, e.g.

0 / 1 for dichotomic variables

Continuous variables

are degrees of freedom associated to observables with a continuous spectrum,

for instance position ො𝑞 and momentum Ƹ𝑝 of a free particle.

Continuous variable (CV) systems

are systems in which the relevant degrees of freedom are continuous variables.

With respect to those degrees of freedom, the quantum states of continuous

variable systems live in an infinite-dimensional Hilbert space

0q

Motivation: CV entanglement

Einstein-Podolsky-Rosen state: contains infinite entanglement

This state is unphysical as it is unnormalisable, but if you can approximate it

then you realise a very powerful resource for quantum technologies ...

0q

Quantum harmonic oscillator

• Canonical commutation relations ො𝑞, Ƹ𝑝 = 𝑖ℏ

• Ladder operators

• Annihilation ො𝑎 =𝑚𝜔

2ℏො𝑞 +

𝑖

𝑚𝜔Ƹ𝑝

• Creation ො𝑎† =𝑚𝜔

2ℏො𝑞 −

𝑖

𝑚𝜔Ƹ𝑝

• Bosonic commutation relations ො𝑎, ො𝑎† = 1

• Number operator: ො𝑛 = ො𝑎† ො𝑎

• Hilbert space = Fock space

• Basis: Fock states 𝑛 ∶ ො𝑛 𝑛 = 𝑛 𝑛

Physical realisations

We need pairs of operators which satisfy the canonical commutation relations

… e.g. amplitude/phase quadratures of light

(q=“electric field”, p=“magnetic field”)

… or collective magnetic moment of atomic

ensembles (q=Sx/, p=Sy/)

Quantized electromagnetic field

• Each oscillator: a mode of the field

• Natural units: ℏ = 𝑐 = 1

• ො𝑎𝑘 =1

2ො𝑞𝑘 + 𝑖 Ƹ𝑝𝑘 , ො𝑎𝑘

† =1

2ො𝑞𝑘 − 𝑖 Ƹ𝑝𝑘

• ො𝑞𝑘 =1

2ො𝑎𝑘 + ො𝑎𝑘

† , Ƹ𝑝𝑘 =1

𝑖 2ො𝑎𝑘 − ො𝑎𝑘

†

Quantized electromagnetic field

• We introduce a vector of canonical operators: 𝑅 = ො𝑞1, Ƹ𝑝1, ො𝑞2, Ƹ𝑝2, … , ො𝑞𝑁, Ƹ𝑝𝑁

• Canonical commutation relations (CCR):

• ො𝑞𝑗 , Ƹ𝑝𝑘 = 𝑖 ℏ 𝛿𝑗𝑘• ො𝑞𝑗 , ො𝑞𝑘 = 0, Ƹ𝑝𝑗 , Ƹ𝑝𝑘 = 0

• Introducing the N-mode symplectic matrix Ω𝑁 = Ω⊕𝑁, with Ω =

0 1−1 0

,

then we can write the CCR more compactly: 𝑅𝑗 , 𝑅𝑘 = 𝑖 Ω𝑁 𝑗𝑘

Canonical commutation relations

• Introducing the N-mode symplectic matrix Ω𝑁 = Ω⊕𝑁, with Ω =

0 1−1 0

,

then we can write the CCR more compactly: 𝑅𝑗 , 𝑅𝑘 = 𝑖 Ω𝑁 𝑗𝑘

• Notice – there is no representation of the CCR algebra through finite

dimensional matrices, but there is an infinite-dimensional representation:

ො𝑞𝑗𝜓 𝒙 = 𝑥𝑗𝜓 𝒙

Ƹ𝑝𝑗𝜓 𝒙 = −𝑖𝜕

𝜕𝑥𝑗𝜓(𝒙)

with 𝜓 𝒙 ∈ 𝐿2 ℝ𝑁

• The Stone-Von Neumann Theorem: for a finite N, all representations of the

CCR algebra are unitarily equivalent to the one above, so we can rewrite

the CCR by exponentiation to get bounded operators (Weyl relations):

𝑒𝑖 ො𝑞𝑗− ො𝑝𝑘 = 𝑒𝑖 ො𝑞𝑗𝑒−𝑖 ො𝑝𝑘𝑒−𝑖

2𝛿𝑗𝑘 = 𝑒−𝑖 ො𝑝𝑘𝑒𝑖 ො𝑞𝑗𝑒

𝑖

2𝛿𝑗𝑘

Exercise: prove this!

Phase space description

Classical

q

p

Quantum

q

p

vacuum

q

p

coherent

q

p

squeezed

here I attempted a measure of position: I localized

the particle but lost information on its momentum!

• Introduce vectors 𝝃, 𝜿 ∈ ℝ2𝑁 of phase-space coordinates

• Weyl displacement operator

• Characteristic function

• Wigner function

• For one mode:

• Other choices (Glauber-Sudarshan P representation, Husimi Q function…)

Quasiprobability distributions

The Wigner function is in 1-to-1 correspondence with the density matrix 𝜌

• General properties of the Wigner function (for N modes):

• W is real ( ⇔ 𝜌 is Hermitian)

• W can be negative (it is a quasi-probability distribution)

• W is normalised: ℝ2𝑁 𝑑𝝃 𝑊𝜌 𝝃 = 1 ( ⇔ tr 𝜌 = 1 )

• State purity: 𝜇 = tr 𝜌2 = 2𝜋 𝑁 ℝ2𝑁 𝑑𝝃 𝑊𝜌 𝝃2

Phase-space description

The Wigner function is in 1-to-1 correspondence with the density matrix 𝜌

• General properties of the Wigner function (for N modes):

• The marginals reproduce the correct probability distributions.

• E.g. for one mode:

• ∞−+∞

𝑑𝑞𝑊𝜌 𝑞, 𝑝 = 𝑝 𝜌 𝑝

• ∞−+∞

𝑑𝑝𝑊𝜌 𝑞, 𝑝 = 𝑞 𝜌|𝑞〉

Phase-space description

Wigner functions: examples

• Fock states 𝑛a) 𝑛 = 0

b) 𝑛 = 1

c) 𝑛 = 5

qp

qp

qp

Gaussian states

qp

are states whose Wigner distribution

is a Gaussian function in phase space

• Recall: Gaussian probability function for one real variable:

• For 2N real variables, forming the phase-space vector 𝝃, we need:

• A vector of means 𝑑 (first moments): 𝑑 = 𝑅 𝜌 = ො𝑞1 , Ƹ𝑝1 , … , ො𝑞𝑁 , Ƹ𝑝𝑁

• A covariance matrix (second moments) 𝝈 of elements 𝜎𝑗𝑘

𝜎𝑗𝑘 = 𝑅𝑗 𝑅𝑘 + 𝑅𝑘 𝑅𝑗 𝜌 − 2 𝑅𝑗 𝜌 𝑅𝑘 𝜌

• Mean: 𝑥0• Variance: 𝑉

generalisation

of 2V …

qp

Gaussian states

are states whose Wigner distribution

is a Gaussian function in phase space

• Completely specified by:

• A vector of means 𝑑 (first moments): 𝑑 = 𝑅 𝜌 = ො𝑞1 , Ƹ𝑝1 , … , ො𝑞𝑁 , Ƹ𝑝𝑁

• A covariance matrix (second moments) 𝝈 of elements 𝜎𝑗𝑘

𝜎𝑗𝑘 = 𝑅𝑗 𝑅𝑘 + 𝑅𝑘 𝑅𝑗 𝜌 − 2 𝑅𝑗 𝜌 𝑅𝑘 𝜌

𝑊𝜌 𝜉 =exp − 𝜉 − 𝑑 𝑇𝝈−1(𝜉 − 𝑑)

𝜋𝑁 det 𝝈

▪ Very natural: ground and thermal states of all physical systems in the

harmonic approximation regime

▪ Relevant theoretical testbeds for the study of structural properties of

entanglement and correlations, thanks to the symplectic formalism

▪ Preferred resources for experimental unconditional implementations of

continuous variable protocols

▪ Crucial role and remarkable control in quantum optics

▪ coherent states

▪ squeezed states

▪ thermal states

Gaussian states

Gaussian operations

Gaussian states can be

efficiently:

▪ displaced(classical currents)

▪ squeezed(nonlinear crystals)

▪ rotated(phase plates, beam splitters)

▪ measured(homodyne detection)

One-mode Gaussian states

• First moments: 𝑑 =ො𝑞

Ƹ𝑝=

ത𝑞ҧ𝑝

• Covariance matrix: 𝝈 =𝜎𝑞𝑞 𝜎𝑞𝑝𝜎𝑞𝑝 𝜎𝑝𝑝

,𝜎𝑞𝑞 = 2 ො𝑞

2 − ത𝑞2 = 2 Δො𝑞2

𝜎𝑝𝑝 = 2 Ƹ𝑝2 − ҧ𝑝2 = 2 Δ Ƹ𝑝2

𝜎𝑞𝑝 = ො𝑞 Ƹ𝑝 + Ƹ𝑝 ො𝑞 − 2ത𝑞 ҧ𝑝

• Coherent state 𝛼 : 𝑑 =2 Re(𝛼)

2 Im(𝛼), 𝝈 =

1 00 1

• Squeezed state 𝑟 : 𝑑 =00

, 𝝈 = 𝑒−2𝑟 00 𝑒2𝑟

q

p

q

p

Multimode Gaussian states

• The first moments can be adjusted by local

displacements, i.e., local unitary operations

• All relevant quantities for quantum information (e.g.

entropy, entanglement) are invariant under local unitaries

• We can imagine all modes are locally centered in the

phase space, i.e., we can set all first moments to zero:

𝑑 = 0 without loss of generality

• All the relevant information is in the covariance matrix

• Covariance matrix has to satisfy a bona fide condition,

𝝈 + 𝑖 Ω𝑁 ≥ 0 in order to describe a physical state 𝜌 ≥ 0

Multimode Gaussian states

• Covariance matrix has a block-form

• 𝝈𝒌: 2x2 reduced covariance matrix of mode 𝑘

• 𝜺𝒋,𝒌: 2x2 matrix of correlations between modes 𝑗 and 𝑘

• Partial trace over mode 𝑗: just eliminate 𝑗th row and column!

1 12 1

12 2 2

1 2

N

TN

T TN N N

𝝈 =

Quadratic Hamiltonians

• 𝐻 =1

2𝑅𝑇𝑯 𝑅 + 𝑅𝑇𝝃 with 𝑯 a symmetric 2𝑁 × 2𝑁 real matrix

• Any Gaussian state can be written as the vacuum or thermal

equilibrium state of a quadratic Hamiltonian:

𝜌 =𝑒−𝛽 𝐻

Tr 𝑒−𝛽 𝐻

• Evolution under quadratic Hamiltonians 𝐻 =1

2𝑅𝑇𝑯 𝑅 :

ሶ𝑅 = 𝑖 𝐻, 𝑅 = ⋯ = Ω𝑁𝑯 𝑅

Gaussian states: alternative definition


Quadratic Hamiltonians

• Evolution: ሶ𝑅 = 𝑖 𝐻, 𝑅 = ⋯ = Ω𝑁𝑯 𝑅

• Solution: 𝑅 𝑡 = 𝑒Ω𝑁𝑯𝑡 𝑅(0)

• Define: 𝑈𝐻 = 𝑒𝑖 𝐻 = 𝑒

𝑖

2𝑅𝑇𝑯 𝑅

, 𝑆𝐻 = 𝑒Ω𝑁𝑯

• Under quadratic Hamiltonians: 𝑅՜𝐻 ෨𝑅 = 𝑈𝐻 𝑅𝑈𝐻

† = 𝑆𝐻 𝑅

• Preservation of CCR: ෨𝑅, ෨𝑅𝑇 = 𝑅, 𝑅𝑇 = 𝑖Ω𝑁

⇒ 𝑆𝐻 ∈ 𝑆𝑝 2𝑁,ℝ ≔ 𝑆 ∈ ℳ2𝑁 ℝ ∶ 𝑆 Ω𝑁𝑆𝑇 = Ω𝑁

Symplectic Group

Symplectic Group

• 𝑆𝑝 2𝑁,ℝ ≔ 𝑆 ∈ ℳ2𝑁 ℝ ∶ 𝑆 Ω𝑁𝑆𝑇 = Ω𝑁

• Action of a symplectic transformation 𝑆: 𝑑 ↦ 𝑆𝑑 , 𝝈 ↦ 𝑆𝝈𝑆𝑇

• The real symplectic group is connected (though not simply

connected) and is non-compact. Every symplectic matrix can

be written as the product of up to two matrix exponentials.

• Generators of 𝑆𝑝 2𝑁,ℝ : matrices 𝐺 = Ω𝑁𝐽 with 𝐽 symmetric

• Antisymmetric 𝐺 generate the compact subgroup (passive transformations) 𝐾 𝑁 = 𝑆𝑝 2𝑁,ℝ ځ 𝑆𝑂 2𝑁

• Symmetric 𝐺 the generate non-compact subgroup (active transformations) 𝑍 𝑁 ⊂ 𝑆𝑝 2𝑁,ℝ

• Every multimode symplectic 𝑆𝑝 2𝑁,ℝ can be decomposed in terms of transformations acting on one or two modes

Symplectic Group

• 𝑆𝑝 2𝑁,ℝ ≔ 𝑆 ∈ ℳ2𝑁 ℝ ∶ 𝑆 Ω𝑁𝑆𝑇 = Ω𝑁

• Define: 𝛿 =0 11 0

, Ω =0 1−1 0

, 𝛾 =1 00 −1

• Generators (one-mode): Ω,−Ω𝛾 = 𝛿, Ω𝛿 = 𝛾

• Generators (two-modes): 0 ΩΩ 0

,0 −𝕀𝕀 0

,0 𝛿𝛿 0

,0 𝛾𝛾 0

• Number of independent generators of the group 𝑆𝑝 2𝑁,ℝ :

3𝑁 + 4𝑁 𝑁 − 1

2= 2𝑁2 +𝑁

• Number of independent generators of the compact subgroup:

𝑁 + 2𝑁 𝑁 − 1

2= 𝑁2

Symplectic transformations

• Example: single-mode squeezing

• 𝑈 𝑟 = exp −𝑟

2ො𝑎𝑘†2 − ො𝑎𝑘

2 ↔ 𝑆 𝑟 =𝑒−𝑟 00 𝑒𝑟

• 𝑟 = 𝑈 𝑟 0 ↔ 𝝈𝑟 = 𝑆 𝑟 𝝈0𝑆𝑇 𝑟 =

𝑒−𝑟 00 𝑒𝑟

1 00 1

𝑒−𝑟 00 𝑒𝑟

= 𝑒−2𝑟 00 𝑒2𝑟

• Example: two-mode beam splitter

• Unitary:

• Symplectic:

𝜏 = cos 𝜃

Symplectic transformations

• Euler (Bloch-Messiah) decomposition for any 𝑆 ∈ 𝑆𝑝 2𝑁,ℝ :

𝑆 = 𝑂𝑍𝑂′

• 𝑂,𝑂′ ∈ 𝐾 𝑁 : orthogonal symplectic matrices (passive)

• 𝑍 = diag 𝑧1,1

𝑧1, … 𝑧𝑁,

1

𝑧𝑁: single-mode squeezings (active)

• Other decompositions: polar, Iwasawa, …

Symplectic diagonalisation

Williamson’s theorem

There exists a global symplectic

transformation which brings the

covariance matrix into diagonal

form, 𝑆𝑇 𝝈 𝑆 = 𝝂

Hilbert space H Phase space G

Unitary (quadratic) operations U Symplectic transformations S

Density matrix r Covariance matrix s

normal mode decomposition

1 12 1

12 2 2

1 2

N

TN

T TN N N

the 𝜈𝑖 ’s are thesymplectic

eigenvalues

1122

NN

nn

nnn

n

O𝑆

0

0

Symplectic diagonalisation

• In Hilbert space: the state with covariance matrix 𝝂 is a tensor product of N thermal states, each at temperature 𝑇𝑘

1 12 1

12 2 2

1 2

N

TN

T TN N N


eigenvalues

1122

NN

nn

nnn

n

O𝑆

0

0

Using the symplectic spectrum

• Bona fide condition: 𝝈 + 𝑖 Ω𝑁 ≥ 0 ⇔ 𝜈𝑘 ≥ 1 ∀ 𝑘 = 1,…𝑁

• Purity of Gaussian states: 𝜇𝜌 = tr 𝜌2 =

2𝜋 𝑁 ℝ2𝑁 𝑑𝝃 𝑊𝜌 𝝃2≡

1

det 𝝈=

1

Π𝑘𝜈𝑘

• Pure Gaussian states: det 𝝈 = 1

• Mixed Gaussian states: det 𝝈 > 1



Gaussian quantum info summary

Adesso et al, J. Phys. A 40, 7821 (2007); Open Syst. Inf. Dyn. 21, 1440001 (2014)

CONTINUOUS VARIABLE

QUANTUM INFORMATION

LECTURE II.

Recall: symplectic eigenvalues

• The symplectic eigenvalues can be calculated from the

standard (orthogonal) spectrum of the matrix 𝑖 Ω𝑁𝝈, which has eigenvalues ±𝜈𝑘

1 12 1

12 2 2

1 2

N

TN

T TN N N


eigenvalues

1122

NN

nn

nnn

n

O0

0𝑆𝑇 𝑆 =

𝝈

Measuring Gaussian information

• Purity: 𝜇𝜌 = tr 𝜌2 =

1

det 𝝈=

1

Π𝑘𝜈𝑘

• Renyi entropies: 𝑆𝑝 𝜌 =log tr 𝜌𝑝

1−𝑝

• For Gaussian states: go to the Williamson form, to find:

• tr 𝜌𝑝 = ς𝑘 𝑔𝑝(𝜈𝑘), with 𝑔𝑝 𝑥 = 2𝑝/ 𝑥 + 1 𝑝 − 𝑥 − 1 𝑝

• Von Neumann Entropy: 𝑆 𝜌 = −tr 𝜌 log 𝜌• Take the limit 𝑝 ՜ 1

• 𝑆 𝜌 = σ𝑘=1𝑁 𝜈𝑘+1

2log

𝜈𝑘+1

2−

𝜈𝑘−1

2log

𝜈𝑘−1

2

• Renyi entropies: 𝑆𝑝 𝜌 =log tr 𝜌𝑝

1−𝑝

• tr 𝜌𝑝 = ς𝑘 𝑔𝑝(𝜈𝑘), with 𝑔𝑝 𝑥 = 2𝑝/ 𝑥 + 1 𝑝 − 𝑥 − 1 𝑝


0 1 2 3 4 50.0

0.5

1.0

1.5

2.0

2.5

3.0

n

Sp

p

1

2

1

2

3

5

10

20

for a single-mode

thermal state:

𝝈 =2ത𝑛 + 1 00 2ത𝑛 + 1


• General remark: the von Neumann entropy is special in

quantum information theory because it is the only one

which satisfies the strong subadditivity inequality:

• However, if one restricts to Gaussian states, then there is

another entropy which satisfies the strong subadditivity:

the Renyi entropy of order 𝑝 = 2: 𝑆2 𝜌 =1

2log det 𝝈

• Interestingly, this entropy is essentially the Shannon

entropy of the Wigner function intended as a continuous

probability distribution in phase space…

G. Adesso et al., Phys. Rev. Lett. 109, 190502 (2012)

Entanglement of Gaussian states

• The continuous variable analogue of a Bell state is…

momentum-squeezed (𝑟)

position-squeezed (𝑟)

Beam Splitter 50:50

Two-mode squeezed state(‘Twin Beam’)

𝝈𝐴𝐵 𝑟

= 𝐵1,2 Τ1 2 ⋅

𝑒2𝑟 00 𝑒−2𝑟

𝟎

𝟎 𝑒−2𝑟 00 𝑒2𝑟

⋅ 𝐵12𝑇 Τ1 2

=

cosh 2𝑟 00 cosh(2𝑟)

sinh(2𝑟) 00 −sinh(2𝑟)


cosh(2𝑟) 00 cosh(2𝑟)


• It approximates the EPR state



Beam Splitter 50:50


• EPR correlations:

𝜁 =1

2Var ො𝑞𝐴 − ො𝑞𝐵 + Var Ƹ𝑝𝐴 + Ƹ𝑝𝐵

=1

2[ ො𝑞𝐴 − ො𝑞𝐵

2 + Ƹ𝑝𝐴 + Ƹ𝑝𝐵2 ]

= …

𝝈𝐴𝐵 𝑟

=






• It approximates the EPR state



Beam Splitter 50:50


• EPR correlations: 𝜁 = 𝑒−2𝑟

• # dB = 10 Log10 𝑒2𝑟

• 𝑟 ≈ 1.15• EPR correlations: 𝜁 = 𝑒−2𝑟 ≈ 0.1

Quantifying entanglement: pure states

• Entropy of Entanglement :

• 𝐸 𝜌𝐴𝐵 = 𝑆 𝜌𝐴 = 𝑆 𝜌𝐵 where 𝜌𝐴 is the marginal state of mode 𝐴, obtained by partial trace over mode 𝐵 (and viceversa for 𝜌𝐵)

• 𝑆 𝜌 = σ𝑘=1𝑁 𝜈𝑘+1

2log

𝜈𝑘+1

2−

𝜈𝑘−1

2log

𝜈𝑘−1

2

• each mode is locally thermal, only one 𝜈𝑘 = cosh 2𝑟 … substitute…

𝐸 𝜌𝐴𝐵 = cosh2 𝑟 log cosh2 𝑟 − sinh2 𝑟 log sinh2 𝑟

𝝈𝐴𝐵 𝑟 =





𝝈𝐴


• Entropy of Entanglement for a two-mode squeezed state

𝐸 𝜌𝐴𝐵 = cosh2 𝑟 log cosh2 𝑟 − sinh2 𝑟 log sinh2 𝑟

0 1 2 3 4 50

2

4

6

8

10

12

14

r

Eeb

its

For large 𝑟, it goes like

~2

ln 2𝑟

𝑟 ≈ 1.15 corresponds to a bit more than 2 ebits


• Multimode states: phase-space Schmidt decomposition

𝝈𝑝 =

𝝈1 𝜺1,2

𝜺1,2𝑇 𝝈2

𝜺1,3 𝜺1,4 𝜺1,5𝜺2,3 𝜺2,4 𝜺2,5

𝜺1,3𝑇 𝜺1,4

𝑇

𝜺1,4𝑇 𝜺2,4

𝑇

𝜺1,5𝑇 𝜺2,5

𝑇

𝝈3 𝜺3,4 𝜺3,5

𝜺3,4𝑇 𝝈4 𝜺4,5

𝜺3,5𝑇 𝜺4,5

𝑇 𝝈5

A

B

𝑁 = 5, 𝑁𝐴 = 2, 𝑁𝐵= 3



Set:

Then, each

is a two-mode squeezed

state between one mode

in A and one mode in B



• 𝐸 𝜌𝐴𝐵 𝑆(𝜌𝐴) = σ𝑘=1𝑁 𝜈𝑘+1

2log

𝜈𝑘+1

2−

𝜈𝑘−1

2log

𝜈𝑘−1

2

Set:

Then, each

is a two-mode squeezed

state between one mode

in A and one mode in B

Detecting entanglement: mixed states

• A mixed state is separable (=not entangled)

iff 𝜌𝐴𝐵 = σ𝑖 𝑝𝑖 𝜓𝑖 𝐴 𝜓𝑖 𝐴 ⊗ 𝜑𝑖 𝐵 𝜑𝑖 𝐵

The Gaussian case

• A bipartite Gaussian state with covariance matrix 𝝈𝐴𝐵 is separable iff there exists bona fide marginal covariance

matrices 𝜸𝐴 and 𝜸𝐵 such that

𝝈𝐴𝐵 ≥ 𝜸𝐴 ⊕𝜸𝐵

Criteria for continuous variable systems

• Separability criteria based on EPR correlations:

• Define rotated quadratures:

ො𝑞𝑗𝜃 = cos 𝜃 ො𝑞𝑗 + sin𝜃 Ƹ𝑝𝑗 , Ƹ𝑝𝑗

𝜃 = cos 𝜃 Ƹ𝑝𝑗 − sin 𝜃 ො𝑞𝑗 ,

• If a bipartite state 𝜌𝐴𝐵 is separable, then • Sum of EPR variances (Duan et al):

Var 𝑧 ො𝑞𝐴𝜃 −

1

𝑧ො𝑞𝐵𝜃 + Var 𝑧 Ƹ𝑝𝐴

𝜃 +1

𝑧𝑝𝐵𝜃 ≥ 𝑧2 +

1

𝑧2

• Product of EPR variances (Giovannetti et al): Var ො𝑞𝐴

𝜃 − ො𝑞𝐵𝜃 × Var Ƹ𝑝𝐴

𝜃 + 𝑝𝐵𝜃 ≥ 1

• If you violate any of the above inequalities, then the state 𝜌𝐴𝐵 is entangled

Criteria for continuous variable systems

• Separability criteria based on partial transposition

• (R Simon 2000) Transposition of 𝜌 ⇔ momentum reflection

𝜌 ՜ 𝜌𝑇 ⇔ 𝑊𝜌 𝑞, 𝑝 ՜ 𝑊𝜌(𝑞, −𝑝)

• PPT Criterion (Peres-Horodecki ‘96): If a bipartite state 𝜌𝐴𝐵 is separable, then its partial transpose is positive-definite 𝜌𝑇𝐴 ≥ 0

• Hierarchy of inequalities for continuous variables involving higher order

moments (Shchukin-Vogel 2006) to detect violation of PPT

• Necessary and sufficient criterion for entanglement in 1-vs-N mode Gaussian

states based on second moments only


PPT criterion for Gaussian states

• 𝜌𝐴𝐵 ՜ 𝜌𝐴𝐵𝑇𝐴 ⇔ 𝝈𝐴𝐵՜ 𝝈𝐴𝐵 = 𝜃𝐴|𝐵𝝈𝐴𝐵𝜃𝐴|𝐵

• Simply flip the sign of all rows and columns referring to 𝑝𝑗 operators

(i.e. the even ones) for the modes 𝑗 belonging to subsystem A

• PPT: if 𝜌𝐴𝐵 (Gaussian) is separable, then 𝝈𝐴𝐵 is bona fide: 𝝈𝐴𝐵 + 𝑖Ω𝑁 ≥ 0

• PPT: in terms of symplectic eigenvalues of the partially transposed

covariance matrix, ǁ𝜈𝑘, then 𝜌𝐴𝐵 separable ⇒ ǁ𝜈𝑘 ≥ 1 ∀𝑘 = 1,… ,𝑁

• PPT: if 𝜌𝐴𝐵 is Gaussian with min 𝑁𝐴, 𝑁𝐵 = 1 then we have a necessary and sufficient condition: 𝜌𝐴𝐵 is entangled iff there exists some ǁ𝜈𝑘 < 1

PPT criterion for Gaussian states

𝜌𝐴𝐵 ՜ 𝜌𝐴𝐵𝑇𝐴 ⇔ 𝝈𝐴𝐵՜ 𝝈𝐴𝐵 = 𝜃𝐴|𝐵𝝈𝐴𝐵𝜃𝐴|𝐵

Simply flip the sign of all rows and columns referring to 𝑝𝑗 operators (i.e.

the even ones) for the modes 𝑗 belonging to subsystem A

• PPT: if 𝜌𝐴𝐵 (Gaussian) is separable, then 𝝈𝐴𝐵 is bona fide: 𝝈𝐴𝐵 + 𝑖Ω𝑁 ≥ 0

• PPT: in terms of symplectic eigenvalues of the partially transposed

covariance matrix, ǁ𝜈𝑘, then 𝜌𝐴𝐵 separable ⇒ ǁ𝜈𝑘 ≥ 1 ∀𝑘 = 1,… ,𝑁

• PPT: if 𝜌𝐴𝐵 is Gaussian with min 𝑁𝐴, 𝑁𝐵 = 1 then we have a necessary and sufficient condition: 𝜌𝐴𝐵 is entangled iff there exists some ǁ𝜈𝑘 < 1

(1 vs N)

Two-mode Gaussian states

• By means of local unitary operations, i.e. 𝑆𝐴 ⊕𝑆𝐵 at the phase space level, the covariance matrix can be transformed in the

above standard form, completely specified by 4 real elements

• 𝑎, 𝑏, 𝑐+, 𝑐−.

• They are in one-to-one correspondence (once we fix the

convention 𝑐+ ≥ 𝑐− ) with the four local symplectic invariants

• det 𝜶 , det 𝜷 , det 𝜸 , det 𝝈


• Apply partial transposition:

~~

~

−

−

det 𝜶 , det 𝜷 ,det 𝜸 , det 𝝈 .

Define 𝚫 == det 𝜶+ det 𝜷+ 2det 𝜸


Define ෩𝚫 == det 𝜶+ det 𝜷− 2det 𝜸

Two-mode Gaussian statesdet 𝜶 , det 𝜷 ,det 𝜸 , det 𝝈 .

Define 𝚫 == det 𝜶+ det 𝜷+ 2det 𝜸


Define ෩𝚫 == det 𝜶+ det 𝜷− 2det 𝜸

• Symplectic eigenvalues of the covariance

matrix 𝝈:

𝜈± =𝚫 ± 𝚫2 − 4det 𝝈

2

• Symplectic eigenvalues of the partial

transpose 𝝈:

𝜈± =෩𝚫 ± ෩𝚫2 − 4det 𝝈

2

In general for two modes, only ǁ𝜈−can be smaller than zero, so we need only to check it!


• Symplectic eigenvalues of the covariance

matrix 𝝈:

𝜈± =𝚫 ± 𝚫2 − 4det 𝝈

2= 1, 1

• Symplectic eigenvalues of the partial

transpose 𝝈:

𝜈± =෩𝚫 ± ෩𝚫2 − 4det 𝝈

2= 𝑒2𝑟 , 𝑒−2𝑟

for a two-mode

squeezed state

Entanglement measures for mixed states

• Entanglement of formation

• Based on a convex-roof procedure:

• 𝐸𝐹 𝜌𝐴𝐵 = inf{𝑝𝑖,|𝜓𝐴𝐵𝑖⟩

}σ𝑖 𝑝𝑖𝐸(|𝜓𝐴𝐵𝑖⟩) , where 𝜌𝐴𝐵 = σ𝑖 𝑝𝑖 𝜓𝐴𝐵𝑖 𝜓𝐴𝐵𝑖

is a pure-state decomposition of 𝜌𝐴𝐵

• Logarithmic negativity

• Based on the violation of the PPT criterion

• 𝐸𝑁 𝜌𝐴𝐵 = log ||𝜌𝐴𝐵𝑇𝐴||1 , where ||𝑜||1 is the trace distance, i.e.

the sum of the absolute values of the operator 𝑜

Entanglement measures for mixed states

for Gaussian states

• (Gaussian) Entanglement of formation

• 𝐸𝐹 𝜌𝐴𝐵 = inf{𝑝𝑖,|𝜓𝐴𝐵𝑖⟩

}σ𝑖 𝑝𝑖𝐸(|𝜓𝐴𝐵𝑖⟩)

• One restricts the decomposition into pure Gaussian states 𝜓𝐴𝐵𝑖• Gives in general an upper bound on 𝐸𝐹 (is it tight? open problem)

• For two-mode symmetric Gaussian states, this is exact: the Gaussian decomposition is optimal (Giedke et al 2003)

• Logarithmic negativity

• 𝐸𝑁 𝜌𝐴𝐵 = log ||𝜌𝐴𝐵𝑇𝐴||1 depends on the symplectic eigenvalues of

the partial transpose which can be smaller than 1 (no bona fide)

𝐸𝑁 𝜌𝐴𝐵 =

0, if ǁ𝜈𝑘 ≥ 1 ∀𝑘

−

𝑘: 𝜈𝑘


symmetric two-mode states

i.e. 𝑎 = 𝑏, i.e. det 𝜶 = det 𝜷

Two-mode Gaussian states: example

momentum-squeezed

thermal (𝑟, ത𝑛)

position-squeezed

thermal (𝑟, ത𝑛)

Beam Splitter 50:50

Two-mode squeezed thermal state

Summary of quantum correlations

classical

discordant

entangled

steerable

nonlocal

Fully device-independent

quantum information

processing

Partially device-independent

quantum information

processing

Teleportation,

dense coding,

…Worst-case

quantum

metrology

CONTINUOUS VARIABLE

QUANTUM INFORMATION

LECTURE III.

Quantum channels

• Quantum channel: a completely positive and trace-

preserving (CPTP) linear map from density matrices to

density matrices

Λ ∶ 𝜌 ↦ 𝜌′ = Λ[𝜌]

• Completely positive (CP):

Λ𝐴 ⊗ 𝕀𝐵 𝜌𝐴𝐵 ≥ 0 ∀𝜌𝐴𝐵 ≥ 0

• Trace-preserving (TP): Tr Λ 𝜌 = Tr 𝜌 = 1

• Gaussian channel: a channel mapping

Gaussian states into Gaussian states

• Gaussian channel: Λ[𝜌] is Gaussian ∀ 𝜌 Gaussian

𝐴

𝐵

𝐴′

𝐵′

Λ

𝜌𝐴𝐵

𝜌 𝜌′Λ

Λ

Quantum channels

• Stinespring dilation: any quantum channel on a system

A can be realised by a unitary operation jointly on A and

an ancillary system B followed by discarding the ancilla

Λ ∶ 𝜌𝐴 ↦ 𝜌𝐴′ = Λ 𝜌𝐴 ≡ Tr𝐵 𝑈𝐴𝐵 𝜌𝐴 ⊗ 𝜏𝐵 𝑈𝐴𝐵

†

𝜌𝐴

𝜏𝐵

𝜌𝐴′

𝜌𝐴 𝜌𝐴′

Λ

𝑈𝐴𝐵

Gaussian channels

• As Gaussian states are completely specified by their

(displacement vector and) covariance matrix, we aim to

describe the action of any Gaussian channel directly at

the level of covariance matrices (up to displacements)

𝜌𝐴

𝜏𝐵

𝜌𝐴′


Λ

𝑈𝐴𝐵

𝝈𝐴 𝝈𝐴′

?

Gaussian channels

• As Gaussian states are completely specified by their

(displacement vector and) covariance matrix, we aim to

describe the action of any Gaussian channel directly at

the level of covariance matrices (up to displacements)

𝝈𝐴′ = Tr𝐵 𝑆𝐴𝐵 𝝈𝐴 ⊕𝝈𝐵 𝑆𝐴𝐵

𝑇

𝝈𝐴

𝝈𝐵

𝝈𝐴′


Λ

𝑆𝐴𝐵


?Gaussian

ancilla symplectic

transformation

Gaussian channels

• Stinespring: 𝝈𝐴′ = Tr𝐵 𝑆𝐴𝐵 𝝈𝐴 ⊕𝝈𝐵 𝑆𝐴𝐵

𝑇 , 𝑆𝐴𝐵 =𝐴 𝐶𝐷 𝐵

𝑆𝐴𝐵 𝝈𝐴 ⊕𝝈𝐵 𝑆𝐴𝐵𝑇 =

𝐴 𝐶𝐷 𝐵

𝝈𝐴𝝈𝐵

𝐴𝑇 𝐷𝑇

𝐶𝑇 𝐵𝑇

=𝐴𝝈𝐴 𝐶𝝈𝐵𝐷𝝈𝐴 𝐵𝝈𝐵

𝐴𝑇 𝐷𝑇

𝐶𝑇 𝐵𝑇= 𝐴𝝈𝐴𝐴

𝑇 + 𝐶𝝈𝐵𝐶𝑇 ⋅

⋅ ⋅⇒ Output covariance matrix: 𝝈𝐴

′ = 𝐴𝝈𝐴𝐴𝑇 + 𝐶𝝈𝐵𝐶

𝑇

• Impose 𝑆𝐴𝐵 is symplectic: 𝑆𝐴𝐵Ω𝐴𝐵𝑆𝐴𝐵𝑇 = Ω𝐴𝐵

𝐴 𝐶𝐷 𝐵

Ω𝐴Ω𝐵

𝐴𝑇 𝐷𝑇

𝐶𝑇 𝐵𝑇=

Ω𝐴Ω𝐵

⇒ 𝐴Ω𝐴𝐴𝑇 + 𝐶Ω𝐵𝐶

𝑇 = Ω𝐴 (∗)

• Impose 𝝈𝐵 is physical (bona fide): 𝝈𝐵 + 𝑖 Ω𝐵 ≥ 0

Gaussian channels

• Output covariance matrix: 𝝈𝐴′ = 𝐴𝝈𝐴𝐴

𝑇 + 𝐶𝝈𝐵𝐶𝑇

• 𝐴Ω𝐴𝐴𝑇 + 𝐶Ω𝐵𝐶

𝑇 = Ω𝐴 (∗)

• 𝝈𝐵 + 𝑖 Ω𝐵 ≥ 0⇒ 𝐶𝝈𝐵𝐶

𝑇 + 𝑖 𝐶Ω𝐵𝐶𝑇 ≥ 0 ⇒ 𝐶𝝈𝐵𝐶

𝑇 + 𝑖Ω𝐴 − 𝑖 𝐴Ω𝐴𝐴𝑇 ≥ 0

Let: 𝑋 ≡ 𝐴, 𝑌 ≡ 𝐶𝝈𝐵𝐶𝑇 and focus only on system A:

𝝈𝐴′ = 𝑋𝝈𝐴𝑋

𝑇 + 𝑌

with: 𝑌 + 𝑖Ω𝐴 − 𝑖𝑋Ω𝐴𝑋𝑇 ≥ 0


Λ


𝑋, 𝑌

Gaussian channels

• The action of a Gaussian channel Λ on any 𝑁-mode Gaussian state described by displacement vector 𝑑 and covariance matrix 𝝈 is specified by two matrices 𝑋, 𝑌 ∈ℳ2𝑁(ℝ) (with 𝑌 = 𝑌

𝑇) acting as follows:

• 𝑑 ↦ 𝑑′ = 𝑋 𝑑

• 𝝈 ↦ 𝝈′ = 𝑋 𝝈 𝑋𝑇 + 𝑌

• The condition 𝑌 + 𝑖Ω𝑁 − 𝑖 𝑋Ω𝑁𝑋𝑇 ≥ 0 is necessary and

sufficient for the channel to be completely positive

𝜌 𝜌′Λ

𝑑, 𝝈 𝑑′, 𝝈′𝑋, 𝑌

Entanglement-breaking channels

• The output Λ𝐴 ⊗ 𝕀𝐵 𝜌𝐴𝐵 is separable for any input 𝜌𝐴𝐵

𝐴

𝐵

𝐴′

𝐵′

Λ

𝜌𝐴𝐵

• A Gaussian channel 𝑋, 𝑌is entanglement-breaking

iff there exists 𝑌1, 𝑌2 s.t.:

• 𝑌 = 𝑌1 + 𝑌2

• ቊ𝑌1 ≥ −𝑖Ω𝑁

𝑌2 ≥ −𝑖𝑋Ω𝑁𝑋𝑇

Single-mode Gaussian channels (𝑁 = 1)

CPTP 𝑌 + 𝑖Ω𝑁 − 𝑖 𝑋Ω𝑁𝑋𝑇 ≥ 0

Ent-breaking 𝑌 + 𝑖Ω𝑁 + 𝑖 𝑋Ω𝑁𝑋𝑇 ≥ 0

Single-mode Gaussian channels• Phase-insensitive: 𝑋, 𝑌 diagonal

• Let 𝑥 = det 𝑋 , 𝑦 = det 𝑌

CPTP 𝑦 ≥ 𝑥 − 1

Ent-breaking 𝑦 ≥ 𝑥 + 1

1

1

𝑥

𝑦

0 amplifiersattenuatorscontravariant

ent-breaking

unphysical unphysicalth

erm

al

ad

dit

ive n

ois

e

Single-mode Gaussian channels

f

0

0

BEAM SPLITTER

(transmittivity)

f

0

0

PARAMETRIC

AMPLIFIER

(squeezing)

AT

TE

NU

AT

ION

AM

PL

IFIC

AT

ION

𝑋 = cos𝜙 𝕀𝑌 = sin2 𝜙 𝕀

𝑋 = cosh 𝜙 𝕀𝑌 = sinh2𝜙 𝕀

(quantum-limited channels)

CONTINUOUS VARIABLE

QUANTUM INFORMATION

LECTURE IV.

Using bipartite entanglement

• Precondition for quantum key distribution (Ekert)

• Quantum dense coding

• Quantum teleportation

• Demonstrations of

nonlocal realism

• …

Quantum technologies 2.0

Continuous variable teleportation

(Braunstein & Kimble ‘98

Furusawa et al ‘98)

Alic

e

Bo

b

𝜓 𝑖𝑛

𝜌𝑜𝑢𝑡

resourc

e 𝜌𝐴𝐵

two-mode

Gaussian state with

entanglement 𝐸𝑁 = 2𝑟

double homodyne

𝑄+ = 𝑞𝑖𝑛 + 𝑞𝐴 / 2

𝑃− = 𝑝𝑖𝑛 − 𝑝𝐴 / 2

local displacement

𝑞𝐵 ՜ 𝑞𝐵 + 𝑔 2𝑄+𝑝𝐵 ՜ 𝑝𝐵 + 𝑔 2𝑃−with tunable gain 𝑔

photocurrents

Continuous variable teleportationA

lice

𝜓 𝑖𝑛

Bo

b

Gaussian channel Λ ≡ (𝑋, 𝑌)𝜌𝑜𝑢𝑡

INPUT MODE

𝑑𝑖𝑛 (first moments)𝝈𝑖𝑛 (covariance matrix)

OUTPUT MODE

𝑑𝑜𝑢𝑡 = 𝑋 𝑑𝑖𝑛𝝈𝑜𝑢𝑡 = 𝑋𝝈𝑖𝑛𝑋

𝑇 + 𝑌


Phase-insensitive Gaussian channel Λ ≡ (𝑋, 𝑌) with: 𝑋 = 𝑔𝕀, 𝑌 = (𝑔2𝑎 − 2𝑔𝑐 + 𝑏)𝕀

𝜌𝑜𝑢𝑡

input

𝑑𝑖𝑛𝝈𝑖𝑛

OUTPUT

𝑑𝑜𝑢𝑡 = 𝑋 𝑑𝑖𝑛𝝈𝑜𝑢𝑡 = 𝑋𝝈𝑖𝑛𝑋

𝑇 + 𝑌RESOURCE

𝑑𝐴𝐵 = 0

𝝈𝐴𝐵 =𝑎 00 𝑎

𝑐 00 −𝑐

𝑐 00 −𝑐

𝑏 00 𝑏

𝜌𝐴𝐵

𝜓 𝑖𝑛

Continuous variable teleportation• Continuous variable teleportation (𝑔 = 1): an additive noise channel

converging to the identity channel for infinite shared entanglement

1

1

𝑥

𝑦

0 amplifiersattenuatorscontravariant

ent-breaking

unphysical unphysicalC

V t

ele

po

rtati

on

ch

an

nel


The output converges to the input only for infinite shared entanglement

In general the fidelity between input and output

quantifies the performance of the teleportation protocol

ℱ = 𝜓𝑖𝑛 𝜌𝑜𝑢𝑡 𝜓𝑖𝑛

Continuous variable teleportationA

lice

Bo

b

𝜓 𝑖𝑛

𝜌𝑜𝑢𝑡

resourc

e 𝜌𝐴𝐵

two-mode

Gaussian state with


double homodyne

𝑄+ = 𝑞𝑖𝑛 + 𝑞𝐴 / 2

𝑃− = 𝑝𝑖𝑛 − 𝑝𝐴 / 2

local displacement

𝑞𝐵 ՜ 𝑞𝐵 + 𝑔 2𝑄+𝑝𝐵 ՜ 𝑝𝐵 + 𝑔 2𝑃−with tunable gain 𝑔

photocurrents


two-mode

Gaussian state with


Alic

e

𝜓 𝑖𝑛 = 𝛼 𝛼 ∈ ℂre

so

urc

e 𝜌𝐴𝐵

Re[𝛼]

Im[𝛼]

Bo

b

𝜌𝑜𝑢𝑡

input alphabet of

coherent states

with prior phase

space distribution

𝑃𝜆 𝛼 =𝜆

𝜋𝑒−𝜆 𝛼

2

with variance 𝜆−1

തℱ𝜆 𝑟 = නℂ

𝑑2𝛼 𝑃𝜆 𝛼 𝛼 𝜌𝑜𝑢𝑡 𝛼

average teleportation fidelity

FIGURE OF MERIT

• Example: teleportation of unknown coherent states

Optimal teleportation fidelity (with a

symmetric Gaussian resource 𝜌𝐴𝐵):തℱ0(𝑟) =

1

1 + 𝑒−2𝑟If fidelity exceeds തℱ0 0 =

1

2

then one has demonstrated

nonclassical teleportation

Continuous variable teleportation• Example: teleportation of unknown coherent states

Optimal teleportation fidelity (with a

symmetric Gaussian resource 𝜌𝐴𝐵):തℱ0(𝑟) =

1

1 + 𝑒−2𝑟If fidelity exceeds തℱ0 0 =

1

2

then one has demonstrated

nonclassical teleportation

A B

Furusawa et al ‘98

Long-term vision

In order to implement quantum

interfaces one needs to be able to:

✓ Entangle multiple nodes

✓ Teleport information through the

channels

✓ Store and retrieve quantum

states from light to matter

Light Atomicensembles

7

Continuous variable teleportation/storage

• Quantum memory for light (coherent states, squeezed

states) onto atoms (Polzik: Nature 2004, Nature Phys 2011)

• Quantum teleportation between light and matter (Nature 2006)

Continuous variable teleportation/storage

• Deterministic teleportation between macroscopic atomic

ensembles at room temperature (Polzik: Nature Phys. 2013)

Hybrid teleportation (discrete/continuous)

• Using continuous variable teleportation with two-mode

Gaussian resources to teleport non-Gaussian, non-classical

states (e.g. Schroedinger’s kittens) or single-photon states (Furusawa: Science 2011; Nature 2013)

Multipartite quantum resources

mom-sqz r1

pos-sqz r2

pos-sqz r2BS t =1/N

pos-sqz r2

pos-sqz r2

BS t =1/(N-1)

BS t =1/2

Multipartite symmetric state

Teleportation networks

mom-sqz r1pos-sqz r2

pos-sqz r2BS =1/N

pos-sqz r2

pos-sqz r2

BS =1/(N-1)

BS =1/2

r r ≡≡ ((rr11++rr22)) 22–– //


pos-sqz r2BS =1/N

pos-sqz r2

pos-sqz r2

BS =1/(N-1)

BS =1/2


pos-sqz r2BS =1/N

pos-sqz r2

pos-sqz r2

BS =1/(N-1)

BS =1/2

r r ≡≡ ((rr11++rr22)) 22–– //

AB

B

B

B B

B

B

B

BB

B

B

B

input state

The optimal fidelity is

a monotonic function

of the genuine

multipartite

entanglement

Teleportation networks (Furusawa, Nature 2004)

Gaussian states

Limitations and no-go’s

• Extremality of Gaussian states (Wolf-Giedke-Cirac PRL 2006)

• “Among all continuous variable states with given first moments and

covariance matrix, the Gaussian state has the lowest entanglement”

• Consequences (-): criteria based on second moments can only give

sufficient conditions to detect entanglement in non-Gaussian states

• Consequences (+): e.g. at fixed

squeezing, you can engineer

non-Gaussian states with higher

entanglement than the twin-beam

• This can be done e.g. by

de-Gaussifying the state

(e.g. photon subtraction/addition)

(Grangier PRL 2007)


• Gaussian entanglement distillation

• “It is impossible to distill Gaussian states with Gaussian operations”

• You need to resort to de-Gaussification (and re-Gaussification)

(Furusawa, Nat. Photon. 2010)


• Gaussian resource distillation (Gaussbusters)

• “It is impossible to distill any Gaussian resource (entanglement,

steering, optical nonclassicality, etc.. by Gaussian free operations”


• Gaussian quantum error correction

• Gaussian quantum bit commitment

• …

• Optimal cloning of a coherent state

Fidelity of the best Gaussian

clone: 2/3=0.6666…

Optimal fidelity (for a non-

Gaussian clone): 0.6826…

One-way CV quantum computation

• Gaussian cluster states can be used, but

• Non-Gaussian measurements are required

One-way CV quantum computation

• Gaussian cluster states can be used, but

• Non-Gaussian measurements are required

10000-temporal-mode cluster state!!!

(Furusawa, …, S. Armstrong,… Nat. Phot. 2013)

Parameter estimation

▪ We need to transmit quantum states through a noisy medium (e.g. an optical fiber)

▪ We can model the channel by means of a master equation

that depends on some parameter, say f

▪ To gain a control over the state transfer one needs to estimate f

▪ This consists in two steps:

1. Devising the optimal input state (probe)

2. Determining the optimal measurement on the output

▪ After repeating N times, one constructs an estimator for f

▪ Optimal estimation minimum variance of the estimator

in out

f

PREPARE

TRANSMIT

MEASURE

ˆ

Quantum Estimation Theory (basics)• For each prepare and measure strategy S, one can construct an

unbiased estimator of minimum variance given by

where the classical Fisher Information is a figure of merit

characterizing the performance of the strategy

• At fixed input probe, the quantum Cramér-Rao bound states that

for any strategy one has

• There exists an optimal POVM yielding maximum sensitivity, that

consists of projections on the eigenstates of the symmetric

logarithmic derivative L, defined implicitly as

• The classical Fisher information associated to such optimal

measurement is known as quantum Fisher information (QFI)

• The QFI can be also computed from the Bures metric of the evolved

states and is thus related to the quantum fidelity between

infinitesimally close states

• Finding the QFI solves the second step, optimizing over the output

measurement. We are left to find the optimal input probe states.

1ˆ Var[ ] ( )N I S

0 0 0( , ) ( , ) ( )I I H

2 /d dr

2

0( ) Tr[ ]rH

in out

f

in out

f

in out

f

0

Bosonic channels

▪ Attenuation channel

▪ Amplification channel

▪ Dephasing channel

in out

f

in out

f

in out

f

0

tan [ ]d

L ad

†tanh [ ]d

L ad

†tan [ ]d

aL ad

† † †[ ] 2L o o o o o o o

Gaussian channels(map Gaussian states

into Gaussian states)

out

out

Ultimate bounds on precision

f

0

0

BEAM SPLITTER

(transmittivity)

f

0

0

PARAMETRIC

AMPLIFIER

(squeezing)

AT

TE

NU

AT

ION

AM

PL

IFIC

AT

ION

…measuring system + environment…

n : mean input energy

max 4H n

max 4 1H n

These bounds are tight! (B. Escher, .. L. Davidovich, Nature Phys. 2011)

The optimal probes are non-GaussianA

TT

EN

UA

TIO

NA

MP

LIF

ICA

TIO

N

n : mean input energy

max 4H n

max 4 1H n

0p

8

p

4

p

23 p

8

p

2

2.0

2.5

3.0

3.5

4.0

f

H

n=1

0p

8

p

4

p

23 p

8

p

2

101214161820

f

H

n=5

0p

8

p

4

p

23 p

8

p

2

40

50

60

70

80

f

H

n=20

0p

8

p

4

p

23 p

8

p

2

100120140160180200

f

H

n=50

Ultimate bound ≡ Fock input and photon counting strategy

Best Gaussian probes

Coherent states and

heterodyne detection

(G. Adesso et al. PRA(R) 2008, PRL 2010)

Non-Gaussian wild world

Continuous variables: some open problems

• Efficient criteria for entanglement of non-Gaussian states

• Efficiently computable entanglement measures/bounds

• Optimal strategies for parameter estimation of CV channels

• …

• Technological improvements for generation, manipulation, measurement

of non-Gaussian states (better sources, better detectors, etc. etc.)

• Hybrid, hybrid, hybrid: use the best of different worlds for integrating

quantum communication networks (e.g. for cryptography, teleportation, computation)

• … .. …

• Just follow your curiosity ☺

CONTINUOUS VARIABLE

QUANTUM INFORMATION

Further reading

• Quantum information with continuous variables• (General) S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77, 513 (2005).

• (Gaussian) C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Rev. Mod. Phys. 84, 621 (2012).

• (Book) "Quantum information with continuous variables" edited by S. L. Braunstein and A. K. Pati (Springer, 2003)

• (Book) "Quantum Information with Continuous Variables of Atoms and Light” edited by N. Cerf, G. Leuchs, and E. S. Polzik, (Imperial College Press, 2007).

• (Book) “Quantum continuous variables: A primer of theoretical methods” by A. Serafini (Taylor and Francis, 2017)

• Introduction to continuous variable entanglement• (General) J. Eisert and M. B. Plenio, Int. J. Quant. Inf. 1, 479 (2003)

• (Gaussian) G. Adesso and F. Illuminati, J. Phys. A: Math. Theor. 40, 7821 (2007)

• (Gaussian) G. Adesso, S. Ragy, A. Lee, Open Syst. Inf. Dyn. 21, 1440001 (2014)

• State engineering / physical implementations• (General/optics) F. Dell’Anno, S. De Siena, and F. Illuminati, Phys. Rep. 428, 53 (2006)

• (Atoms/interfaces) K. Hammerer, A. S. Sørensen, and E. S. Polzik, Rev. Mod. Phys. 82, 1041 (2010)

• (Atoms/interfaces) H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, Rev. Mod. Phys. 85, 553 (2013)

Thank you!

Gerardo Adesso

School of Mathematical Sciences

University of Nottingham (UK)

[email protected]

https://quantingham.wordpress.com

https://twitter.com/QCG_Nottingham

.
mailto:[email protected]

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