CONTINUOUS VARIABLE QUANTUM INFORMATION · Gaussian states q p are states whose Wigner distribution...

CONTINUOUS VARIABLE

QUANTUM INFORMATION

Gerardo Adesso

School of Mathematical Sciences

The University of Nottingham

gerardo.adesso@gmail.com

http://quantumcorrelations.weebly.com

LECTURE I. • Introduction and examples

• Phase space formalism

• Gaussian states

LECTURE II. • Gaussian states: measures of information

• Gaussian states: measures of correlations

• Applications to quantum communication

LECTURE III. • Overview of continuous variable protocols

• Non-Gaussian resources

• Open problems and perspectives

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 𝑝 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

Motivation: CV entanglement

Einstein-Podolski-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 protocols...

Quantum harmonic oscillator

• Canonical commutation relations 𝑞 , 𝑝 = 𝑖ℏ

• Ladder operators

• Annihilation 𝑎 =𝑚𝜔

2ℏ𝑞 +

𝑚𝜔 𝑝

• Creation 𝑎 † =𝑚𝜔

2ℏ𝑞 −

𝑚𝜔 𝑝

• 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/<Sz>, p=Sy/<Sz>)

Quantized electromagnetic field

• Each oscillator: a mode of the field

• Natural units: ℏ = 𝑐 = 1

• 𝑎 𝑘 =1

2𝑞 𝑘 + 𝑖 𝑝 𝑘 , 𝑎 𝑘

† =1

2𝑞 𝑘 − 𝑖 𝑝 𝑘

• 𝑞 𝑘 =1

2𝑎 𝑘 + 𝑎 𝑘

† , 𝑝 𝑘 =1

𝑖 2𝑎 𝑘 − 𝑎 𝑘

Quantized electromagnetic field

• We introduce a vector of canonical operators: 𝑅 = 𝑞 1, 𝑝 1, 𝑞 2, 𝑝 2, … , 𝑞 𝑁, 𝑝 𝑁

• Canonical commutation relations: • 𝑞 𝑗 , 𝑝 𝑘 = 𝑖 ℏ 𝛿𝑗𝑘

• 𝑞 𝑗 , 𝑞 𝑘 = 0, 𝑝 𝑗 , 𝑝 𝑘 = 0

• Introducing the N-mode symplectic matrix Ω𝑁 = Ω⊕𝑁, with Ω =0 1−1 0

then we can write the commutation relations compactly: 𝑅 𝑗 , 𝑅 𝑘 = 𝑖 Ω𝑁 𝑗𝑘

Phase space description

Classical

Quantum

vacuum

coherent

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: 𝑑𝝃ℝ2𝑁 𝑊𝜌 𝝃 = 1 ( ⇔ tr 𝜌 = 1 )

• State purity: 𝜇 = tr 𝜌2 = 2𝜋 𝑁 𝑑𝝃ℝ2𝑁 𝑊𝜌 𝝃

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:

• 𝑑𝑞+∞

−∞ 𝑊𝜌 𝑞, 𝑝 = 𝑝 𝜌 𝑝

• 𝑑𝑝+∞

−∞ 𝑊𝜌 𝑞, 𝑝 = 𝑞 𝜌|𝑞⟩

Phase-space description

Wigner functions: examples

• Fock states 𝑛

a) 𝑛 = 0

b) 𝑛 = 1

c) 𝑛 = 5

Gaussian states

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 𝑅 (first moments): 𝑅 = 𝑅 𝜌 = 𝑞 1 , 𝑝 1 , … , 𝑞 𝑁 , 𝑝 𝑁

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

𝜎𝑗𝑘 = 𝑅 𝑗𝑅 𝑘 + 𝑅 𝑘𝑅 𝑗 𝜌 − 2 𝑅 𝑗 𝜌 𝑅 𝑘 𝜌

• Mean: 𝑥0

• Variance: 𝑉

generalisation

of 2V …

Gaussian states

are states whose Wigner distribution

is a Gaussian function in phase space

• Completely specified by:

• A vector of means 𝑅 (first moments): 𝑅 = 𝑅 𝜌 = 𝑞 1 , 𝑝 1 , … , 𝑞 𝑁 , 𝑝 𝑁

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

𝜎𝑗𝑘 = 𝑅 𝑗𝑅 𝑘 + 𝑅 𝑘𝑅 𝑗 𝜌 − 2 𝑅 𝑗 𝜌 𝑅 𝑘 𝜌

𝑊𝜌 𝜉 =exp − 𝜉 − 𝑅 𝑇𝝈−1(𝜉 − 𝑅 )

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)

Gaussian operations

efficiently:

Gaussian operations

efficiently:

Gaussian operations

efficiently:

Gaussian operations

efficiently:

One-mode Gaussian states

• First moments: 𝑅 =𝑞

𝑝 =

𝑞 𝑝

• Covariance matrix: 𝝈 =𝜎𝑞𝑞 𝜎𝑞𝑝𝜎𝑞𝑝 𝜎𝑝𝑝

, 𝜎𝑞𝑞 = 2 𝑞 2 − 𝑞 2 = 2 Δ𝑞 2

𝜎𝑝𝑝 = 2 𝑝 2 − 𝑝 2 = 2 Δ𝑝 2

𝜎𝑞𝑝 = 𝑞 𝑝 + 𝑝 𝑞 − 2𝑞 𝑝

• Coherent state 𝛼 : 𝑅 =2 Re(𝛼)

2 Im(𝛼), 𝝈 =

1 00 1

• Squeezed state 𝑟 : 𝑅 =00

, 𝝈 = 𝑒−2𝑟 00 𝑒2𝑟

Multimode Gaussian states

• The first moments can be adjusted by local

displacements, i.e., local unitary operations

• All relevant quantitites for quantum information (e.g. state

purity, correlations, entanglement, etc.) 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:

𝑅 = 0 without loss of generality

• Then, it is the shape of those Gaussians which matters:

• All the relevant information is in the covariance matrix

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

T TN N N

𝝈 =

Gaussian states

• Covariance matrix has to satisfy a bona fide condition,

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

• Unitary operations 𝑈 = exp (𝑖 𝜃 𝐻 ) where the Hamiltonians

𝐻 are at most quadratic in the canonical operators, i.e.,

𝐻 = 𝑓 𝑎 𝑘†, 𝑎 𝑘

†𝑎 𝑙,𝑎 𝑘†𝑎 𝑙

† + herm. conj. , are associated to:

• Symplectic transformations 𝑆 which act by congruence on

covariance matrices, 𝜎 ↦ 𝑆 𝜎 𝑆𝑇, where 𝑆 ∈ 𝑆𝑝2𝑁,ℝ by

definition satisfy 𝑆𝑇Ω𝑁𝑆 = Ω𝑁 (it follows that det 𝑆 = 1)

Gaussian operations: preserve Gaussianity

Gaussian operations

• Example: single-mode squeezing

• 𝑈 𝑟 = exp −𝑟

2𝑎 𝑘†2 − 𝑎 𝑘

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 diagonalisation

• Williamson’s theorem

There exists a global symplectic

transformation which brings the

covariance matrix into diagonal

form, 𝑆 𝝈 𝑆𝑇 = 𝝂

Hilbert space H Phase space G

Unitary operations U Symplectic operations S

Density matrix r Covariance matrix s

normal mode decomposition

1 12 1

12 2 2

T TN N N

the 𝜈𝑖 ’s are the symplectic

eigenvalues

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

T TN N N

eigenvalues

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 𝑑𝝃ℝ2𝑁 𝑊𝜌 𝝃

det 𝝈=

Π𝑘𝜈𝑘

• Pure Gaussian states: det 𝝈 = 1

• Mixed Gaussian states: det 𝝈 > 1

Exercise: prove this!

Summary of Gaussian states

G. Adesso & F. Illuminati, J. Phys. A: Math. Gen. 40, 7821 (2007)

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

T TN N N

eigenvalues

0 𝑆𝑇 𝑆 =

Measuring Gaussian information

• Purity: 𝜇𝜌 = tr 𝜌2 =1

det 𝝈=

Π𝑘𝜈𝑘

• Renyi entropies: 𝑆𝑝 𝜌 =log tr 𝜌𝑝

1−𝑝

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

• tr 𝜌𝑝 = 𝑔𝑝(𝜈𝑘)𝑘 , 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 = 𝜈𝑘+1

𝜈𝑘+1

𝜈𝑘−1

2𝑁𝑘=1

• Renyi entropies: 𝑆𝑝 𝜌 =log tr 𝜌𝑝

1−𝑝

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

0 1 2 3 4 50.0

for a single-mode

thermal state:

𝝈 =2𝑛 + 1 0

0 2𝑛 + 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 1 2 ⋅

𝑒2𝑟 00 𝑒−2𝑟

𝟎 𝑒−2𝑟 00 𝑒2𝑟

⋅ 𝐵12𝑇 1 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 𝑞 𝐴 − 𝑞 𝐵 + Var 𝑝 𝐴 + 𝑝 𝐵

2[ 𝑞 𝐴 − 𝑞 𝐵

2 + 𝑝 𝐴 + 𝑝 𝐵2 ]

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 = 𝜈𝑘+1

𝜈𝑘+1

𝜈𝑘−1

2𝑁𝑘=1

• 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

For large 𝑟, it goes like

ln 2 𝑟

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

𝜺1,4𝑇 𝜺2,4

𝜺1,5𝑇 𝜺2,5

𝝈3 𝜺3,4 𝜺3,5

𝜺3,4𝑇 𝝈4 𝜺4,5

𝜺3,5𝑇 𝜺4,5

𝑇 𝝈5

𝑁 = 5, 𝑁𝐴 = 2, 𝑁𝐵= 3

Then, each

is a two-mode squeezed

state between one mode

in A and one mode in B

• 𝐸 𝜌𝐴𝐵 𝑆(𝜌𝐴) = 𝜈𝑘+1

𝜈𝑘+1

𝜈𝑘−1

2𝑁𝑘=1

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 = 𝑝𝑖 𝜓𝑖 𝐴 𝜓𝑖 𝐴 ⊗ 𝜑𝑖 𝐵 𝜑𝑖 𝐵𝑖

Criteria for continuous variable systems

• Separability criteria based on uncertainty principle / EPR correlations:

• Define rotated quadratures: 𝑞 𝑗𝜃 = cos 𝜃 𝑞 𝑗 + sin 𝜃 𝑝 𝑗 , 𝑝 𝑗

𝜃 = cos 𝜃 𝑝 𝑗 − sin 𝜃 𝑞 𝑗 ,

• If a bipartite state 𝜌𝐴𝐵 is separable, then

• Sum of EPR variances (Duan et al): Var 𝑧 𝑞 𝐴𝜃 −

𝑧𝑞 𝐵𝜃 + Var 𝑧 𝑝 𝐴

𝜃 +1

𝑧𝑝𝐵𝜃 ≥ 𝑧2 +

• Product of EPR variances (Giovannetti et al): Var 𝑞 𝐴𝜃 − 𝑞 𝐵

𝜃 × Var 𝑝 𝐴𝜃 + 𝑝𝐵

𝜃 ≥ 1

• Entropic criteria (UFRJ, Walborn, Toscano et al): 𝐻𝛼 𝑞 𝐴𝜃 − 𝑞 𝐵

𝜃 + 𝐻𝛼 𝑝 𝐴𝜃 + 𝑝𝐵

𝜃 ≥ 𝑐𝛼

• 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 ⇔ phase-space 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→ 𝝈 𝐴𝐵 = 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 𝝈 𝐴𝐵 is bona fide: 𝝈 𝐴𝐵 + 𝑖Ω𝑁 ≥ 0

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

covariance matrix, 𝜈 𝑘, then 𝜌𝐴𝐵 separable ⇒ 𝜈 𝑘 ≥ 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 𝜈 𝑘 < 1

PPT criterion for Gaussian states

𝜌𝐴𝐵 → 𝜌𝐴𝐵𝑇𝐴 ⇔ 𝝈𝐴𝐵→ 𝝈 𝐴𝐵 = 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 𝝈 𝐴𝐵 is bona fide: 𝝈 𝐴𝐵 + 𝑖Ω𝑁 ≥ 0

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

covariance matrix, 𝜈 𝑘, then 𝜌𝐴𝐵 separable ⇒ 𝜈 𝑘 ≥ 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 𝜈 𝑘 < 1

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 𝚫 = = det 𝜶+ det 𝜷− 2det 𝜸

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

Define 𝚫 = = det 𝜶+ det 𝜷+ 2det 𝜸

Define 𝚫 = = det 𝜶+ det 𝜷− 2det 𝜸

• Symplectic eigenvalues of the covariance

matrix 𝝈:

𝜈± =𝚫 ± 𝚫2 − 4det 𝝈

• Symplectic eigenvalues of the partial

transpose 𝝈 :

𝜈 ± =𝚫 ± 𝚫 2 − 4det 𝝈

In general for two modes, only 𝜈 −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 ± =𝚫 ± 𝚫 2 − 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⟩

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

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⟩

} 𝑝𝑖𝐸(|𝜓𝐴𝐵𝑖⟩) 𝑖

• 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 𝜈 𝑘 ≥ 1 ∀𝑘

− log 𝜈 𝑘𝑘: 𝜈 𝑘<1

symmetric two-mode states

i.e. 𝑎 = 𝑏, i.e. det 𝜶 = det 𝜷

Two-mode Gaussian states: example

momentum- squeezed

thermal (𝑟, 𝑛 )

position- squeezed

thermal (𝑟, 𝑛 )

Beam Splitter 50:50

Two-mode squeezed thermal state

CONTINUOUS VARIABLE

QUANTUM INFORMATION

LECTURE III.

Using bipartite entanglement

• Precondition for quantum key distribution (Ekert)

• Quantum dense coding

• Quantum teleportation

• Demonstrations of

nonlocality

• …

Continuous variable teleportation

Two-mode entangled state

input state

(Braunstein & Kimble ‘98

Furusawa et al ‘98)

Continuous variable teleportation

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 network

The optimal fidelity (maximized over local unitaries on the shared symmetric resource state) is a monotonic function of the entanglement distributed between A and B, when the input is a generic coherent state

ℱ =1

1 + 𝜈 −

Benchmark: experiments must achieve fidelities higher than the classical

threshold to demonstrate a genuine quantum transfer based on entanglement

To determine such a threshold: a nontrivial problem of quantum estimation

“ ” Fidelity of the optimal

“measure-and-prepare”

strategy:

classical threshold

Quantum teleportation benchmarks

Cheating teleportation without entanglement

p Input: coherent states with totally unknown displacement

(Hammerer et al. PRL 2005)

(first experiment: Furusawa et al. Science 1998)

Benchmark: 50%

p Input: arbitrary single-mode Gaussian state

Benchmark?

**OPEN PROBLEM**

Quantum teleportation benchmarks

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 Atomic ensembles

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 to appear 2013)

Hybrid teleportation (discrete/continuous)

• The other way around: using qubit teleporters in parallel to

teleport input continuous variable states (Andersen-Ralph, theory,

PRL 2013)

Multipartite quantum resources

mom-sqz r1

pos-sqz r2

pos-sqz r2 BS t =1/N

pos-sqz r2

BS t =1/(N-1)

BS t =1/2

Multipartite symmetric state

mom-sqz r1pos-sqz r2

pos-sqz r2BS =1/N

pos-sqz r2

BS =1/(N-1)

BS =1/2

r r ≡≡ ((rr11++rr22)) 22–– //

pos-sqz r2BS =1/N

pos-sqz r2

BS =1/(N-1)

BS =1/2

pos-sqz r2BS =1/N

pos-sqz r2

BS =1/(N-1)

BS =1/2

r r ≡≡ ((rr11++rr22)) 22–– //

• Genuine N-partite entanglement

Teleportation networks

pos-sqz r2BS =1/N

pos-sqz r2

BS =1/(N-1)

BS =1/2

r r ≡≡ ((rr11++rr22)) 22–– //

pos-sqz r2BS =1/N

pos-sqz r2

BS =1/(N-1)

BS =1/2

pos-sqz r2BS =1/N

pos-sqz r2

BS =1/(N-1)

BS =1/2

r r ≡≡ ((rr11++rr22)) 22–– //

input state

Teleportation networks (Furusawa, Nature 2004)

• More general (non-symmetric) Gaussian tripartite states (P. Nussenzveig, USP, Science 2009)

Open problems with Gaussian states

• Entanglement of formation for general Gaussian states

• Strong monogamy of multipartite entanglement for arbitrary Gaussian states

• Quantum teleportation benchmarks for Gaussian states* *not open for long

• Quantum benchmarks for cloning, amplifying, etc…

• Additivity of the output entropy for Gaussian channels

• Nonlocality: are all mixed Gaussian entangled states nonlocal? Maximal violations of Bell inequalities?

• Are Gaussian measurements optimal for quantum discord?

• Technological challenges: increase squeezing level etc.

• …

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 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

PREPARE

TRANSMIT

MEASURE

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

0( ) Tr[ ]rH

Bosonic channels

Dissipation channel

Amplification channel

Dephasing channel

tan [ ]d

†tanh [ ]d

†tan [ ]d

† † †[ ] 2L o o o o o o o

Gaussian channels (map Gaussian states

into Gaussian states)

Ultimate bounds on precision

BEAM SPLITTER

(transmittivity)

PARAMETRIC

AMPLIFIER

(squeezing)

…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-Gaussian D

n : mean input energy

max 4H n

max 4 1H n

10 12 14 16 18 20

n = 20

100 120 140 160 180 200

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: more open problems

• Resource theory for non-Gaussianity

• The cheap states/operations are Gaussian (like separable

states/LOCC for entanglement)

• A measure of non-Gaussianity for a general state 𝜌 is the minimum

relative entropy between 𝜌 and the closest Gaussian state (Genoni et

al 2009)

• This measure is well motivated, invariant under symplectic

operations, obviously zero for Gaussian states, and moreover it can

be computed…

• The closest Gaussian state is the one with the same first and second

moments as 𝜌 (Marian & Marian 2013)

• So what is missing?

• An operational interpretation for non-Gaussianity

Continuous variables: more 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

CONTINUOUS VARIABLE QUANTUM INFORMATION · Gaussian states q p are states whose Wigner distribution...

Documents

Kriging models with Gaussian processes - covariance function estimation and …fbachoc/Bachoc_Forge... · 2015-09-14 · Kriging models with Gaussian processes - covariance function

Wigner oscillations and fractional Wigner oscillations in ...exotic.phy.bme.hu/SeminarPresentations/Traveso_2015.pdfFractional Wigner oscillations. WIGNER CRYSTAL Electron liquid Usually

Integrals Involving H-function of Several Complex Variables · Integrals Involving H-function of Several Complex Variables AshiqHussain Khan, ... he Gaussian hypergeometric function

An application of Gaussian radial based function neural

Analytic Wigner distribution function for a split

Electrocardiogram Synthesis Using a Gaussian Combination ...cinc.mit.edu/archives/2007/pdf/0621.pdf2.2.3. Training Algorithm for Gaussian fitting At this level, a Gaussian function

Gaussian process models and covariance function designtugaut.perso.math.cnrs.fr/pdf/APSSE/2014.11.28.pdf · 2014. 12. 1. · Gaussian process models and covariance function design

Glimcher Decision Making. Signal Detection Theory With Gaussian Assumption Without Gaussian Assumption Equivalent to Maximum Likelihood w/o Cost Function

Training Radial Basis Function Networks€¦ · Training radial basis function networks: Radii of radial basis functions. Special case: Gaussian activation function ∂out(l) v ∂σv

Covariance function estimation in Gaussian process …statmath.wu.ac.at/research/talks/resources/Bachoc_WU_2015.pdf · Covariance function estimation in Gaussian process regression

Gaussian Processes - Marc Deisenroth · Gaussian Processes Marc Deisenroth @AIMS, Rwanda, October 16, 2018 9. GP Prior Treat a function as a long vector of function values: f rf 1,

Wigner Distribution Function and Integral Imaging

Curve Smoothing Using Gaussian Function

Gaussian Process Function Data Analysis R …Gaussian Process Function Data Analysis R Package ‘GPFDA’ Jian Qing Shi & Yafeng Cheng School of Mathematics and Statistics Newcastle

WIGNER FUNCTION APPROACH TO OSCILLATING …...WIGNER FUNCTION APPROACH TO OSCILLATING SOLUTIONS OF THE 1D-QUINTIC NONLINEAR SCHRODINGER EQUATION ALEX MAHALOV AND SERGEI K. SUSLOV Abstract

Wigner function approach Nov 2006

Gaussian uctuations of the determinant of Wigner Matricesbourgade/papers/CLTforDeterminants.pdf · determinants has then been the subject of recent active research. 1.1 Non-Hermitian

Fractional Wigner

Wigner Distribution Function and Integral Imaging · 2020-01-04 · Wigner Distribution and Integral Imaging 2.710 May 6, 2009 Wigner Distribution Function • Describes a signal

Introduction to Gaussian Processes - University of Torontohinton/csc2515/notes/gp_slides_fall08.pdfThe covariances were set using a kernel function: ... by Gaussian noise) creates