CONTINUOUS VARIABLE QUANTUM INFORMATION · Gaussian states q p are states whose Wigner distribution...
Transcript of 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
http://quantumcorrelations.weebly.com
Plan
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
0 q
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...
0 q
Quantum harmonic oscillator
• Canonical commutation relations 𝑞 , 𝑝 = 𝑖ℏ
• Ladder operators
• Annihilation 𝑎 =𝑚𝜔
2ℏ𝑞 +
𝑖
𝑚𝜔 𝑝
• Creation 𝑎 † =𝑚𝜔
2ℏ𝑞 −
𝑖
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
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: 𝑑𝝃ℝ2𝑁 𝑊𝜌 𝝃 = 1 ( ⇔ tr 𝜌 = 1 )
• State purity: 𝜇 = tr 𝜌2 = 2𝜋 𝑁 𝑑𝝃ℝ2𝑁 𝑊𝜌 𝝃
2
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
q p
q p
q p
Gaussian states
q p
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 …
q p
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
Gaussian states can be
efficiently:
displaced (classical currents)
squeezed (nonlinear crystals)
rotated (phase plates, beam splitters)
measured (homodyne detection)
Gaussian operations
Gaussian states can be
efficiently:
displaced (classical currents)
squeezed (nonlinear crystals)
rotated (phase plates, beam splitters)
measured (homodyne detection)
Gaussian operations
Gaussian states can be
efficiently:
displaced (classical currents)
squeezed (nonlinear crystals)
rotated (phase plates, beam splitters)
measured (homodyne detection)
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: 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
q
p
q
p
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
1 2
N
TN
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
1 2
N
TN
T TN N N
the 𝜈𝑖 ’s are the symplectic
eigenvalues
1 1
2
2
N
N
n n
n n n
n
O
𝑆
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
the 𝜈𝑖 ’s are the symplectic
eigenvalues
1 1
2
2
N
N
n n
n n n
n
O
𝑆
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 𝑑𝝃ℝ2𝑁 𝑊𝜌 𝝃
2≡
1
det 𝝈=
1
Π𝑘𝜈𝑘
• Pure Gaussian states: det 𝝈 = 1
• Mixed Gaussian states: det 𝝈 > 1
Exercise: prove this!
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
1 2
N
TN
T TN N N
the 𝜈𝑖 ’s are the symplectic
eigenvalues
1 1
2
2
N
N
n n
n n n
n
O 0
0 𝑆𝑇 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 = 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
2log
𝜈𝑘+1
2−
𝜈𝑘−1
2log
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
Measuring Gaussian information
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 =2𝑛 + 1 0
0 2𝑛 + 1
Measuring Gaussian information
• 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
𝟎 𝑒−2𝑟 00 𝑒2𝑟
⋅ 𝐵12𝑇 1 2
=
cosh 2𝑟 00 cosh (2𝑟)
sinh (2𝑟) 00 −sinh (2𝑟)
sinh (2𝑟) 00 −sinh (2𝑟)
cosh (2𝑟) 00 cosh (2𝑟)
Entanglement of Gaussian states
• It approximates the EPR state
momentum- squeezed (𝑟)
position- squeezed (𝑟)
Beam Splitter 50:50
Two-mode squeezed state (‘Twin Beam’)
• EPR correlations:
𝜁 =1
2Var 𝑞 𝐴 − 𝑞 𝐵 + Var 𝑝 𝐴 + 𝑝 𝐵
=1
2[ 𝑞 𝐴 − 𝑞 𝐵
2 + 𝑝 𝐴 + 𝑝 𝐵2 ]
= …
𝝈𝐴𝐵 𝑟
=
cosh 2𝑟 00 cosh (2𝑟)
sinh (2𝑟) 00 −sinh (2𝑟)
sinh (2𝑟) 00 −sinh (2𝑟)
cosh (2𝑟) 00 cosh (2𝑟)
Entanglement of Gaussian states
• It approximates the EPR state
momentum- squeezed (𝑟)
position- squeezed (𝑟)
Beam Splitter 50:50
Two-mode squeezed state (‘Twin Beam’)
• 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
2log
𝜈𝑘+1
2−
𝜈𝑘−1
2log
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 =
cosh 2𝑟 00 cosh (2𝑟)
sinh (2𝑟) 00 −sinh (2𝑟)
sinh (2𝑟) 00 −sinh (2𝑟)
cosh (2𝑟) 00 cosh (2𝑟)
𝝈𝐴
Quantifying entanglement: pure states
• 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 2 𝑟
𝑟 ≈ 1.15 corresponds to
a bit more than 2 ebits
Quantifying entanglement: pure states
• 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
Quantifying entanglement: pure states
• Multimode states: phase-space Schmidt decomposition
Set:
Then, each
is a two-mode squeezed
state between one mode
in A and one mode in B
Quantifying entanglement: pure states
• Multimode states: phase-space Schmidt decomposition
• 𝐸 𝜌𝐴𝐵 𝑆(𝜌𝐴) = 𝜈𝑘+1
2log
𝜈𝑘+1
2−
𝜈𝑘−1
2log
𝜈𝑘−1
2𝑁𝑘=1
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 = 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 −
1
𝑧𝑞 𝐵𝜃 + Var 𝑧 𝑝 𝐴
𝜃 +1
𝑧𝑝𝐵𝜃 ≥ 𝑧2 +
1
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
Two-mode Gaussian states
• Apply partial transposition:
~ ~
~
−
−
det 𝜶 , det 𝜷 , det 𝜸 , det 𝝈 .
Define 𝚫 = = det 𝜶+ det 𝜷+ 2det 𝜸
det 𝜶 , det 𝜷 , det 𝜸 , det 𝝈 .
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
det 𝜶 , det 𝜷 , det 𝜸 , det 𝝈 .
Define 𝚫 = = 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 ± =𝚫 ± 𝚫 2 − 4det 𝝈
2
In general for two modes, only 𝜈 −can be smaller than zero, so we need only to check it!
Two-mode Gaussian states
• 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
Two-mode Gaussian states
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
A
input state
B
B
A B
(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
q
p Input: coherent states with totally unknown displacement
(Hammerer et al. PRL 2005)
(first experiment: Furusawa et al. Science 1998)
Benchmark: 50%
q
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
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 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
pos-sqz r2
BS t =1/(N-1)
BS t =1/2
Multipartite symmetric state
Multipartite quantum resources
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–– //
mom-sqz r1pos-sqz r2
pos-sqz r2BS =1/N
pos-sqz r2
pos-sqz r2
BS =1/(N-1)
BS =1/2
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–– //
• Genuine N-partite entanglement
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–– //
mom-sqz r1pos-sqz r2
pos-sqz r2BS =1/N
pos-sqz r2
pos-sqz r2
BS =1/(N-1)
BS =1/2
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–– //
A B
B
B
B B
B
B
B
B B
B
B
B
input state
Teleportation networks (Furusawa, Nature 2004)
Multipartite quantum resources
• 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)
Limitations and no-go’s
• 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)
Limitations and no-go’s
• 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
0
Bosonic channels
Dissipation channel
Amplification channel
Dephasing channel
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)
DIS
SIP
AT
ION
A
MP
LIF
ICA
TIO
N
…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
ISS
IPA
TIO
N
AM
PL
IFIC
AT
ION
n : mean input energy
max 4H n
max 4 1H n
0 p
8
p
4
p
2 3 p
8
p
2
2.0
2.5
3.0
3.5
4.0
f
H
n = 1
0 p
8
p
4
p
2 3 p
8
p
2
10 12 14 16 18 20
f
H
n = 5
0 p
8
p
4
p
2 3 p
8
p
2
40
50
60
70
80
f
H
n = 20
0 p
8
p
4
p
2 3 p
8
p
2
100 120 140 160 180 200
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: 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
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).
• 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)
• State engineering / physical implementations
• (General/optics) F. Dell’Anno, S. De Siena, and F. Illuminati, Phys. Rep. 428, 53 (2006)
• (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
The University of Nottingham
http://quantumcorrelations.weebly.com
https://www.facebook.com/QuantumCorrelations
Let’s thank the organisers
of this fantastic school !!!
Antonio Zelaquett Khoury (Fluminense Federal University - UFF)
Eduardo Novais (Federal University of ABC - UFABC)
Fabricio Toscano (Federal University of Rio de Janeiro - UFRJ)
Ivan Oliveira (Brazilian Center for Physics Research- CBPF)
Marcos Cesar de Oliveira (State University of Campinas - UNICAMP)
Raul O. Vallejos (Brazilian Center for Physics Research- CBPF)
Roberto Imbuzeiro Oliveira (National Institute of Pure and Applied Math - IMPA)