SLIDE 1 Andrea Mari NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56127 Pisa, Italy
Majorization and entropy at the output
- f bosonic Gaussian channels
- V. Giovannetti
(SNS, Pisa)
(Steklov Math. Inst., Moscow)
(QuIC, Bruxelles and MPI, Garching)
(QuIC, Bruxelles)
SLIDE 2
Outline
General question behind this talk Overview of recent breakthroughs on the field Gaussian states and Gaussian channels Proof of the majorization conjecture An old result about the quantum beam splitter Implications and outlook Long introduction Proof of the result
SLIDE 3
Outline
General question behind this talk Overview of recent breakthroughs on the field Gaussian states and Gaussian channels Proof of the majorization conjecture An old result about the quantum beam splitter Implications and outlook Long introduction Proof of the result
SLIDE 4 Physical transmission line:
- ptical fiber, free space communication,
satellite link, micro-waveguide, etc. Noisy output signal Radiation modes:
microwaves, radio waves, etc. WARNING for typical QIP attendees: this talk may have some implications in the real world.
SLIDE 5 General question behind this talk:
Physical transmission line:
- ptical fiber, free space communication,
satellite link, micro-waveguide, etc.
“ What is the minimum “noise” or “disorder” achievable at the
- utput state, optimizing over all possible input states? ”
Noisy output signal Important implications in communication theory:
what are the code-words which are less disturbed by the channel ?
Radiation modes:
microwaves, radio waves, etc.
SLIDE 6
What does it mean that is more “disordered” then ?
(i) Von Neumann entropy criterion (ii) Quantum state majorization criterion = eigenvalues of and arranged in decreasing order Remark: condition (ii) is very strong, indeed it can be shown that, for every concave function In particular,
SLIDE 7 (i) Minimum output entropy conjecture: (ii) Majorization conjecture: Coherent states: Vacuum state the output entropy is minimized by coherent input states
- utput of coherent input states majorize all other output
states
(phase-insensitive) Giovannetti, et al., PRA, (2004) Holevo, Werner, PRA, (1997)
SLIDE 8 (i) Minimum output entropy conjecture: (ii) Majorization conjecture: Coherent states: Vacuum state the output entropy is minimized by coherent input states
- utput of coherent input states majorize all other output
states
(phase-insensitive)
PROOF GIVEN IN THIS TALK
Giovannetti, et al., PRA, (2004) Holevo, Werner, PRA, (1997)
SLIDE 9
Outline
General question behind this talk Overview of recent breakthroughs on the field Gaussian states and Gaussian channels Proof of the majorization conjecture An old result about the quantum beam splitter Implications and outlook Long introduction Proof of the result
SLIDE 10 A hierarchy of conjectures (situation before 8/12/2013)
Majorization conjecture Minimum output entropy conjecture Gaussian additivity conjecture Gaussian optimal encoding conjecture CLASSICAL CAPACITY (explicit formula for phase-insensitive Gaussian channels)
“the max of classical capacity is obtained by Gaussian input states” “The output Holevo information is additive (entangled code-words are not required)” “the output entropy is minimized by coherent input states” “the output of coherent input states majorize all other output states”
Holevo, Werner, PRA, (1997) Giovannetti, et al., PRA, (2004) Giovannetti, et al., PRA, (2004)
SLIDE 11 A hierarchy of conjectures (situation before 8/12/2013)
Majorization conjecture Minimum output entropy conjecture Gaussian additivity conjecture Gaussian optimal encoding conjecture CLASSICAL CAPACITY (explicit formula for phase-insensitive Gaussian channels)
“the max of classical capacity is obtained by Gaussian input states” “The output Holevo information is additive (entangled code-words are not required)” “the output entropy is minimized by coherent input states” “the output of coherent input states majorize all other output states”
Holevo, Werner, PRA, (1997) Giovannetti, et al., PRA, (2004)
Entropy photon-number inequality conjecture
Guha, Shapiro, Erkmen,
- Proc. Inf. Th., IEEE, (2008).
König, Smith, Nat. Photon, (2013)
Entropy power inequality conjecture bounds for classical capacity
Giovannetti, et al., PRA, (2004)
SLIDE 12 A hierarchy of conjectures (situation before 8/12/2013)
Majorization conjecture Minimum output entropy conjecture Gaussian additivity conjecture Gaussian optimal encoding conjecture CLASSICAL CAPACITY (explicit formula for phase-insensitive Gaussian channels)
“the max of classical capacity is obtained by Gaussian input states” “The output Holevo information is additive (entangled code-words are not required)” “the output entropy is minimized by coherent input states” “the output of coherent input states majorize all other output states”
Holevo, Werner, PRA, (1997) Giovannetti, et al., PRA, (2004)
Entropy photon-number inequality conjecture
Guha, Shapiro, Erkmen,
- Proc. Inf. Th., IEEE, (2008).
König, Smith, Nat. Photon, (2013)
Entropy power inequality conjecture bounds for classical capacity Additivity and minimum of
conjecture Phase-space majorization conjecture
Giovannetti, et al., PRA, (2004)
Strong converse theorem for classical capacity of Gaussian channels
Bardhan et al., IEEE Inf. Th., (2014) Giovannetti, et al., PRA, (2004) Lieb, Solovej, arXiv:1208.3632
“output Rényi entropies
additive and minimized by coherent input states” “the Q-function of a coherent states majorizes every other Q-function”
SLIDE 13 A hierarchy of conjectures (situation before 8/12/2013)
Majorization conjecture Minimum output entropy conjecture Gaussian additivity conjecture Gaussian optimal encoding conjecture CLASSICAL CAPACITY (explicit formula for phase-insensitive Gaussian channels)
“the max of classical capacity is obtained by Gaussian input states” “The output Holevo information is additive (entangled code-words are not required)” “the output entropy is minimized by coherent input states” “the output of coherent input states majorize all other output states”
Holevo, Werner, PRA, (1997) Giovannetti, et al., PRA, (2004)
Entropy photon-number inequality conjecture
Guha, Shapiro, Erkmen,
- Proc. Inf. Th., IEEE, (2008).
König, Smith, Nat. Photon, (2013)
Entropy power inequality conjecture bounds for classical capacity Additivity and minimum of
conjecture Phase-space majorization conjecture
Giovannetti, et al., PRA, (2004)
Strong converse theorem for classical capacity of Gaussian channels
Bardhan et al., IEEE Inf. Th., (2014) Giovannetti, et al., PRA, (2004) Lieb, Solovej, arXiv:1208.3632
“output Rényi entropies
additive and minimized by coherent input states” “the Q-function of a coherent states majorizes every other Q-function”
SLIDE 14 A hierarchy of conjectures (situation before 8/12/2013)
Majorization conjecture Minimum output entropy conjecture Gaussian additivity conjecture Gaussian optimal encoding conjecture CLASSICAL CAPACITY (explicit formula for phase-insensitive Gaussian channels)
“the max of classical capacity is obtained by Gaussian input states” “The output Holevo information is additive (entangled code-words are not required)” “the output entropy is minimized by coherent input states” “the output of coherent input states majorize all other output states”
Holevo, Werner, PRA, (1997) Giovannetti, et al., PRA, (2004)
Entropy photon-number inequality conjecture
Guha, Shapiro, Erkmen,
- Proc. Inf. Th., IEEE, (2008).
König, Smith, Nat. Photon, (2013)
Entropy power inequality conjecture bounds for classical capacity Additivity and minimum of
conjecture Phase-space majorization conjecture
Giovannetti, et al., PRA, (2004)
Strong converse theorem for classical capacity of Gaussian channels
Bardhan et al., IEEE Inf. Th., (2014) Giovannetti, et al., PRA, (2004) Lieb, Solovej, arXiv:1208.3632
“output Rényi entropies
additive and minimized by coherent input states” “the Q-function of a coherent states majorizes every other Q-function”
SLIDE 15 A hierarchy of conjectures (situation before 8/12/2013)
Majorization conjecture Minimum output entropy conjecture Gaussian additivity conjecture Gaussian optimal encoding conjecture CLASSICAL CAPACITY (explicit formula for phase-insensitive Gaussian channels)
“the max of classical capacity is obtained by Gaussian input states” “The output Holevo information is additive (entangled code-words are not required)” “the output entropy is minimized by coherent input states” “the output of coherent input states majorize all other output states”
Holevo, Werner, PRA, (1997) Giovannetti, et al., PRA, (2004)
Entropy photon-number inequality conjecture
Guha, Shapiro, Erkmen,
- Proc. Inf. Th., IEEE, (2008).
König, Smith, Nat. Photon, (2013)
Entropy power inequality conjecture bounds for classical capacity Additivity and minimum of
conjecture Phase-space majorization conjecture
Giovannetti, et al., PRA, (2004)
Strong converse theorem for classical capacity of Gaussian channels
Bardhan et al., IEEE Inf. Th., (2014) Giovannetti, et al., PRA, (2004) Lieb, Solovej, arXiv:1208.3632
“output Rényi entropies
additive and minimized by coherent input states” “the Q-function of a coherent states majorizes every other Q-function”
SLIDE 16 proofs (8/12/2013)
Majorization conjecture Minimum output entropy conjecture Gaussian additivity conjecture Gaussian optimal encoding conjecture CLASSICAL CAPACITY (explicit formula for phase-insensitive Gaussian channels)
“the max of classical capacity is obtained by Gaussian input states” “The Holevo information is additive (entangled code-words are not required)” “the output entropy is minimized by coherent input states” “the output of coherent input states majorize all other output states”
Holevo, Werner, PRA, (1997) Giovannetti, et al., PRA, (2004)
Entropy photon-number inequality conjecture
Guha, Shapiro, Erkmen,
- Proc. Inf. Th., IEEE, (2008).
bounds for classical capacity Additivity and minimum of
conjecture
Giovannetti, et al., PRA, (2004)
Strong converse theorem for classical capacity of Gaussian channels
Bardhan et al., IEEE Inf. Th., (2014) Giovannetti, et al., PRA, (2004)
A hierarchy of conjectures
Phase-space majorization conjecture
König, Smith, Nat. Photon, (2013)
Entropy power inequality conjecture
G i
a n n e t t i , H
e v
G a r c i a
a t r
, a r X i v : 1 3 1 2 . 2 2 5 1 , C
m . M a t h . P h y s . , ( 2 1 4 )
Lieb, Solovej, arXiv:1208.3632
“output Rényi entropies
additive and minimized by coherent input states” “the Q-function of a coherent states majorizes every other Q-function”
SLIDE 17 proofs (12/12/2013)
Majorization conjecture Minimum output entropy conjecture Gaussian additivity conjecture Gaussian optimal encoding conjecture CLASSICAL CAPACITY (explicit formula for phase-insensitive Gaussian channels)
“the max of classical capacity is obtained by Gaussian input states” “The Holevo information is additive (entangled code-words are not required)” “the output entropy is minimized by coherent input states” “the output of coherent input states majorize all other output states”
Holevo, Werner, PRA, (1997) Giovannetti, et al., PRA, (2004)
Entropy photon-number inequality conjecture
Guha, Shapiro, Erkmen,
- Proc. Inf. Th., IEEE, (2008).
bounds for classical capacity Additivity and minimum of
conjecture
Giovannetti, et al., PRA, (2004)
Strong converse theorem for classical capacity of Gaussian channels
Bardhan et al., IEEE Inf. Th., (2014) Giovannetti, et al., PRA, (2004)
A hierarchy of conjectures
M a r i , G i
a n n e t t i , H
e v
a r X i v : 1 3 1 2 . 6 2 2 5 N a t . C
m . ( 2 1 4 )
Phase-space majorization conjecture
König, Smith, Nat. Photon, (2013)
Entropy power inequality conjecture
G i
a n n e t t i , H
e v
G a r c i a
a t r
, a r X i v : 1 3 1 2 . 2 2 5 1 , C
m . M a t h . P h y s . , ( 2 1 4 )
Lieb, Solovej, arXiv:1208.3632
“output Rényi entropies
additive and minimized by coherent input states” “the Q-function of a coherent states majorizes every other Q-function”
SLIDE 18 proofs (21/12/2013)
Majorization conjecture Minimum output entropy conjecture
“the output entropy is minimized by coherent input states” “the output of coherent input states majorize all other output states”
Giovannetti, et al., PRA, (2004)
Entropy photon-number inequality conjecture
Guha, Shapiro, Erkmen,
- Proc. Inf. Th., IEEE, (2008).
bounds for classical capacity Additivity and minimum of
conjecture
“output Rényi entropies
additive and minimized by coherent input states”
Giovannetti, et al., PRA, (2004)
Strong converse theorem for classical capacity of Gaussian channels
Bardhan et al., IEEE Inf. Th., (2014) Giovannetti, et al., PRA, (2004)
A hierarchy of conjectures
M a r i , G i
a n n e t t i , H
e v
a r X i v : 1 3 1 2 . 6 2 2 5 N a t . C
m . ( 2 1 4 )
Phase-space majorization conjecture
König, Smith, Nat. Photon, (2013)
Entropy power inequality conjecture Gaussian additivity conjecture Gaussian optimal encoding conjecture CLASSICAL CAPACITY (explicit formula for phase-insensitive Gaussian channels)
“the max of classical capacity is obtained by Gaussian input states” “The Holevo information is additive (entangled code-words are not required)”
Holevo, Werner, PRA, (1997)
G i
a n n e t t i , G a r c i a
a t r
, C e r f , H
e v
r X i v : 1 3 1 2 . 6 2 2 5 , N a t . P h
. ( 2 1 4 ) G i
a n n e t t i , H
e v
G a r c i a
a t r
, a r X i v : 1 3 1 2 . 2 2 5 1 , C
m . M a t h . P h y s . , ( 2 1 4 )
“the Q-function of a coherent states majorizes every other Q-function”
Lieb, Solovej, arXiv:1208.3632
SLIDE 19 proofs (16/01/2014)
Majorization conjecture Minimum output entropy conjecture
“the output entropy is minimized by coherent input states” “the output of coherent input states majorize all other output states”
Giovannetti, et al., PRA, (2004)
Entropy photon-number inequality conjecture
Guha, Shapiro, Erkmen,
- Proc. Inf. Th., IEEE, (2008).
bounds for classical capacity Additivity and minimum of
conjecture
“output Rényi entropies
additive and minimized by coherent input states”
Giovannetti, et al., PRA, (2004)
Strong converse theorem for classical capacity of Gaussian channels
Giovannetti, et al., PRA, (2004)
A hierarchy of conjectures
M a r i , G i
a n n e t t i , H
e v
a r X i v : 1 3 1 2 . 6 2 2 5 N a t . C
m . ( 2 1 4 )
Phase-space majorization conjecture
König, Smith, Nat. Photon, (2013)
Entropy power inequality conjecture Gaussian additivity conjecture Gaussian optimal encoding conjecture CLASSICAL CAPACITY (explicit formula for phase-insensitive Gaussian channels)
“the max of classical capacity is obtained by Gaussian input states” “The Holevo information is additive (entangled code-words are not required)”
Holevo, Werner, PRA, (1997)
G i
a n n e t t i , G a r c i a
a t r
, C e r f , H
e v
r X i v : 1 3 1 2 . 6 2 2 5 , N a t . P h
. ( 2 1 4 ) G i
a n n e t t i , H
e v
G a r c i a
a t r
, a r X i v : 1 3 1 2 . 2 2 5 1 , C
m . M a t h . P h y s . , ( 2 1 4 ) Bardhan, Garcia-Patron, Wilde, Winter arXiv:1401.4161, IEEE Inf. Th., (2014)
“the Q-function of a coherent states majorizes every other Q-function”
Lieb, Solovej, arXiv:1208.3632
SLIDE 20 proofs (03/02/2014)
Majorization conjecture Minimum output entropy conjecture
“the output entropy is minimized by coherent input states” “the output of coherent input states majorize all other output states”
Giovannetti, et al., PRA, (2004)
Entropy photon-number inequality conjecture
Guha, Shapiro, Erkmen,
- Proc. Inf. Th., IEEE, (2008).
bounds for classical capacity Additivity and minimum of
conjecture
“output Rényi entropies
additive and minimized by coherent input states”
Giovannetti, et al., PRA, (2004)
Strong converse theorem for classical capacity of Gaussian channels
Giovannetti, et al., PRA, (2004)
A hierarchy of conjectures
M a r i , G i
a n n e t t i , H
e v
a r X i v : 1 3 1 2 . 6 2 2 5 N a t . C
m . ( 2 1 4 )
Phase-space majorization conjecture
König, Smith, Nat. Photon, (2013)
Entropy power inequality conjecture Gaussian additivity conjecture Gaussian optimal encoding conjecture CLASSICAL CAPACITY (explicit formula for phase-insensitive Gaussian channels)
“the max of classical capacity is obtained by Gaussian input states” “The Holevo information is additive (entangled code-words are not required)”
Holevo, Werner, PRA, (1997)
G i
a n n e t t i , G a r c i a
a t r
, C e r f , H
e v
r X i v : 1 3 1 2 . 6 2 2 5 , N a t . P h
. ( 2 1 4 ) De Palma, Mari, Giovannetti, arXiv: 1402.0404, Nat. Phot. (2014) De Palma, Mari, Lloyd, Giovannetti, arXiv:1408.6410 G i
a n n e t t i , H
e v
G a r c i a
a t r
, a r X i v : 1 3 1 2 . 2 2 5 1 , C
m . M a t h . P h y s . , ( 2 1 4 ) Bardhan, Garcia-Patron, Wilde, Winter arXiv:1401.4161, IEEE Inf. Th., (2014)
“the Q-function of a coherent states majorizes every other Q-function”
Lieb, Solovej, arXiv:1208.3632
SLIDE 21 proofs (16/05/2014) A hierarchy of conjectures
Majorization conjecture Minimum output entropy conjecture
“the output entropy is minimized by coherent input states” “the output of coherent input states majorize all other output states”
Giovannetti, et al., PRA, (2004)
Entropy photon-number inequality conjecture
Guha, Shapiro, Erkmen,
- Proc. Inf. Th., IEEE, (2008).
bounds for classical capacity Additivity and minimum of
conjecture
“output Rényi entropies
additive and minimized by coherent input states”
Giovannetti, et al., PRA, (2004)
Strong converse theorem for classical capacity of Gaussian channels
Giovannetti, et al., PRA, (2004)
M a r i , G i
a n n e t t i , H
e v
a r X i v : 1 3 1 2 . 6 2 2 5 N a t . C
m . ( 2 1 4 )
Phase-space majorization conjecture
König, Smith, Nat. Photon, (2013)
Entropy power inequality conjecture Gaussian additivity conjecture Gaussian optimal encoding conjecture CLASSICAL CAPACITY (explicit formula for phase-insensitive Gaussian channels)
“the max of classical capacity is obtained by Gaussian input states” “The Holevo information is additive (entangled code-words are not required)”
Holevo, Werner, PRA, (1997)
G i
a n n e t t i , G a r c i a
a t r
, C e r f , H
e v
r X i v : 1 3 1 2 . 6 2 2 5 , N a t . P h
. ( 2 1 4 ) De Palma, Mari, Giovannetti, arXiv: 1402.0404, Nat. Phot. (2014) De Palma, Mari, Lloyd, Giovannetti, arXiv:1408.6410 G i
a n n e t t i , H
e v
G a r c i a
a t r
, a r X i v : 1 3 1 2 . 2 2 5 1 , C
m . M a t h . P h y s . , ( 2 1 4 ) Bardhan, Garcia-Patron, Wilde, Winter arXiv:1401.4161, IEEE Inf. Th., (2014)
“the Q-function of a coherent states majorizes every other Q-function”
Lieb, Solovej, arXiv:1208.3632
- V. Giovannetti, A. S. Holevo, A. Mari
arXiv:1405.4066
SLIDE 22 proofs (state of the art)
Majorization conjecture Minimum output entropy conjecture
“the output entropy is minimized by coherent input states” “the output of coherent input states majorize all other output states”
Giovannetti, et al., PRA, (2004)
Entropy photon-number inequality conjecture
Guha, Shapiro, Erkmen,
- Proc. Inf. Th., IEEE, (2008).
bounds for classical capacity Additivity and minimum of
conjecture
“output Rényi entropies
additive and minimized by coherent input states”
Giovannetti, et al., PRA, (2004)
Strong converse theorem for classical capacity of Gaussian channels
Giovannetti, et al., PRA, (2004)
A hierarchy of conjectures
M a r i , G i
a n n e t t i , H
e v
a r X i v : 1 3 1 2 . 6 2 2 5 N a t . C
m . ( 2 1 4 )
Phase-space majorization conjecture
König, Smith, Nat. Photon, (2013)
Entropy power inequality conjecture Gaussian additivity conjecture Gaussian optimal encoding conjecture CLASSICAL CAPACITY (explicit formula for phase-insensitive Gaussian channels)
“the max of classical capacity is obtained by Gaussian input states” “The Holevo information is additive (entangled code-words are not required)”
Holevo, Werner, PRA, (1997)
G i
a n n e t t i , G a r c i a
a t r
, C e r f , H
e v
r X i v : 1 3 1 2 . 6 2 2 5 , N a t . P h
. ( 2 1 4 ) De Palma, Mari, Giovannetti, arXiv: 1402.0404, Nat. Phot. (2014) De Palma, Mari, Lloyd, Giovannetti, arXiv:1408.6410 G i
a n n e t t i , H
e v
G a r c i a
a t r
, a r X i v : 1 3 1 2 . 2 2 5 1 , C
m . M a t h . P h y s . , ( 2 1 4 ) Bardhan, Garcia-Patron, Wilde, Winter arXiv:1401.4161, IEEE Inf. Th., (2014)
“the Q-function of a coherent states majorizes every other Q-function”
Lieb, Solovej, arXiv:1208.3632
- V. Giovannetti, A. S. Holevo, A. Mari
arXiv:1405.4066
SLIDE 23
Outline
General question behind this talk Overview of recent breakthroughs on the field Gaussian states and Gaussian channels Proof of the majorization conjecture An old result about the quantum beam splitter Implications and outlook Long introduction Proof of the result
SLIDE 24 Quantum bosonic systems
e.g. single mode of electromagnetic radiation
Eigenstates (Fock states): vacuum Coherent states: Displacement (or Weyl) operator
SLIDE 25
Phase space quasi-distributions
Characteristic function (or Fourier-Weyl transform): Wigner function: (closest analogue to a classical phase-space density but can assume negative values) Q-function: (“not so close analogue” to a phase-space distribution but at least it is positive) One can use it to define entropy (and majorization) as for classical random variables Wherl entropy:
SLIDE 26
Gaussian states
Gaussian states = states with Gaussian phase space quasi-distribution vacuum state coherent states thermal state
SLIDE 27
Gaussian channels
Gaussian channels = CPT operations mapping Gaussian states into Gaussian states In general: Gaussian Gaussian unitary channels: quadratic in
SLIDE 28
Gaussian channels
Gaussian channels = CPT operations mapping Gaussian states into Gaussian states. Phase-insensitive Gaussian channels = those commuting with phase space rotations In general: Gaussian Gaussian unitary channels: quadratic in Most common channels are phase-insensitive: quantum limited attenuator, thermal attenuator, quantum limited amplifier, thermal amplifier, additive Gaussian noise. (gauge-covariant )
For a single mode,
SLIDE 29
Gaussian phase-insensitive channels
Quantum limited attenuator (beam splitter) Thermal attenuator (thermal beam splitter) thermal state with
SLIDE 30
Gaussian phase-insensitive channels
Quantum limited amplifier Thermal amplifier thermal state with
SLIDE 31
Gaussian phase-insensitive channels
Quantum limited amplifier Complementary of quantum limited amplifier (this is not phase-insensitive) Important property: for a pure input the complementary output states and have the same spectrum. phase-conjugation
SLIDE 32 Gaussian phase-insensitive channels
Additive classical noise channel Random displacement with variance Noiseless phase conjugation (this is not phase-insensitive and not CPT)
- Transposition in Fock basis
- Equivalent to time inversion
Important property: for any state , have the same spectrum. and
SLIDE 33
Gaussian phase-insensitive channels
Summary in terms of characteristic functions Attenuator Complementary of q. lim. amplifier Additive classical noise Amplifier Noiseless phase-conjugation (phase-contravariant) (phase-contravariant, not CPT)
SLIDE 34 Gaussian phase-insensitive channels
Every phase-insensitive Gaussian channel is equivalent to a quantum limited attenuator followed by a quantum limited amplifier.
Giovannetti, et al., PRA, (2004) Garcia-Patron et al. PRL, (2012)
SLIDE 35 Gaussian phase-insensitive channels
Example: complementary of quantum limited amplifier Every phase-insensitive Gaussian channel is equivalent to a quantum limited attenuator followed by a quantum limited amplifier. Every phase-contravaraint Gaussian channel is equivalent to a quantum limited attenuator followed by a quantum limited amplifier, and a noiseless phase-conjugation.
Giovannetti, et al., PRA, (2004) Garcia-Patron et al. PRL, (2012)
Giovannetti, Holevo, Garcia-Patron, arXiv:1312.2251, Comm. Math. Phys., (2014)
SLIDE 36
Outline
General question behind this talk Overview of recent breakthroughs on the field Gaussian states and Gaussian channels Proof of the majorization conjecture An old result about the quantum beam splitter Implications and outlook Long introduction Proof of the result
SLIDE 37
An old result about the quantum beam splitter
Aharanov et al., Ann. of Phys., (1966).
SLIDE 38 An old result about the quantum beam splitter
Radio transmission
concerto
A C B
Radio transmission
concerto
B
Radio transmission
concerto
C
Case 1 Case 2 Is it possible to distinguish between case 1 and case 2 ?
Aharanov et al., Ann. of Phys., (1966)
SLIDE 39 An old result about the quantum beam splitter
Radio transmission
concerto
A C B
Radio transmission
concerto
B
Radio transmission
concerto
C
Case 1 Case 2 Is it possible to distinguish between case 1 and case 2 ? It is always possible unless the transmitted signals are encoded on coherent states
Aharanov et al., Ann. of Phys., (1966)
SLIDE 40 An old result about the quantum beam splitter
The “golden property” of a beam splitter: the only input states producing pure output states for a quantum limited attenuator are coherent states.
Aharanov et al., Ann. of Phys., (1966) Asbòth, Calsamiglia, Ritsch, PRL, (2005) Jiang, Lang, Caves, PRA, (2013)
SLIDE 41 An old result about the quantum beam splitter
The “golden property” of a beam splitter: the only input states producing pure output states for a quantum limited attenuator are coherent states.
def characteristic function
solutions are exp functions
proof:
Aharanov et al., Ann. of Phys., (1966) Asbòth, Calsamiglia, Ritsch, PRL, (2005) Jiang, Lang, Caves, PRA, (2013)
SLIDE 42
Outline
General question behind this talk Overview of recent breakthroughs on the field Gaussian states and Gaussian channels Proof of the majorization conjecture An old result about the quantum beam splitter Implications and outlook Long introduction Proof of the result
SLIDE 43 Proof of the majorization conjecture
(phase-insensitive)
Majorization conjecture: the output of coherent input states majorize all
SLIDE 44 Proof of the majorization conjecture
(phase-insensitive)
Majorization conjecture: the output of coherent input states majorize all
We prove an equivalent and actually stronger proposition: Minimization of stricly concave output funcitonals: For every nonnegative, unitary invariant, and strictly concave functional and for every quantum state , moreover the equality sign is obtained only if is a coherent state. Strict concavity: equality holds only if
SLIDE 45
Proof of the majorization conjecture
is unitary invariant and strictly concave. Then the minimization problem is trivially optimized by pure states and only by them.
SLIDE 46 Proof of the majorization conjecture
is pure, if and only if, is coherent. is unitary invariant and strictly concave. Then the minimization problem is trivially optimized by pure states and only by them. Special case: quantum limited attenuator Then, is minimized only by coherent input states. “golden property”
Aharanov et al., Ann. of Phys., (1966)
SLIDE 47 Proof of the majorization conjecture
is pure, if and only if, is coherent. is unitary invariant and strictly concave. Then the minimization problem is trivially optimized by pure states and only by them. Special case: quantum limited attenuator Then, is minimized only by coherent input states. General case: phase-invariant Gaussian channel Decomposition property maps coherent states into coherent states It is enough to prove the theorem only for the quantum limited amplifier
Garcia-Patron et al. PRL, (2012)
“golden property”
Aharanov et al., Ann. of Phys., (1966)
SLIDE 48
Proof of the majorization conjecture
is strictly concave it is minimized only by pure input states. (for “invertible” channels)
SLIDE 49
Proof of the majorization conjecture
Quantum limited amplifier applied to pure states same spectrum is strictly concave it is minimized only by pure input states. (for “invertible” channels)
SLIDE 50 Proof of the majorization conjecture
Quantum limited amplifier applied to pure states same spectrum Decomposition of the complementary channel same channel !
Giovannetti, Holevo, Garcia-Patron, arXiv:1312.2251, Comm. Math. Phys., (2014)
is strictly concave it is minimized only by pure input states. (for “invertible” channels)
SLIDE 51 Proof of the majorization conjecture
Quantum limited amplifier applied to pure states same spectrum Decomposition of the complementary channel same channel !
Giovannetti, Holevo, Garcia-Patron, arXiv:1312.2251, Comm. Math. Phys., (2014)
up to this point same proof of is strictly concave it is minimized only by pure input states. (for “invertible” channels)
SLIDE 52 Proof of the majorization conjecture
Quantum limited amplifier applied to pure states same spectrum Decomposition of the complementary channel same channel ! Take to be an optimal minimizer, then also must be optimal. But optimal states must be pure
Giovannetti, Holevo, Garcia-Patron, arXiv:1312.2251, Comm. Math. Phys., (2014)
is strictly concave it is minimized only by pure input states. (for “invertible” channels)
SLIDE 53 Proof of the majorization conjecture
Quantum limited amplifier applied to pure states same spectrum Decomposition of the complementary channel same channel ! Take to be an optimal minimizer, then also must be optimal. But optimal states must be pure this is possible if and only if is coherent. “golden property”
Aharanov et al., Ann. of Phys., (1966)
Giovannetti, Holevo, Garcia-Patron, arXiv:1312.2251, Comm. Math. Phys., (2014)
is strictly concave it is minimized only by pure input states. (for “invertible” channels)
SLIDE 54
We have just proved that:
Minimization of stricly concave output funcitonals: For every nonnegative, unitary invariant, and strictly concave functional and for every quantum state , moreover the equality sign is obtained only if is a coherent state.
SLIDE 55
We have just proved that:
Minimization of stricly concave output funcitonals: For every nonnegative, unitary invariant, and strictly concave functional and for every quantum state , moreover the equality is sign is obtained only if is a coherent state. Majorization conjecture: the output of coherent input states majorize all other output states
SLIDE 56
Outline
General question behind this talk Overview of recent breakthroughs on the field Gaussian states and Gaussian channels Proof of the majorization conjecture An old result about the quantum beam splitter Implications and outlook Long introduction Proof of the result
SLIDE 57
We have just proved that:
Minimization of stricly concave output funcitonals: For every nonnegative, unitary invariant, and strictly concave functional and for every quantum state , moreover the equality sign is obtained only if is a coherent state. Majorization conjecture: the output of coherent input states majorize all other output states
SLIDE 58
We have just proved that:
Minimization of stricly concave output funcitonals: For every nonnegative, unitary invariant, and strictly concave functional and for every quantum state , moreover the equality sign is obtained only if is a coherent state. Majorization conjecture: the output of coherent input states majorize all other output states (Strong) minimum output entropy conjecture: the output entropy is minimized only by coherent input states proof: take
SLIDE 59 We have just proved that:
Minimization of stricly concave output funcitonals: For every nonnegative, unitary invariant, and strictly concave functional and for every quantum state , moreover the equality sign is obtained only if is a coherent state. Majorization conjecture: the output of coherent input states majorize all other output states (Strong) minimum output entropy conjecture: the output entropy is minimized only by coherent input states proof: take (Strong) minimum output Rényi entropy conjecture: the output Rényi entropy is minimized
- nly by coherent input states
proof: take
SLIDE 60 We have just proved that:
Minimization of stricly concave output funcitonals: For every nonnegative, unitary invariant, and strictly concave functional and for every quantum state , moreover the equality sign is obtained only if is a coherent state. Majorization conjecture: the output of coherent input states majorize all other output states (Strong) minimum output entropy conjecture: the output entropy is minimized only by coherent input states proof: take (Strong) minimum output Rényi entropy conjecture: the output Rényi entropy is minimized
- nly by coherent input states
proof: take Phase space majorization The (generalized) Q-function of a coherent state majorizes every other (generalized) Q-function
Lieb, Solovej, arXiv (2012)
proof: Giovannetti, Holevo, Mari
arXiv:1405.4066
SLIDE 61 General question behind this talk:
Physical transmission line:
- ptical fiber, free space communication,
satellite link, micro-waveguide, etc. Noisy output signal Important implications in communication theory:
what are the code-words which are less disturbed by the channel ?
Radiation modes:
microwaves, radio waves, etc.
Conclusions
“ What is the minimum “noise” or “disorder” achievable at the
- utput state, optimizing over all possible input states? ”
SLIDE 62 General question behind this talk:
Physical transmission line:
- ptical fiber, free space communication,
satellite link, micro-waveguide, etc. Noisy output signal Important implications in communication theory:
what are the code-words which are less disturbed by the channel ?
Input coherent states produce the least “disordered” output states
Radiation modes:
microwaves, radio waves, etc.
Answer:
Conclusions
“ What is the minimum “noise” or “disorder” achievable at the
- utput state, optimizing over all possible input states? ”
SLIDE 63 proofs (state of the art)
Majorization conjecture Minimum output entropy conjecture
“the output entropy is minimized by coherent input states” “the output of coherent input states majorize all other output states”
Giovannetti, et al., PRA, (2004)
Entropy photon-number inequality conjecture
Guha, Shapiro, Erkmen,
- Proc. Inf. Th., IEEE, (2008).
bounds for classical capacity Additivity and minimum of
conjecture
“output Rényi entropies
additive and minimized by coherent input states”
Giovannetti, et al., PRA, (2004)
Strong converse theorem for classical capacity of Gaussian channels
Giovannetti, et al., PRA, (2004)
A hierarchy of conjectures
M a r i , G i
a n n e t t i , H
e v
a r X i v : 1 3 1 2 . 6 2 2 5 N a t . C
m . ( 2 1 4 )
Phase-space majorization conjecture
König, Smith, Nat. Photon, (2013)
Entropy power inequality conjecture Gaussian additivity conjecture Gaussian optimal encoding conjecture CLASSICAL CAPACITY (explicit formula for phase-insensitive Gaussian channels)
“the max of classical capacity is obtained by Gaussian input states” “The Holevo information is additive (entangled code-words are not required)”
Holevo, Werner, PRA, (1997)
G i
a n n e t t i , G a r c i a
a t r
, C e r f , H
e v
r X i v : 1 3 1 2 . 6 2 2 5 , N a t . P h
. ( 2 1 4 ) De Palma, Mari, Giovannetti, arXiv: 1402.0404, Nat. Phot. (2014) De Palma, Mari, Lloyd, Giovannetti, arXiv:1408.6410 G i
a n n e t t i , H
e v
G a r c i a
a t r
, a r X i v : 1 3 1 2 . 2 2 5 1 , C
m . M a t h . P h y s . , ( 2 1 4 ) Bardhan, Garcia-Patron, Wilde, Winter arXiv:1401.4161, IEEE Inf. Th., (2014)
“the Q-function of a coherent states majorizes every other Q-function”
Lieb, Solovej, arXiv:1208.3632
- V. Giovannetti, A. S. Holevo, A. Mari
arXiv:1405.4066
Quantum Capacity there are still gray problems...
SLIDE 64
Supplementary material
For every non-negative, strictly concave function f(x), Proof ( ): follows trivially from concavity assume Proof ( ): then there exists a minimum integer such that 1 x violates the initial inequality construct the function Problem: is concave but not strictly concave. It's not a big problem, just make it strictly concave: small enough
