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TITLE PAGE TITLE HERE TITLE HERE DIVISIBILITY OF QUBIT CHANNELS - - PowerPoint PPT Presentation

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TITLE PAGE TITLE HERE TITLE HERE DIVISIBILITY OF QUBIT CHANNELS AND DYNAMICAL MAPS David Carlos Mario Ziman Davalos Pineda PLAN 1. Channel divisibility 2. dynamical maps 3. Subsets of 1. 4. How 2. passes through 3. QUANTUM CHANNELS

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TITLE HERE TITLE HERE TITLE PAGE Mario Ziman DIVISIBILITY OF QUBIT CHANNELS AND DYNAMICAL MAPS

David Davalos Carlos Pineda

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PLAN

  • 1. Channel divisibility
  • 2. dynamical maps
  • 4. How 2. passes through 3.
  • 3. Subsets of 1.
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QUANTUM CHANNELS

C H A N N E L time

Φ completely positive

trace-preserving linear map

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DIVISIBILITY

C H A N N E L time C H A N N E L

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TYPES OF DIVISIBILITY TYPES OF DIVISIBILITY

in

fjnitely simal infnitely divisible infnitesimal divisible I n d I v I s I b l e

DIVISIBLE

M.Wolf, J. Eisert, T. Cubitt, I. Cirac, Phys. Rev. Lett., 101, 150402 (2008) M.Wolf, I. Cirac, Comm. Math. Phys. 279, 147–168 (2008)

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identity

SET OF CHANNELS

  • convex set

Φ

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identity

SET OF CHANNELS

Φ

infnitely-divisible infnitesimal-divisible divisible indivisible

C∞ ⊂Cinf ⊂Cdiv Cindivisible, e.g. qNOT

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identity

DYNAMICAL MAP

SET OF CHANNELS

t Φ →

t

Φ

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identity

DYNAMICAL MAP

SET OF CHANNELS

t Φ →

t

Δ=Φ·Ψ-1

Φ

Ψ Δ

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identity

DYNAMICAL MAP

SET OF CHANNELS

t Φ →

t

Δ=Φ·Ψ-1

Φ Ψ Δ Δ t → Ψt

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identity

DYNAMICAL MAP

SET OF CHANNELS

t Φ →

t

t → Ψt

semigroup

t → εt

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identity

DYNAMICAL MAP

SET OF CHANNELS

t Φ →

t

DIVISIBILITY OF DYNAMICAL MAPS

t → Ψt

semigroup

t → εt

CP P

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DYNAMICAL MAP

identity

L-divisible CP-divisible P-divisible IN-divisible for all time intervals CP P NP

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CHANNEL TYPES

identity

CL L-divisible CCP CP-divisible CP P-divisible CNP NP-divisible

Achievability by dynamical maps + closure

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CHANNEL TYPES

CL L-divisible CCP CP-divisible CP P-divisible CNP NP-divisible CL CCP CP CNP=C

Cindivisible Cdiv divisible Cinfinfnitesimal-divisible C∞ infnitely-divisible C∞ Cinf Cdiv

=

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CHANNEL TYPES

CL L-divisible CCP CP-divisible CP P-divisible CNP NP-divisible CL CCP CP CNP=C

Cindivisible Cdiv divisible Cinfinfnitesimal-divisible C∞ infnitely-divisible C∞ Cinf Cdiv

=

C∞⊂Cinf ⊂Cdiv CL ⊂CCP ⊂CP ⊂ =

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QUBIT UNITAL

identity

CP ⇔ det≥0 CP

i n d i v i s i b l e Cindivisible = faces

not CP CP

but not divisible

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QUBIT UNITAL

CL CCP \ CL

no tetrahedron symmetries plus tetrahedron symmetries

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QUBIT UNITAL

identity

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QUBIT NONUNITAL

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DYNAMICAL MAP

identity

L-divisible CP-divisible P-divisible NP-divisible dynamical phases

  • intervals
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CHANNELS

identity

L-divisible CP-divisible divisible indivisible dynamical phases

  • pointwise
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QUESTION

identity

Which transitions are allowed?

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QUBIT UNITAL

identity

All types of borders exist.

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QUBIT DYNAMICAL MAP

identity qNOT

CL CP Cdiv

Time evolution to quantum NOT

  • T. Rybár, S. N. Filippov, M. Ziman, V. Bužek.
  • J. Phys. B, 45, 154006 (2012)
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QUBIT DYNAMICAL MAP

Jaynes-Cumming model

  • E. T. Jaynes and F. W. Cummings. Proc. IEEE 51, 89 (1963)
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THANK YOU THANK YOU FOR YOUR FOR YOUR ATTENTION ATTENTION GAME OVER

  • D. Davalos, M. Ziman, C. Pineda, Quantum 3, 144 (2019)