Everything is Quantum Channel
sections 4.5 and 4.6

Ludmila Botelho

Before we start...

$$\rho_A \longmapsto \rho_B=\sum_i E_i \rho E_i^{\dagger}$$

Kraus representation

$$\sum_i E_i E_i^{\dagger} = \mathbb{I}_A$$

  • Noisy Evolution as the Loss of a Measurement Outcome

Kraus Operators!

\(=\mathcal{N}(\rho)\)

Interpretation of quantum channels

Environment

System

Unitary Operator

  • Noisy Evolution from a Unitary Interaction

  • Noisy Evolution from a Unitary Interaction

Interpretation of quantum channels

  • Noisy Evolution from a Unitary Interaction

Interpretation of quantum channels

  • Preparation Channel

Everything is a quantum channel

Kraus operators:

Kraus operators:

  • Appending Channel

  • Preparation Channel

parallel concatenation

  • Trace-out

Everything is a quantum channel

  • Discarding Channel

Kraus operators:

Kraus operators:

  • Unitary and Isometric Channels

Everything is a quantum channel

  • Unitary and Isometric Channels

Everything is a quantum channel

  • Reversing Unitary channel

Everything is a quantum channel

  • Reversing isometric channel

Everything is a quantum channel

  • Reversing isometric channel

Completely positive and trace preserving

Everything is a quantum channel

Everything is a quantum channel

  • Classical-to-Classical Channels

$$\rho$$

Krauss Operators

Everything is a quantum channel

  • Classical-to-Classical Channels

Everything is a quantum channel

  • Classical-to-Quantum Channels

preparation + trace-out?

Everything is a quantum channel

  • Quantum-to-classical Channels

Klaus operators?

(measurement channel)

Everything is a quantum channel

  • Quantum-to-classical Channels

Completeness?

Everything is a quantum channel

  • Entanglement-Breaking Channels

Everything is a quantum channel

  • Entanglement-Breaking Channels

Completeness?

Proof of "if" part

Everything is a quantum channel

  • Entanglement-Breaking Channels

Proof of "only if" part

unit-rank

Completeness?

Everything is a quantum channel

  • Entanglement-Breaking Channels

Everything is a quantum channel

  • Entanglement-Breaking Channels

Everything is a quantum channel

  • Quantum Instruments

Everything is a quantum channel

  • Quantum Instruments

Everything is a quantum channel

  • Quantum Instruments

Everything is a quantum channel

  • Quantum Instruments

independent!

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