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
$$\sum_i E_i E_i^{\dagger} = \mathbb{I}_A$$
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Noisy Evolution as the Loss of a Measurement Outcome
Kraus Operators!
\(=\mathcal{N}(\rho)\)
Interpretation of quantum channels
Environment
System
Unitary Operator
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Noisy Evolution from a Unitary Interaction
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Noisy Evolution from a Unitary Interaction
Interpretation of quantum channels
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Noisy Evolution from a Unitary Interaction
Interpretation of quantum channels
Preparation Channel
Everything is a quantum channel
Kraus operators:
Kraus operators:
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Appending Channel
Preparation Channel
parallel concatenation
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Trace-out
Everything is a quantum channel
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Discarding Channel
Kraus operators:
Kraus operators:
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Unitary and Isometric Channels
Everything is a quantum channel
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Unitary and Isometric Channels
Everything is a quantum channel
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Reversing Unitary channel
Everything is a quantum channel
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Reversing isometric channel
Everything is a quantum channel
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Reversing isometric channel
Completely positive and trace preserving
Everything is a quantum channel
Everything is a quantum channel
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Classical-to-Classical Channels
$$\rho$$
Krauss Operators
Everything is a quantum channel
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Classical-to-Classical Channels
Everything is a quantum channel
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Classical-to-Quantum Channels
preparation + trace-out?
Everything is a quantum channel
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Quantum-to-classical Channels
Klaus operators?
(measurement channel)
Everything is a quantum channel
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Quantum-to-classical Channels
Completeness?
Everything is a quantum channel
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Entanglement-Breaking Channels
Everything is a quantum channel
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Entanglement-Breaking Channels
Completeness?
Proof of "if" part
Everything is a quantum channel
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Entanglement-Breaking Channels
Proof of "only if" part
unit-rank
Completeness?
Everything is a quantum channel
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Entanglement-Breaking Channels
Everything is a quantum channel
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Entanglement-Breaking Channels
Everything is a quantum channel
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Quantum Instruments
Everything is a quantum channel
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Quantum Instruments
Everything is a quantum channel
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Quantum Instruments
Everything is a quantum channel
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Quantum Instruments
independent!