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
-
Quantum Instruments





Everything is a quantum channel
-
Quantum Instruments








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Quantum Channels
By ludmilaasb
Quantum Channels
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