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Canonical static concept record

Partial trace

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Summary

For a trace-class operator on $H_A\otimes H_B$ , the unique trace-class operator on $H_A$ satisfying $\operatorname{Tr}[(X\otimes I)T]=\operatorname{Tr}[X\,\operatorname{Tr}_B T]$ for bounded $X$ .

Record metadata

Canonical name: Partial trace
Concept ID: partial_trace
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Analysis
Domains: Analysis
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Operator construction

Carrier(s)

Hilbert spaces $H_A,H_B$
trace-class operator $T$ on $H_A\otimes H_B$

Data

linear map $\operatorname{Tr}_B T$

Axioms / constraints

trace-pairing characterization with bounded operators on $H_A$

Canonically induces

reduced density operator

Notes

The partial trace is not defined on every bounded operator in infinite dimensions.

Concept sources

Wikipedia — Partial trace — Partial trace · encyclopedic reference · source ID wp-partial_trace
nLab — partial trace — partial trace · mathematical reference · source ID nlab-partial_trace
Cambridge — Quantum Mechanics — Quantum Mechanics · University of Cambridge lecture notes · source ID tong-quantum

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Tensor-product Hilbert space Partial trace

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: partialtrace_from_tensor
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: trace one subsystem

Authored explanation

The partial trace maps trace-class operators on $H_A\otimes H_B$ to trace-class reduced operators on one factor, characterized by the trace pairing with bounded operators on that factor.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Wikipedia — Tensor product of Hilbert spaces — Tensor product of Hilbert spaces · encyclopedic reference · source ID wp-tensor_product_hilbert_space
nLab — tensor product of Hilbert spaces — tensor product of Hilbert spaces · mathematical reference · source ID nlab-tensor_product_hilbert_space
Cambridge — Quantum Mechanics — Quantum Mechanics · University of Cambridge lecture notes · source ID tong-quantum
Wikipedia — Partial trace — Partial trace · encyclopedic reference · source ID wp-partial_trace
nLab — partial trace — partial trace · mathematical reference · source ID nlab-partial_trace

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

Partial trace Density operator

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: partial_trace_to_density_operator_canonical_construction
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: constructs a reduced state

Authored explanation

Taking the partial trace of a composite-system density operator canonically constructs the positive trace-one density operator of the retained subsystem.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Wikipedia — Partial trace — Partial trace · encyclopedic reference · source ID wp-partial_trace
nLab — partial trace — partial trace · mathematical reference · source ID nlab-partial_trace
Cambridge — Quantum Mechanics — Quantum Mechanics · University of Cambridge lecture notes · source ID tong-quantum
Wikipedia — Quantum state — Quantum state · encyclopedic reference · source ID wp-physics-quantum_state
nLab — mixed state — mixed state · mathematical physics reference · source ID nlab-physics-mixed_quantum_state

Partial trace Quantum decoherence

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: partialtrace_to_decoherence
Relation type: Mathematical formulation mathematical-formulation
Direction: source → target
Endpoint roles: source: Mathematically formulates; target: Mathematically formulated using
Authored annotation: reduced density operator

Authored explanation

Tracing over environmental degrees of freedom gives the reduced state in which decoherence is quantified.

How to interpret this relation type

A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.

Relation sources

Wikipedia — Partial trace — Partial trace · encyclopedic reference · source ID wp-partial_trace
nLab — partial trace — partial trace · mathematical reference · source ID nlab-partial_trace
Cambridge — Quantum Mechanics — Quantum Mechanics · University of Cambridge lecture notes · source ID tong-quantum
Wikipedia — Quantum decoherence — Quantum decoherence · encyclopedic reference · source ID wp-physics-quantum_decoherence
nLab — decoherence — decoherence · mathematical physics reference · source ID nlab-physics-quantum_decoherence
Schlosshauer — Quantum Decoherence — Quantum Decoherence · peer-reviewed review · source ID schlosshauer-decoherence-2019

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Partial trace

Summary

Record metadata

Carrier(s)

Data

Axioms / constraints

Canonically induces

Notes

Concept sources

Incoming relations (arrows to this concept)

Tensor-product Hilbert space Partial trace

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)

Partial trace Density operator

Authored explanation

How to interpret this relation type

Relation sources

Partial trace Quantum decoherence

Authored explanation

How to interpret this relation type

Relation sources