Partial trace over a finite tensor factor

Depends on

• Ordered tensor registers, their inner product, and wire permutations
• The basis-independent trace of an endomorphism of a finite-dimensional vector space
• Coordinate columns $[v]_{\mathcal B}$ and matrices $[T]_{\mathcal B}^{\mathcal C}$ of linear maps relative to ordered bases
• A finite list $v : n \to V$ is an ordered basis if and only if every $x \in V$ equals $\sum_{i<n} \lambda_i v_i$ for exactly one $\lambda : n \to F$; those scalars are the coordinates of $x$ in that ordered basis
• Linear map between vector spaces over the same field

Used by

• Five-local history-state adiabatic path Algorithm
• One finite abelian hidden-subgroup Fourier sample Algorithm
• Broadcasting Family Definition
• Complementary channels and coherent information Definition
• Ensemble and purification Definition
• Finite quantum mutual, conditional, and conditional mutual information Definition
• Finite-dimensional quantum erasure channel Definition
• Isometric Stinespring representation Definition
• Operational quantum, private, and entanglement-assisted capacities Definition
• Partial trace over the first tensor factor Definition
• Uniform verifier interface with a quantum witness Definition
• A Bell pair has negative quantum conditional entropy Example
• Bell-state Schmidt decomposition and reduced density Example
• Qubit dephasing channel in Kraus and Choi forms Example
• Register endianness and a two-qubit partial trace Example
• Strong subadditivity for a pure GHZ tripartite state Example
• A finite instrument has an isometric coherent record Lemma
• A finite verifier induces an effect on its witness Lemma
• Coherent information equals output entropy minus environment entropy Lemma
• Compression dimension bounds entanglement fidelity Lemma
• Dihedral hidden-state pair combination Lemma
• Discarding the hiding register yields a coset-state mixture Lemma
• Every finite extension is obtained from a purification by a channel Lemma
• Fidelity classical measurement minimum Lemma
• Fourier sample from a generalized-dihedral reflection coset state Lemma
• History state and padded output Lemma
• Joint state support is contained in the tensor product of its marginal supports Lemma
• Optimal Cloner Attains Bound Lemma
• Partial trace contracts the trace norm on Hermitian operators Lemma
• Partial trace contracts trace distance Lemma
• Partial traces of a finite bipartite state are states Lemma
• Positive Marginals Imply Joint Support Lemma
• The finite Weyl twirl is the normalized partial trace Lemma
• An isometry followed by partial trace is a channel Theorem
• Channel data processing for finite quantum relative entropy Theorem
• Circuit-to-adiabatic simulation Theorem
• Every finite state ensemble is obtainable from a purification Theorem
• Every finite-dimensional channel has a finite isometric dilation Theorem
• Fidelity Data Processing Theorem
• Fidelity Purification Characterization Theorem

…and 22 more results.

Dependency tree · two levels

32 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• John Watrous, The Theory of Quantum Information (standard reference, not scraped)
• John Preskill, Quantum Computation, Chapter 2: Foundations I: States and Ensembles (standard reference, not scraped)