Trace is a linear functional on $M_n(F)$

Depends on

• The trace $\operatorname{tr}(A)$ as the sum of the diagonal entries
• The vector space $M_{m \times n}(F) := F^{\,m \times n}$ of $m$ by $n$ matrices over a field, with entrywise operations
• A finite sum in a commutative monoid indexed by an arbitrary finite set

Used by

• Matrix-unit rotation for a k–Mₙ(k) Morita pair Example
• Computational-basis instruments and conditional states on two registers Lemma
• Discarding the hiding register yields a coset-state mixture Lemma
• Finite square-root measurement packing bound for cq states Lemma
• Flagged trace ratio returns a Gibbs-state observable mean Lemma
• Normalized Hermitian observable defines a two-outcome POVM Lemma
• Positive-probability Pauli branches are density operators Lemma
• The transposition class sum acts on a complex Specht module by total content Lemma
• Typical-projector bounds for finite quantum sources and cq ensembles Lemma
• Degree-zero Hochschild homology is bimodule coinvariants Proposition
• A pure witness suffices for a QMA completeness threshold Theorem
• An isometry followed by partial trace is a channel Theorem
• Characters add on direct sums, multiply on tensor products, and conjugate on duals Theorem
• Conditional state for a projective measurement Theorem
• Every finite instrument induces a unique POVM Theorem
• Every nonzero finite-dimensional complex inner-product space admits a density operator Theorem
• Finite ensemble barycenters are density operators Theorem
• Fourier samples are uniform in the annihilator Theorem
• Holevo–Schumacher–Westmoreland capacity theorem for finite cq channels Theorem
• Instrument branch Kraus families satisfy total normalization Theorem
• Matrix multiplicative-weights regret bound Theorem
• Phase estimation on a superposition of eigenspaces Theorem
• Positive-probability instrument branches are density operators Theorem
• POVM outcome weights are probabilities Theorem
• Projective measurement outcome weights are probabilities Theorem
• Rank-one density operators are pure-state projectors Theorem
• The class-function inner product ⟨χ_V,χ_W⟩ equals dimHom_G(W,V) Theorem
• The encoded Local Hamiltonian problem belongs to QMA Theorem
• The finite Holevo measurement bound Theorem
• The finite POVM law is a probability kernel Theorem
• The nonselective output of an instrument is a density operator Theorem
• The square-root Lüders maps form an instrument Theorem
• Trace preservation is Kraus normalization Theorem
• Weak duality for the finite SDP pair Theorem

Dependency tree · two levels

14 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

• S. Axler, Linear Algebra Done Right, 4th ed., §8D (standard reference, not scraped)