Alphabeta MathHow statement and proof provenance work
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Trace Distance, Fidelity, and Information Limits — Examples
1 · Prerequisites
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Congruences, the Integers Modulo n and the Chinese Remainder Theorem
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Cyclic Groups and Direct Products
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Divisibility, Greatest Common Divisors and Bézout's Identity
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Finite Counting, Factorials and Binomial Coefficients
- Finite-Dimensional Complex Inner-Product Spaces and Matrix Interfaces
- Foundations of the Real Numbers for Analysis
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Hilbert Space Geometry and Riesz Representation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Limits of Real Functions
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Measurements, POVMs, and Instruments
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Order, Zorn's Lemma, and the Axiom of Choice
- Polynomial Rings, the Division Algorithm and Roots
- Primes, Euclid's Lemma and the Fundamental Theorem of Arithmetic
- Pure and Mixed States, Ensembles, and Schmidt Decomposition
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
- Suprema and Infima
- Sylow's Theorems, p-Groups and Nilpotent Groups
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- The ZFC Axioms and the Basic Set Constructions
- Topology of ℝ
- Trace Distance, Fidelity, and Information Limits
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Summary
The orthogonal pure qubit states and have trace distance one, and their computational-basis PVM achieves total variation one. The calculation checks the factor in the state-distance convention against a finite measurement that distinguishes the states perfectly.
3 · Logical flowchart
4 · Definitions, theorems and proofs
None yet.
5 · Examples, counterexamples and false statements
Trace distance of orthogonal pure states
Example
Let with orthonormal computational basis , and set and . Their trace distance is maximal, and the computational-basis projective measurement distinguishes them with total variation one.
Facts & Assumptions
Given: The two orthogonal computational-basis unit vectors in .
The rank-one operators associated with orthonormal unit vectors are rank-one orthogonal projections (Pure state and ray, Ray-to-projector is well-defined and bijective).
A PVM with effects has outcome law (Finite POVMs and their outcome law).
Verification
In the basis , , whose singular values are . Thus and [F2] gives .
The computational-basis PVM has effects and . By [F3], the outcome laws are on and on , with total variation . This equals the trace distance from step 1.1.