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Quantum Key Distribution and Composable Secrecy — Examples
1 · Prerequisites
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Channels, Kraus, Choi, and Stinespring Representations
- 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
- Free Modules, Exact Sequences, Projective and Injective Modules
- 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
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- 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
- Quantum Key Distribution and Composable Secrecy
- Registers, Tensor Products, and Partial Trace
- 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
- Tensor Products of Modules
- 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 first calculation keeps a abort probability and shows that the real-ideal trace distance and secrecy error are both when the accepted key is always but the parties agree. The second compares a uniformly secret but always mismatched key with a perfectly matched but deterministic key, showing that secrecy and correctness are distinct conditions.
3 · Logical flowchart
4 · Definitions, theorems and proofs
None yet.
5 · Examples, counterexamples and false statements
Acceptance probability weights an otherwise imperfect one-bit key
Example
Take , a one-element transcript alphabet , and a one-dimensional adversary system . A finite protocol aborts with probability and, on acceptance, both parties always output key . Then the correctness error is , the secrecy error is , and the abort-aware real-ideal trace distance is exactly .
Facts & Assumptions
Given: The key alphabet, transcript, adversary space, and output law in the Example.
Finite authenticated key-distribution protocol: the output has a normalized classical state on keys, transcript and adversary; the abort flag is shared.
Abort-aware real and ideal QKD outputs and security errors: the ideal accepted key is uniform and shared; the secrecy error is the acceptance probability times the normalized accepted secrecy distance.
Trace Distance: for density operators, .
Trace Norm Spectral Formula and Trace Norm Unitary Invariance And Positive Formula: for a self-adjoint diagonal matrix, the trace norm is the sum of the absolute diagonal entries.
QKD composable secrecy is bounded by correctness plus secrecy: the real-ideal trace-distance error is bounded by the sum of correctness and secrecy errors.
Verification
Given: , with the one-dimensional Eve factor suppressed.
The ideal state keeps the abort block and replaces the accepted block by , so .
Their difference is . The two basis vectors are orthogonal by [F1], so its nonzero eigenvalues are and ; [F3]–[F4] give .
The accepted keys always agree, so . The accepted Alice state is , whose difference from has nonzero diagonal entries and trace distance . Thus . The theorem's sum bound [F5] holds with equality.
Secrecy and correctness are separate requirements
Example
For , no abort, a one-element transcript, and a one-dimensional adversary system, consider two finite output laws. In the first, Alice's key is uniform but Bob always outputs its complement: secrecy error is and correctness error is . In the second, both parties always output : correctness error is and secrecy error is . Thus neither condition implies the other.
Facts & Assumptions
Given: The two normalized classical output states described in the Example.
Abort-aware real and ideal QKD outputs and security errors: the correctness error is the probability of unequal accepted outputs, and secrecy compares Alice's accepted key with a uniform key independent of transcript and Eve.
Trace Distance: trace distance of density operators is half the trace norm of their difference.
Trace Norm Spectral Formula: for an orthogonal diagonal decomposition, the trace norm is the sum of absolute eigenvalues.
QKD composable secrecy is bounded by correctness plus secrecy: the real-ideal distance is at most the sum of the correctness and secrecy errors.
Verification
Given: Let and , with Eve's one-dimensional factor suppressed.
For , Alice's marginal is , so it is already uniform and independent of the trivial transcript-adversary state; hence . Every output pair differs, so . The real state is supported on while the ideal state is supported on the orthogonal set . Their difference therefore has positive and negative parts each of trace one, giving by [F2]–[F3].
For , the parties always agree, so . Alice's accepted state is , and has eigenvalues ; hence . The real-ideal output difference has the same two eigenvalues, so . Both cases satisfy [F4] with equality, while each has exactly one of the two component errors zero.