Quantum Markov Categories

Arthur J. Parzygnat · 2020 · arxiv arXiv:2001.08375

Prereqs: 🍞 Fritz 2020 (Markov categories, copy, determinism). Some comfort with matrix multiplication helps.

Quantum channels form a Markov category with a dagger — a canonical way to reverse any morphism. Bayesian inversion becomes adjunction. The no-cloning theorem forbids copying quantum states. Copy is the very operation that detects determinism in Fritz's setup.

Quantum channels as morphisms

A quantum channel maps density matrices to density matrices: the quantum analogue of a stochastic matrix. It's completely positive and trace-preserving (CPTP). In the classical limit, density matrices reduce to probability distributions and quantum channels reduce to Markov kernels.

Scheme

; Classical analogue: a stochastic channel as a matrix
; Quantum: density matrices instead of distributions

; A 2x2 stochastic matrix (classical channel)
(define channel '((0.7 0.3) (0.2 0.8)))

; Apply to a distribution (classical state)
(define (apply-channel chan state)
  (map (lambda (row)
    (apply + (map * row state)))
  chan))

(define prior '(0.6 0.4))
(define posterior (apply-channel channel prior))

(display "prior:     ") (display prior) (newline)
(display "posterior: ") (display posterior) (newline)
; Classical channels are the commutative special case.
; Quantum channels: replace distributions with density
; matrices, stochastic matrices with CPTP maps.

The dagger — reversing channels

A dagger category has an involution † that reverses morphisms: if f: A → B then f†: B → A. For quantum channels, the dagger is the adjoint (conjugate transpose). For classical channels with a reference state, the dagger recovers Bayesian inversion.

Level	Structure	Example
Bayesian inversion	a.e. inverse	classical Bayes
Dagger	exact inverse	unitary quantum
Conjugation	up to automorphism	modular theory

Scheme

; Dagger: transpose of a matrix (classical analogue)
; For quantum: conjugate transpose of the channel

(define (transpose mat)
  (apply map list mat))

(define channel '((0.7 0.3) (0.2 0.8)))
(define dagger (transpose channel))

(display "channel:  ") (display channel) (newline)
(display "dagger:   ") (display dagger) (newline)
; In the quantum case, dagger is the unique
; adjoint with respect to the Hilbert-Schmidt
; inner product. Bayesian inversion is a special case.

Bayesian inversion from the dagger

Given a prior state π and a channel f, the Bayesian inverse f†_π is the unique channel such that the joint distribution factors correctly: π ⊗ f = f†_π ⊗ (f applied to π). In the quantum case, this is Petz recovery.

Scheme

; Bayesian inversion via dagger and prior
; f†_π(b) = Bayes' rule with prior π

(define (bayes-inverse channel prior)
  ; For a 2x2 classical channel:
  ; f†_π(j|i) = f(i|j) * π(j) / Σ_k f(i|k) * π(k)
  (let* ((n (length prior))
         (marginal (map (lambda (row)
           (apply + (map * row prior)))
           channel)))
    (let build-rows ((i 0) (rows '()))
      (if (= i n) (reverse rows)
        (build-rows (+ i 1)
          (cons
            (let build-cols ((j 0) (cols '()))
              (if (= j n) (reverse cols)
                (build-cols (+ j 1)
                  (cons
                    (let ((f-ij (list-ref (list-ref channel i) j))
                          (pi-j (list-ref prior j))
                          (marg-i (list-ref marginal i)))
                      (if (> marg-i 0)
                        (/ (* f-ij pi-j) marg-i)
                        0))
                    cols))))
            rows))))))

(define channel '((0.9 0.1) (0.2 0.8)))
(define prior '(0.3 0.7))
(define inv (bayes-inverse channel prior))

(display "channel: ") (display channel) (newline)
(display "prior: ") (display prior) (newline)
(display "Bayes inverse: ") (display inv)

; Bayesian inversion via dagger and prior
; f†_π(b) = Bayes' rule with prior π

(define (bayes-inverse channel prior)
  ; For a 2x2 classical channel:
  ; f†_π(j|i) = f(i|j) * π(j) / Σ_k f(i|k) * π(k)
  (let* ((n (length prior))
         (marginal (map (lambda (row)
           (apply + (map * row prior)))
           channel)))
    (let build-rows ((i 0) (rows '()))
      (if (= i n) (reverse rows)
        (build-rows (+ i 1)
          (cons
            (let build-cols ((j 0) (cols '()))
              (if (= j n) (reverse cols)
                (build-cols (+ j 1)
                  (cons
                    (let ((f-ij (list-ref (list-ref channel i) j))
                          (pi-j (list-ref prior j))
                          (marg-i (list-ref marginal i)))
                      (if (> marg-i 0)
                        (/ (* f-ij pi-j) marg-i)
                        0))
                    cols))))
            rows))))))

(define channel '((0.9 0.1) (0.2 0.8)))
(define prior '(0.3 0.7))
(define inv (bayes-inverse channel prior))

(display "channel: ") (display channel) (newline)
(display "prior: ") (display prior) (newline)
(display "Bayes inverse: ") (display inv)

No-broadcasting theorem

Classical distributions can be copied (broadcast to multiple receivers without distortion). Quantum states cannot: the no-broadcasting theorem. In Fritz's framework, copy detects determinism. In the quantum version, copy doesn't exist for non-commutative states. That's quantum weirdness showing up as missing structure.

Scheme

; Classical: copy works — broadcast a distribution
(define (copy-classical dist)
  ; (p, q) -> joint where both marginals equal (p, q)
  (let ((n (length dist)))
    (let build-rows ((i 0) (rows '()))
      (if (= i n) (reverse rows)
        (build-rows (+ i 1)
          (cons
            (let build-cols ((j 0) (cols '()))
              (if (= j n) (reverse cols)
                (build-cols (+ j 1)
                  (cons (if (= i j) (list-ref dist i) 0) cols))))
            rows))))))

(define dist '(0.3 0.7))
(define copied (copy-classical dist))

(display "original: ") (display dist) (newline)
(display "copied (joint): ") (display copied) (newline)
(display "marginal 1: ")
(display (map (lambda (row) (apply + row)) copied)) (newline)
; Both marginals equal the original.
; Quantum: this construction FAILS for non-diagonal
; density matrices. No copy = no broadcasting.

Notation reference

Paper	Scheme	Meaning
f†	(transpose channel)	Dagger (adjoint/reverse)
f†_π	(bayes-inverse f π)	Bayesian inverse given prior π
CPTP	; completely positive, trace-preserving	Quantum channel
ν	(copy-classical ...)	Copy (exists classically, not quantum)
ρ	'(0.3 0.7)	State (distribution or density matrix)

Neighbors

Other paper pages

🍞 Fritz 2020 — the classical Markov category (copy exists everywhere)
🍞 Cho, Jacobs 2015 — effectus theory (Born rule, quantum-classical bridge)
🍞 Smithe 2021 — Bayesian lenses (uses the same inversion structure)
🍞 Di Lavore 2025 — partial Markov categories (observations and restrictions)

Foundations (Wikipedia)

Translation notes

All examples use classical stochastic matrices as stand-ins for quantum channels. The paper works with completely positive trace-preserving maps on matrix algebras, where the dagger is the Hilbert-Schmidt adjoint and Bayesian inversion is Petz recovery. For example: the Bayesian inverse on this page computes Bayes' rule for a 2×2 stochastic matrix. In the paper, the same construction works for arbitrary CPTP maps on C*-algebras, involving the modular automorphism group to recover the Petz recovery map. Factorization structure (joint = forward × backward) is identical. Operator algebra is not.

Every example is Analogy — classical stand-ins for quantum constructions.

Ready for the real thing? arxiv

Read the paper. Start at §3 for quantum Markov categories, §5 for Bayesian inversion and the dagger, §7 for no-broadcasting.

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