Discard ZX-Calculus: A Mixed-State Extension

Updated 8 July 2026

Discard ZX-calculus is a diagrammatic framework that extends pure ZX-calculus to mixed-state systems by doubling pure morphisms and introducing discard maps.
It employs generators such as Z- and X-spiders, cups, caps, and a dedicated discard spider to locally enforce trace preservation and completely positive dynamics.
Its rewrite rules based on unitarity, cyclicity, and discard enable rigorous diagrammatic reasoning for infrared safety, decoherence, and Lindblad evolution.

Searching arXiv for the cited work and closely related discard ZX-calculus papers. Discard ZX-calculus is a graphical calculus for mixed-state quantum mechanics obtained from the pure ZX-calculus by a CPM construction in which every pure morphism is doubled into ket and bra wires and a new generator, the discard map, models partial trace. In the formulation used in “Infrared Safety from ZX-Diagrams” (Rey, 29 Jun 2026), it is a diagrammatic language for completely positive trace-preserving maps, Schur channels, Lindblad evolution, and non-Markovian process tensors; related work also presents discard extensions that are complete for mixed-state stabiliser quantum mechanics in odd prime dimensions (Booth et al., 2022) and studies a decohered fragment that is universal and complete for affinely supported probability distributions over $\mathbb{F}_2^n$ (Carette et al., 6 Aug 2025).

1. Definition and categorical setting

In the CPM/discard setting, the underlying category is a $^\dagger$ -compact closed category: the ordinary pure ZX-calculus provides objects as wires and morphisms as spiders and their compositions, and the CPM construction passes to a category $\mathrm{CPM}(C)$ whose objects are the same wires but whose morphisms are completely positive maps (Rey, 29 Jun 2026). In diagrammatic language, each pure morphism is doubled, one copy for ket and one for bra, and a special discard map $\varepsilon:A\to I$ is introduced on each wire.

The categorical interpretation given in the source material makes the discard operation structurally constrained. It obeys naturality, $\varepsilon_A\circ(f\otimes f^\ast)=\varepsilon_B$ for any isometry or unitary $f:A\to B$ , and monoidality, $\varepsilon_{A\otimes B}=\varepsilon_A\otimes\varepsilon_B$ (Rey, 29 Jun 2026). In this setting, spiders, cups, caps, and unitaries all lift to CP maps via the doubling construction.

A closely related generalization appears in odd prime dimensions, where the extended syntax adds a discard or environment node $\ground:1\to 0$ interpreted as partial trace in $\mathrm{CPM}(\mathrm{Stab}_p)$ (Booth et al., 2022). There, the extended PROP is described as the free environment structure over the pure part, and completeness for mixed-state stabiliser quantum mechanics follows by a discard construction once the pure calculus is complete and there are enough isometries. This suggests that discard ZX-calculus is not merely an ad hoc augmentation of pure ZX, but a systematic categorical passage from pure to mixed-state process theories.

2. Generators, syntax, and denotational semantics

The generators presented in the mixed-state setting include Z-spiders, X-spiders, wires, cups, caps, and the discard spider (Rey, 29 Jun 2026). A green Z-spider with phase $\alpha$ and arity $^\dagger$ 0 is interpreted as the linear map

$^\dagger$ 1

where $^\dagger$ 2 iff all $^\dagger$ 3 (Rey, 29 Jun 2026). The red X-spider is identical but in the $^\dagger$ 4 basis, with $^\dagger$ 5.

Cups and caps provide the compact-closed structure. The cap is $^\dagger$ 6, $^\dagger$ 7, and the cup is $^\dagger$ 8, $^\dagger$ 9 (Rey, 29 Jun 2026). The discard spider is diagrammatically a blue dot with one input on the ket wire and one on the bra wire; semantically it is the trace map $\mathrm{CPM}(C)$ 0, and in doubled-wire notation it joins the top and bottom wires of a density operator. The semantic assignment also includes the explicit tensor form

$\mathrm{CPM}(C)$ 1

A more general prop presentation additionally lists identities, swaps, cups, caps, Hadamard, discard $\mathrm{CPM}(C)$ 2, and global scalars, with arrows denoting CP maps between Hilbert spaces of dimension $\mathrm{CPM}(C)$ 3 (Carette et al., 6 Aug 2025). In that semantics, $\mathrm{CPM}(C)$ 4, and soundness means that each rewrite rule denotes the same CP map on both sides.

3. Rewrite theory and the role of discard

Three rules are singled out as fundamental in the treatment of soft QED: the unitarity rule, cyclicity of trace, and the discard rule (Rey, 29 Jun 2026). Unitarity is the ordinary equation $\mathrm{CPM}(C)$ 5. Cyclicity is the diagrammatic sliding of boxes around a discard loop, corresponding algebraically to $\mathrm{CPM}(C)$ 6. The discard rule expresses trace preservation for any CPTP channel $\mathrm{CPM}(C)$ 7: $\mathrm{CPM}(C)$ 8 or diagrammatically, discarding after $\mathrm{CPM}(C)$ 9 is the same as discarding immediately. In one equation,

$\varepsilon:A\to I$ 0

so doubling a unitary and then discarding is equivalent to direct discard.

The same theme appears in the odd-prime stabiliser extension, where discard is required to erase generating pure isometries such as the green unit, red unit, green comultiplication, red comultiplication, and Hadamard box (Booth et al., 2022). The added equations are of the form

$\varepsilon:A\to I$ 1

for those generators. Their semantic justification is the partial-trace identity

$\varepsilon:A\to I$ 2

whenever $\varepsilon:A\to I$ 3 is an isometry.

The general mixed-state theory summarized in the decohered ZX work states that discard ZX-calculus is complete and universal for mixed-state quantum mechanics, and that every CP map admits a purification into a unitary plus discard (Carette et al., 6 Aug 2025). Within the provided material, a common misconception is that discard only adds a terminal symbol to a pure graphical language. The data instead emphasize that the discard spider and the axiom $\varepsilon:A\to I$ 4 for CPTP $\varepsilon:A\to I$ 5 are essential for representing partial traces, enforcing trace preservation locally, and capturing decoherence as the action of discarding environmental wires (Rey, 29 Jun 2026).

4. Diagrammatic identities and infrared safety

The central application developed in “Infrared Safety from ZX-Diagrams” is a nonperturbative, categorical proof of the Bloch–Nordsieck cancellation of infrared divergences in QED (Rey, 29 Jun 2026). Soft photons are treated as an open quantum system: the resolved charged particles and hard photons form the system, while photons below a detector resolution form the environment. The reduced hard channel is a CPTP map, and the soft-photon theorem replaces the full $\varepsilon:A\to I$ 6-matrix by a controlled displacement operator whose Feynman–Vernon influence functional satisfies the equal-history normalization

$\varepsilon:A\to I$ 7

In the ZX-calculus, this normalization becomes a single diagrammatic identity: the doubled displacement diagram collapses to the bare wire under unitarity, cyclicity, and discard (Rey, 29 Jun 2026). For one fixed hard branch $\varepsilon:A\to I$ 8, the algebraic content is

$\varepsilon:A\to I$ 9

The derivation is explicitly decomposed into three steps: cyclicity moves $\varepsilon_A\circ(f\otimes f^\ast)=\varepsilon_B$ 0 around the discard loop to the ket wire, unitarity fuses $\varepsilon_A\circ(f\otimes f^\ast)=\varepsilon_B$ 1 into the identity, and the remaining bare wire with discard yields trace one.

For off-diagonal hard-state elements, the same diagram produces the coherent-state overlap

$\varepsilon_A\circ(f\otimes f^\ast)=\varepsilon_B$ 2

with modulus

$\varepsilon_A\circ(f\otimes f^\ast)=\varepsilon_B$ 3

This is used in the paper as a first-principles account of soft-cloud decoherence. The proof is presented as a categorical consistency check on the companion open-system treatment of soft QED, confirming that the physical derivation is logically complete and free of hidden assumptions about the infrared limit.

5. CPTP channels, Schur maps, and Lindblad evolution

The calculus represents arbitrary channels $\varepsilon_A\circ(f\otimes f^\ast)=\varepsilon_B$ 4 as doubled boxes, and trace preservation can be checked locally by attaching discard and simplifying: $\varepsilon_A\circ(f\otimes f^\ast)=\varepsilon_B$ 5 exactly when $\varepsilon_A\circ(f\otimes f^\ast)=\varepsilon_B$ 6 (Rey, 29 Jun 2026). This local form of CPTP verification is one of the operational advantages emphasized in the source material.

The soft-shell coarse-graining of infrared modes is encoded as a CPTP Schur channel. For a shell $\varepsilon_A\circ(f\otimes f^\ast)=\varepsilon_B$ 7, the isometry is

$\varepsilon_A\circ(f\otimes f^\ast)=\varepsilon_B$ 8

and the channel is

$\varepsilon_A\circ(f\otimes f^\ast)=\varepsilon_B$ 9

Projecting the input on $f:A\to B$ 0, applying $f:A\to B$ 1 and $f:A\to B$ 2 on soft wires, and then discarding yields the scalar

$f:A\to B$ 3

so that

$f:A\to B$ 4

that is, a Schur or entrywise map (Rey, 29 Jun 2026).

For an infinitesimal shell $f:A\to B$ 5, the expansion

$f:A\to B$ 6

yields, after tracing out the shell, an infinitesimal CPTP map whose limit is exactly of GKSL form (Rey, 29 Jun 2026): $f:A\to B$ 7 with diagonal jump operators

$f:A\to B$ 8

The diagrammatic derivation is organized by term type: the $f:A\to B$ 9 contribution gives the identity superoperator, cross-branch contractions through discard give the jump term $\varepsilon_{A\otimes B}=\varepsilon_A\otimes\varepsilon_B$ 0, and same-branch loops give the anticommutator subtraction together with a Hamiltonian phase.

The source material explicitly contrasts discard ZX-calculus with pure ZX-calculus (Rey, 29 Jun 2026). Pure ZX-calculus models states as vectors $\varepsilon_{A\otimes B}=\varepsilon_A\otimes\varepsilon_B$ 1, unitaries, and pure Z/X-spiders, but it cannot directly handle mixed states or partial trace. Discard ZX-calculus is obtained by the CPM construction: every pure morphism is doubled, and a new generator $\varepsilon_{A\otimes B}=\varepsilon_A\otimes\varepsilon_B$ 2 is introduced to model trace. The resulting extension is described as seamless for open quantum systems, CPTP maps, Schur channels, Lindblad dynamics, and non-Markovian process tensors, while retaining soundness and completeness.

A further development is the decohered ZX-calculus, a fragment obtained by decohering the usual generators of discard ZX-calculus (Carette et al., 6 Aug 2025). In this fragment, pre- and post-composition by the decoherence map $\varepsilon_{A\otimes B}=\varepsilon_A\otimes\varepsilon_B$ 3 kills off-diagonal matrix entries in the computational basis, and generators collapse to positive matrix embeddings. The fragment is stated to be universal and complete for affinely supported probability distributions over $\varepsilon_{A\otimes B}=\varepsilon_A\otimes\varepsilon_B$ 4, with a normal form combining an affine part implemented by a row-reduced-echelon binary matrix and a Fourier part consisting of a scalar and nonzero parameters. This clarifies how hybrid classical–quantum processes are handled inside the broader discard framework.

The odd-prime stabiliser extension provides another variant (Booth et al., 2022). There, adding discard yields a calculus complete for mixed-state stabiliser quantum mechanics in odd prime dimensions and, after imposing an additional scalar equation, a complete axiomatization of affine co-isotropic relations. A plausible implication is that discard constructions are robust across several diagrammatic settings, not only the qubit theory emphasized in many ZX presentations.

The data also mention ZH-calculus as a distinct graphical calculus optimized for classical non-linearity, with compact encodings of AND, Toffoli, CCZ, and $\varepsilon_{A\otimes B}=\varepsilon_A\otimes\varepsilon_B$ 5-controlled- $\varepsilon_{A\otimes B}=\varepsilon_A\otimes\varepsilon_B$ 6 operations (Backens et al., 2018). That comparison is not a critique of discard ZX-calculus; rather, it marks a separation of tasks. Discard ZX-calculus is presented as the framework for mixed states, partial trace, decoherence, and open-system reasoning, whereas ZH-calculus is presented as especially economical when classical Boolean non-linearity is the dominant concern.