How does finite-group asymmetry combine with other quantum resource theories?

When two constraints cost more than one

Quantum resource theories describe what a state can be converted into when the allowed operations are restricted, and the central operational quantity is the asymptotic conversion rate: how many copies of a target state can be produced per copy of an input state when many independent copies are processed and the error is allowed to vanish. Realistic settings rarely impose one restriction at a time, so the paper asks whether imposing a finite-group symmetry constraint on top of another resource theory changes that rate, or whether the two restrictions simply combine without extra cost.

The answer is conditional, and the condition is the paper's organizing idea. The authors argue that the effect of combining the constraints is governed by the accessibility of symmetry information under the underlying resource theory: whether the information needed for the conversion can be extracted from the input and used to control the free operations. If this can be done with asymptotically vanishing error, the combined rate equals the smaller of the rates of the two constituent theories. If it cannot, the combination can block conversion entirely.

Symmetry information as the deciding resource

A state's symmetry subgroup collects the group elements that leave it unchanged, and the distinct states in its orbit are indexed by the cosets of that subgroup. To convert an input into a target under covariant operations, a protocol must effectively know which group element relates the two, because that label determines which rotated version of a free operation is appropriate. The paper's mechanism is therefore discrimination followed by conditional control: first identify the relevant orbit label, then use the outcome to select the subsequent free operation.

The authors formalize the second half of this mechanism through control admissibility, which asks whether a measurement outcome can be used to select subsequent free operations without requiring a physical memory that stores the outcome. They also give a memory-assisted formulation as an alternative operational reading. The important point for the rate question is that the discrimination step need not consume a finite fraction of the input: the symmetry information can be extracted from a sublinear number of copies, leaving the remaining copies available for the resource conversion itself.

Why the combined rate equals the smaller rate

The general conversion theorem states that asymptotically reliable discrimination of the symmetry orbit, together with compatibility of the conversion with the residual symmetry, is sufficient to achieve the desired rate under the combined constraints. Group averaging then produces a covariant free channel without changing the first-order rate. This is why the combined rate is the smaller of the two constituent rates rather than a new, lower value: the symmetry constraint adds a discrimination overhead that vanishes asymptotically, not a persistent tax on every copy.

The paper reports explicit copy costs for achieving discrimination error ε. For LOCC and stabilizer operations the protocols use O(log(1/ε)) copies, and for thermal operations O((log(1/ε))²) copies. The authors state that these bounds are sufficient to make the discrimination cost sublinear in the total number of input copies, and they explicitly note that they have not established optimality of these bounds.

Where rate preservation holds and where it fails

The condition holds for LOCC on arbitrary mixed multipartite states, for magic-state conversion of multiqubit systems under Clifford symmetries, and for Gibbs-preserving operations. In these settings, whenever the input's symmetry subgroup is contained in the output's, imposing finite-group covariance does not change the asymptotic conversion rate under LOCC, SEP, stabilizer operations, or completely stabilizer-preserving operations. The results hold without requiring an explicit closed-form expression for the unconstrained rate, which matters because the optimal LOCC conversion rate for general mixed multipartite states is not known in closed form.

The failure mode is equally sharp. The paper establishes an exact obstruction: if the input's symmetry subgroup is not contained in the output's, no positive asymptotic conversion rate is possible under covariant operations. Thermal operations occupy a distinct category because time-translation covariance limits which orbit information is accessible; only the orbit information that survives collective energy dephasing is available to the discrimination step, so successful conversion requires an additional compatibility condition between the finite-group action and time translations on the input and output.

Symmetry on channels versus on elementary operations

The paper separates two ways of imposing symmetry. At the map level, the resulting channel must belong both to the original free-operation class and to the class of covariant channels. At the primitive level, each elementary operation used to realize the channel must itself respect the symmetry. These are different requirements, and the paper treats them separately rather than assuming one implies the other.

For LOCC, the authors establish implementation and rate results at the primitive level under stated assumptions on ancillary systems and classical communication, and they note that the globally symmetric setting requires distinguishing freely available shared invariant reference states from only local invariant ancillary preparation. For thermal operations, they show that the map-level and primitive-level operation classes coincide, which yields the corresponding rate formula. For magic, the map-level result allows arbitrary Clifford representations, whereas the primitive-level result concerns Pauli representations and uses symmetric Clifford extensions with joint system-memory Pauli measurements whose observables commute with the full symmetry representation.

Finite symmetry does not create new passive states

The paper also examines work extraction by symmetry-respecting unitaries. Ergotropy quantifies the work extractable from a finite system, and complete passivity identifies states from which no work can be extracted by the allowed operations. The authors prove that an arbitrary fixed finite-group symmetry commuting with the Hamiltonian leaves the asymptotic ergotropy per copy unchanged, so finite-group symmetry introduces no additional completely passive states.

The accompanying corollary makes the statement concrete: a state is completely passive under invariant unitaries if and only if it is completely passive in the ordinary sense, meaning it is a ground state or a Gibbs state with nonnegative inverse temperature. This holds even when the ground eigenspace is degenerate. The result is a statement about the asymptotic per-copy quantity, not about every finite-copy protocol.

Rate preservation is not operational triviality

The authors are explicit that asymptotic rate preservation does not make the symmetry constraint operationally trivial. Even when the rate is unchanged, the discrimination step consumes copies and introduces an error at finite block length, so finite-copy conversion can remain sensitive to the symmetry constraint even after its first-order asymptotic effect disappears. A reader comparing protocols at realistic copy numbers should not read the asymptotic equality as a statement about small blocks.

A second distinction concerns what the theorems actually deliver. The rate equalities are not an explicit solution of the underlying resource-conversion problem; for general mixed multipartite states the optimal LOCC rate is unknown. The role of the symmetry analysis is to determine the additional restriction generated by the symmetry constraint, not to solve the underlying resource theory. The paper also fixes the group and the single-copy Hilbert spaces in the asymptotic limit, so the results are statements about that fixed setting rather than about families of growing groups.

The subgroup test is the first thing to check

For a reader deciding whether a finite-group symmetry constraint will hurt a conversion, the paper's priority is clear: first check whether the input's symmetry subgroup is contained in the output's. If it is not, the covariant conversion rate is zero, regardless of how favorable the unconstrained rate would have been. If it is, then for LOCC, magic under Clifford symmetries, and Gibbs-preserving operations the constrained rate matches the unconstrained one, and the remaining question is only the finite-copy discrimination overhead.

Thermal operations are the setting where this simple test is not enough. There, time-translation covariance restricts accessible orbit information, so the relevant condition becomes the compatibility between the finite-group action and time translations on the input and output, identified through the subgroup associated with collectively energy-dephased blocks. That split, between a subgroup inclusion test for most operation classes and a two-symmetry compatibility test for time-covariant thermodynamics, is the paper's most decision-relevant contribution.

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