Clifford Ergotropy in Quantum Work Extraction

• Clifford ergotropy is the quantification of work extraction from quantum states when only Clifford unitaries are allowed, emphasizing limitations imposed by restricted control.
• It employs a Pauli-space formulation and universal bounds to relate extractable energy with nonstabilizerness, or ‘magic’, across various quantum systems.
• The framework reveals a discrete control landscape and second-law-like behavior wherein high magic suppresses accessible work, validated through explicit one- and two-qubit formulas.

Searching arXiv for the cited papers and closely related work on Clifford ergotropy and nonstabilizerness/ergotropy. Clifford ergotropy is the amount of extractable energy from a closed quantum system when the admissible control unitaries are restricted to the Clifford group rather than the full unitary group. In this sense it is a resource-theoretic refinement of standard ergotropy: it quantifies not only how much work is in principle stored in a state, but how much of that work remains operationally accessible when control is limited to stabilizer-compatible dynamics. The central result is that this restriction couples thermodynamic performance to nonstabilizerness, or “magic,” through universal upper bounds governed by the infinite-order filtered stabilizer Rényi entropy. For one- and two-qubit systems the framework yields explicit formulas and a discrete control-landscape structure absent in unrestricted unitary control, while in many-body settings it leads to a second-law-type statement: typical high-magic pure states are effectively inert under Clifford-only work extraction (Maity et al., 11 May 2026).

1. Definition and operational setting

Consider a finite-dimensional quantum system with Hamiltonian $H$ and state $\rho$. The energy expectation is $\mathrm{Tr}(\rho H)$. Standard ergotropy is the maximum work extractable by unitary control and may be written as

$$E(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$

Equivalently,

$E(\rho,H)=\mathrm{Tr}[(\rho-\pi_\rho)H],$

where the passive state $\pi_\rho$ is obtained by assigning the largest eigenvalues of $\rho$ to the lowest energy levels of $H$. If $\rho=\sum_{k=1}^d p_k |p_k\rangle\langle p_k|$ with $p_1\ge p_2\ge\cdots$, and the Hamiltonian eigenvalues satisfy $\rho$0, then the passive energy is $\rho$1, so

$\rho$2

For pure states this reduces to

$\rho$3

with $\rho$4 the ground-state energy (Maity et al., 11 May 2026).

On $\rho$5 qubits, the Clifford group $\rho$6 consists of unitaries that map Pauli strings to Pauli strings up to a sign while preserving all Pauli commutation and anticommutation relations:

$\rho$7

The operational setting considered is closed unitary dynamics with fixed and time-independent $\rho$8, and with controls limited to Clifford gates such as Hadamard, phase, and CNOT, without ancillas, measurements, or classical randomness. Clifford ergotropy is then defined by restricting the variational optimization to $\rho$9:

$\mathrm{Tr}(\rho H)$0

The associated ergotropy gap,

$\mathrm{Tr}(\rho H)$1

measures the work unlocked specifically by non-Clifford resources. For pure $\mathrm{Tr}(\rho H)$2,

$\mathrm{Tr}(\rho H)$3

This formalism makes explicit that ergotropy depends not only on the state and Hamiltonian, but also on the admissible control set (Maity et al., 11 May 2026).

2. Pauli-space formulation and the role of magic

The framework is especially transparent in the Pauli basis. Let $\mathrm{Tr}(\rho H)$4, let $\mathrm{Tr}(\rho H)$5 denote the $\mathrm{Tr}(\rho H)$6-qubit Pauli strings with $\mathrm{Tr}(\rho H)$7, and use the orthogonality relation $\mathrm{Tr}(\rho H)$8. Any state and traceless Hamiltonian admit expansions

$\mathrm{Tr}(\rho H)$9

and

$$E(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$0

where $$E(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$1 is the number of nonzero Pauli coefficients of $$E(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$2. Because Clifford conjugation acts by signed permutation,

$$E(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$3

the energy on a Clifford orbit becomes

$$E(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$4

The optimization defining $$E(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$5 is therefore a discrete signed permutation problem over Pauli coefficients, with the nontrivial constraint that the permutations must arise from a Clifford unitary and hence preserve Pauli commutation structure (Maity et al., 11 May 2026).

The relevant magic monotone is the filtered stabilizer Rényi entropy (FSRE). Let $$E(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$6 be the ordered absolute values of the non-identity Pauli coefficients, $$E(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$7, normalized so that

$$E(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$8

For order $$E(\rho,H)=\mathrm{Tr}(\rho H)-\min_{U\in U(d)} \mathrm{Tr}(U\rho U^\dagger H).$$9,

$E(\rho,H)=\mathrm{Tr}[(\rho-\pi_\rho)H],$0

The infinite-order quantity is

$E(\rho,H)=\mathrm{Tr}[(\rho-\pi_\rho)H],$1

so that

$E(\rho,H)=\mathrm{Tr}[(\rho-\pi_\rho)H],$2

With this filtering and normalization, stabilizer states satisfy $E(\rho,H)=\mathrm{Tr}[(\rho-\pi_\rho)H],$3. Since Clifford unitaries only sign-flip and permute the Pauli coefficients, $E(\rho,H)=\mathrm{Tr}[(\rho-\pi_\rho)H],$4 is invariant on Clifford orbits:

$E(\rho,H)=\mathrm{Tr}[(\rho-\pi_\rho)H],$5

Operationally, $E(\rho,H)=\mathrm{Tr}[(\rho-\pi_\rho)H],$6 measures the largest non-identity Pauli weight. Larger magic corresponds to smaller $E(\rho,H)=\mathrm{Tr}[(\rho-\pi_\rho)H],$7, and hence to weaker peak overlap that Clifford control can exploit (Maity et al., 11 May 2026).

3. Universal bounds and magic-dependent suppression

The main quantitative result is a family of universal upper bounds on Clifford ergotropy. Let $E(\rho,H)=\mathrm{Tr}[(\rho-\pi_\rho)H],$8 be the sorted absolute Hamiltonian coefficients, with zeros appended. If one relaxes the Clifford constraint and allows arbitrary permutations of the absolute Pauli coefficients, then the minimum attainable energy would be $E(\rho,H)=\mathrm{Tr}[(\rho-\pi_\rho)H],$9. Since the true Clifford optimization is more constrained,

$\pi_\rho$0

which yields

$\pi_\rho$1

Using Hölder’s inequality and $\pi_\rho$2,

$\pi_\rho$3

where

$\pi_\rho$4

For pure states, the ergotropy gap obeys

$\pi_\rho$5

This lower bound can be loose, and may even be negative when $\pi_\rho$6 is too small compared with $\pi_\rho$7, but it becomes informative when the ground-state energy is extensive and $\pi_\rho$8 is small (Maity et al., 11 May 2026).

Substituting $\pi_\rho$9 into (2) gives the central magic-dependent estimate,

$\rho$0

The bound decreases monotonically with $\rho$1. The physical interpretation is that high magic broadens the Pauli spectrum and suppresses the largest coefficient $\rho$2; since Clifford operations can only permute and sign-flip Pauli components, they cannot reproduce the continuous spectral rearrangements available to arbitrary unitaries. The result reverses a common intuition imported from fault-tolerant quantum computation: although magic is a resource for universality, in the present thermodynamic setting increasing magic suppresses the work accessible to Clifford-only control (Maity et al., 11 May 2026).

The proof mechanism is entirely structural. Clifford conjugation induces only signed permutations of Pauli coefficients compatible with the symplectic geometry of the Pauli group. Replacing this constrained optimization by an arbitrary rearrangement of absolute values gives the first bound, and Hölder’s inequality yields the second. Tightness depends on the Hamiltonian structure: the bounds are tight for single-qubit Hamiltonians diagonal in a Pauli basis, and become looser when many Hamiltonian terms compete and Clifford constraints obstruct simultaneous alignment of the dominant state and Hamiltonian coefficients (Maity et al., 11 May 2026).

4. One- and two-qubit structure

For a single qubit with

$\rho$3

the Clifford orbit of $\rho$4 is $\rho$5. Hence

$\rho$6

where

$\rho$7

Therefore

$\rho$8

while the unrestricted ergotropy is

$\rho$9

The gap is

$H$0

In this case the universal bound is saturated because $H$1 and $H$2. The same example also connects Clifford ergotropy to stabilizer fidelity:

$H$3

and for pure states,

$H$4

where $H$5. For mixed states,

$H$6

The one-qubit case therefore gives an exact identification of the Clifford penalty with the discrepancy between Bloch-vector length and the largest Cartesian component (Maity et al., 11 May 2026).

For two qubits, the optimization acquires a genuinely discrete control-landscape character. For the transverse-field Ising-type Hamiltonian

$H$7

and initial pure state $H$8, with

$H$9

direct optimization over the two-qubit Clifford group shows that $\rho=\sum_{k=1}^d p_k |p_k\rangle\langle p_k|$0 and $\rho=\sum_{k=1}^d p_k |p_k\rangle\langle p_k|$1 exhibit sharp, cusp-like changes as $\rho=\sum_{k=1}^d p_k |p_k\rangle\langle p_k|$2 varies: at $\rho=\sum_{k=1}^d p_k |p_k\rangle\langle p_k|$3 for $\rho=\sum_{k=1}^d p_k |p_k\rangle\langle p_k|$4, and at $\rho=\sum_{k=1}^d p_k |p_k\rangle\langle p_k|$5 for $\rho=\sum_{k=1}^d p_k |p_k\rangle\langle p_k|$6. These cusps are discrete changes in the optimal Clifford operator induced by the finite, nonconvex Clifford set. By contrast, the unrestricted ergotropy $\rho=\sum_{k=1}^d p_k |p_k\rangle\langle p_k|$7 varies smoothly because the passive-state construction is governed by continuous unitary rearrangements. The tighter rearrangement bound in (1) tracks the cusps, whereas the Hölder bound in (2) becomes looser when $\rho=\sum_{k=1}^d p_k |p_k\rangle\langle p_k|$8 (Maity et al., 11 May 2026).

These few-qubit examples establish two qualitative features that persist at larger scale. First, the Clifford restriction does not merely reduce the optimum quantitatively; it changes the geometry of the control problem from continuous to discrete. Second, magic enters through a Pauli-space obstruction rather than through energy-space populations alone, so identical average energies can correspond to very different Clifford-extractable work (Maity et al., 11 May 2026).

5. Many-body consequences and a Clifford second law

For product states $\rho=\sum_{k=1}^d p_k |p_k\rangle\langle p_k|$9, let $p_1\ge p_2\ge\cdots$0 denote the largest non-identity Pauli coefficient on site $p_1\ge p_2\ge\cdots$1. Then

$p_1\ge p_2\ge\cdots$2

and the bounds specialize to

$p_1\ge p_2\ge\cdots$3

and

$p_1\ge p_2\ge\cdots$4

For $p_1\ge p_2\ge\cdots$5 one has $p_1\ge p_2\ge\cdots$6. In the classical Ising chain with periodic boundary conditions,

$p_1\ge p_2\ge\cdots$7

the ground energy is $p_1\ge p_2\ge\cdots$8 and $p_1\ge p_2\ge\cdots$9, so

$\rho$00

This is an extensive lower bound on the ergotropy gap arising solely from the Clifford restriction. For the transverse-field Ising chain,

$\rho$01

the large-$\rho$02 estimate is

$\rho$03

with $\rho$04 the complete elliptic integral of the second kind. This lower bound is positive for $\rho$05 and $\rho$06 (Maity et al., 11 May 2026).

The strongest statements concern typical pure states. Let $\rho$07 be Haar-random. Then for any $\rho$08,

$\rho$09

so with overwhelming probability $\rho$10 and

$\rho$11

For short-range Hamiltonians, $\rho$12, while

$\rho$13

Using (3),

$\rho$14

Thus no macroscopic work can be extracted via Clifford operations from typical high-magic pure states. This is the sense in which the theory yields a second-law statement for closed dynamics under Clifford-restricted controls (Maity et al., 11 May 2026).

A related finite-energy-density statement uses Haar-random states in a microcanonical shell. Let $\rho$15 be Haar-random within the shell at energy $\rho$16, and $\rho$17 the corresponding microcanonical density matrix. Measure concentration implies

$\rho$18

with $\rho$19. Choosing $\rho$20 makes the difference exponentially small, so

$\rho$21

For normal macroscopic systems, one has

$\rho$22

hence

$\rho$23

The second-law claim is therefore not an unrestricted statement about entropy alone; it is specifically a typicality result for high-magic states under Clifford-limited control (Maity et al., 11 May 2026).

6. Related stabilizer-restricted work extraction, computational aspects, and limitations

A related line of work on quantum batteries studies how nonstabilizerness and ergotropy co-evolve during charging and discharging protocols. In a composite spin-$\rho$24 charger–battery system, a one-to-one functional relation between the ergotropy stored in the battery and the total nonstabilizerness of the composite state emerges when the charging dynamics preserves a $\rho$25 symmetry generated by total $\rho$26, whereas the correspondence generally fails for Ising-type interactions or fully Haar-random circuits that do not conserve excitation number. The same work also analyzes a notion of Clifford-restricted ergotropy for stabilizer states and shows that Clifford charging from a stabilizer initial state can store finite ergotropy without generating any magic, while the maximum average charging power can depend non-monotonically on the initial nonstabilizerness and may even be maximized at zero magic (Konar et al., 5 May 2026).

This complementary result helps delimit the scope of Clifford ergotropy as introduced in (Maity et al., 11 May 2026). High magic suppresses extractable work under Clifford-only discharging, but magic is not universally necessary for storing ergotropy or achieving high charging power in Clifford protocols. In the stabilizer-only setting studied for quantum batteries, the reduced battery state has a flat nonzero spectrum supported on a stabilizer subspace, and the passive energy can be obtained by filling the lowest-energy levels until the stabilizer support is exhausted. In that case the Clifford-restricted ergotropy is determined by the stabilizer support structure, and at long times by the stabilizer rank through

$\rho$27

with $\rho$28 (Konar et al., 5 May 2026).

From an algorithmic perspective, exact computation of $\rho$29 is difficult because it requires minimizing

$\rho$30

over the Clifford group, whose size scales as

$\rho$31

Brute-force search is therefore practical only for very small $\rho$32; exhaustive search was used for the one- and two-qubit analyses. The paper identifies three scalable heuristics: greedy Clifford synthesis that maps large $\rho$33 onto large $\rho$34 while preserving commutation relations, locality-based restriction to Clifford frames acting nontrivially only on qubits supporting the largest Hamiltonian terms, and symplectic-group searches using the representation of Clifford elements in $\rho$35. These provide approximations but no general optimality guarantees (Maity et al., 11 May 2026).

The formalism also has clear assumptions and limitations. It is defined for closed unitary dynamics, fixed traceless Hamiltonian $\rho$36, and controls restricted to unitary elements of the Clifford group only. The universal bounds can be loose when many Hamiltonian terms compete or when $\rho$37 does not capture the detailed relative structure of the Pauli spectra of $\rho$38 and $\rho$39. The lower bound on $\rho$40 for pure states can become negative when $\rho$41 is small compared with $\rho$42. The many-body second-law statements rely on typicality and high magic; they need not apply to non-typical high-entropy but low-magic states, including entangled antipodal pair stabilizer states, for which Clifford operations may extract full ergotropy if the Hamiltonian’s ground state is stabilizer. Open directions include improving bounds, developing efficient optimization methods beyond $\rho$43–$\rho$44, extending the framework to nonunitary stabilizer operations and open-system dynamics, characterizing analogous quantities for other gate sets and resource theories, and understanding large-scale control-landscape transitions and their relation to many-body magic (Maity et al., 11 May 2026).