One-Shot Yield-Cost Relations in General Quantum Resource Theories

Although it is well known that the amount of resources that can be asymptotically distilled from a quantum state or channel does not exceed the resource cost needed to produce it, the corresponding relation in the non-asymptotic regime hitherto has not been well understood. Here, we establish a quantitative relation between the one-shot distillable resource yield and dilution cost in terms of transformation errors involved in these processes. Notably, our bound is applicable to quantum state and channel manipulation with respect to any type of quantum resource and any class of free transformations thereof, encompassing broad types of settings, including entanglement, quantum thermodynamics, and quantum communication. We also show that our techniques provide strong converse bounds relating the distillable resource and resource dilution cost in the asymptotic regime. Moreover, we introduce a class of channels that generalize twirling maps encountered in many resource theories, and by directly connecting it with resource quantification, we compute analytically several smoothed resource measures and improve our one-shot yield--cost bound in relevant theories. We use these operational insights to exactly evaluate important measures for various resource states in the resource theory of magic states.


I. INTRODUCTION
Resource distillation and dilution are two fundamental resource manipulation tasks concerned, respectively, with the purification of noisy quantum resources and with the synthesis of general resources from pure ones. There have been many efforts to investigate the optimal rates of these tasks, known respectively as distillable resource yield and resource dilution cost, under various physical settings in the asymptotic regime with vanishing error [1][2][3][4][5][6] and the non-asymptotic regime with non-zero error [7][8][9][10][11][12]. In the asymptotic regime, an operational argument paralleling the second law of thermodynamics implies that the distillable resource is bounded from above by the dilution cost, because otherwise one could produce an unbounded amount of resources by repeating the distillation-dilution cycle ad infinitum [2,5,[13][14][15][16][17][18][19]. (See also Refs. [20][21][22] for related discussions of second-order asymptotics in the context of entanglement theory.) However, in the non-asymptotic regime, the relation between the distillable resource and the resource cost is more subtle, especially when errors incurred in the transformations are also taken into account.
The recent work of Ref. [23] established a quantitative relation in the non-asymptotic regime for entanglement transformations that use local operations and classical communication, and it recovered the well-known asymptotic relation as a limit of the one-shot bound. A natural question following these findings is to what extent we can strengthen the relations obtained there and establish the corresponding bounds in more general settings beyond the distillation and dilution of entangled states.
One possibility is to extend the result for entanglement manipulation to the manipulation of other types of quantum resources, such as quantum superposition [9,10,24,25] and thermal non-equilibrium [3,26]. The ultimate form of this extension is to consider general resource theories [27], which encompass diverse types of physical resources in a single framework. Results obtained for general resource theories are not only readily applicable to practical settings of interest but also provide foundational insights into what properties are universally shared by wide classes of quantum resources, which might seem very different from each other on the surface. Despite the apparent difficulty of getting meaningful results out of this abstract setup, recent years have seen significant development in the understanding of operational aspects of general resource theories, including the settings of resource erasure [28,29], discrimination tasks [30][31][32][33][34][35], resource quantification [27,36,37], and resource transformations [4,12,16,19,31,[38][39][40][41][42].
Another possible extension is to the setting of quantum channel manipulation. Although research in resource theories has largely focused on quantum states, there are various situations in which quantum dynamics, generally represented by quantum channels, play the role of the main objects of study. As a framework to accommodate these settings, resource theories of quantum channels have recently been under rapid development [11,31,. These dynamical resources include resource theories of states as special cases, because quantum states can be equivalently represented as quantum channels that prepare them. Thus, channel theories are strictly more general than state theories, and results that hold for arbitrary quantum channel resources are automatically carried over to state theories. Importantly, we can impose additional structures to channels under study so that the resulting resource theory can describe various settings involving quantum channels with pre-determined structures, such as the scenario of Bell non-locality represented by no-signalling boxes [70][71][72], and measurement incompatibility [73], where a channel describes a set of POVMs [63]. Although the performance of channel distillation and one-shot channel manipulation in general resource theories has recently been studied [58,59,63,67,68], the relation between the one-shot distillable resource and resource cost under arbitrary free channel transformations has still been elusive.
In this work, we establish a fundamental quantitative relation between the one-shot distillable resource and dilution cost that is applicable to general resource theories of quantum channels, accomplishing the two extensions in a single framework. We further extend the one-shot bound to the asymptotic regime and obtain strong converse bounds between transformation rates, revealing also new relations in the asymptotic manipulation of quantum states. Our framework not only accommodates well-known quantum resources that have been previously studied using related approaches -such as quantum entanglement and coherence -but also other settings that involve dynamical resources as central objects of study, e.g., quantum communication. Notably, a major part of our main results does not even assume the convexity or closedness of the set of free channels, unlike most of the works dealing with general resource theories. Moreover, our bounds hold for all chosen sets of free operations satisfying only minimal requirements, thus accommodating all physically motivated choices of free operations, including ones that may be equipped with complex structures.
An important technical aspect of our result is that, for the bounds to be most effective, they require the collapse of a class of resource measures based on Rényi divergences. Focusing on resource theories of states, we provide an operational interpretation for this and related conditions in terms of a special class of free operations that reflects the essential features of the "twirling" operation commonly encountered in various theories [74,75]. We use this class of operations to obtain analytical expressions for a number of smoothed resource measures, leading also to an improved yield-cost bound for the case of state manipulation. Finally, to demonstrate the applicability of our results, we use these operational insights in the resource theory of magic [54,76,77] and compute resource measures for several special states of interest.
We focus on discussing the main results and their implications in the main text, while deferring all proofs and technical details to the Appendices.

II. RESOURCE THEORIES OF STATES
We begin by introducing the framework of resource theories of quantum states, in which quantum states are the central objects under study [78]. Quantum information processing necessarily involves the manipulation of quantum states but commonly is performed under some physical restrictions, which limits the accessible set of quantum states and operations. A primary example is a scenario in which two parties, Alice and Bob, are physically separated. Unless they are connected by a quantum communication channel, they can only apply local quantum operations and exchange classical messages. This restriction limits the accessible set of operations to the class known as local operations and classical communication (LOCC). Importantly, LOCC channels cannot create entanglement for free but can only prepare states that are separable [79].
The framework of resource theories allows us to deal formally with such restrictions [27]. It provides a recipe to quantify the amount of resource that cannot be created due to the restriction and characterize feasible resource manipulation using only accessible operations. Each resource theory comes with a set F of free states and a set O of free operations. They are subsets of quantum states and quantum channels (completely positive trace-preserving maps), respectively, and are considered to be accessible for free under a given setting. The minimal requirement for free operations is that they should not create resourceful states out of free states, i.e., Λ ∈ O ⇒ Λ( ) ∈ F, ∀ ∈ F. Physical considerations usually impose more constraints, but it is useful to consider the set of free operations consisting of all the operations satisfying this minimal requirement; we call this the maximal set of free operations and write it as O max . Consequently, any valid set of free operations is a subset of O max .
A major strength of resource theories is that they allow us to evaluate the resourcefulness of a given state quantitatively. It is made possible by introducing resource measures, which are functions from quantum states to real numbers that represent the amount of resource contained in the state. The requirement for resource measures is that they output the same value for free states, and they do not increase under the application of free operations.
Here, we consider two resource measures that can be defined for an arbitrary set F of free states and an arbitrary set O of free operations. The first one is known as the generalized robustness, or alternatively, max-relative entropy measure [80][81][82][83] defined as max,F ( ) ≔ inf log(1 + ) where D is the set of all quantum states. The other type of resource measure relevant to this work is min-relative entropy measure [82], defined as where min ( ) ≔ − log Tr[Π ], with Π denoting the projector onto the support of .
These quantities lay out a platform for establishing relations between resource manipulation and resource quantification. Resource distillation and dilution particularly stand out as important subclasses of resource manipulation tasks. Resource distillation is a protocol to transform a given state to a state in the family T of reference states using free operations, while resource dilution is the opposite task in which a reference state is to be transformed to the desired state. The optimal performance of these tasks is characterized by the one-shot distillable resource and dilution cost, defined respectively as where ( , ) ≔ √ √ 2 1 is the fidelity [84], and ℜ F refers to an arbitrary resource monotone. For simplicity, we take ℜ F = min,F in the above definition throughout this manuscript.
To provide more insight into these definitions and connect them with notions of distillation and dilution familiar from commonly encountered resource theories, let us consider the case of quantum entanglement. Here, F denotes all separable quantum states, and the family T of reference states can be taken simply as copies of the maximally entangled qubit Bell state: where |Φ 2 ≔ 1 √ 2 (|00 + |11 ) and we use the shorthand notation Φ 2 = |Φ 2 Φ 2 |. Combined with the fact that [85] min, the quantities O and O can be alternatively understood as asking, respectively, how many copies of |Φ 2 can be obtained from a given state, or how many copies of |Φ 2 are necessary to produce a given state. We use more general definitions of distillation yield and resource cost to accommodate potentially more complex families of reference states and resource theories where a simple choice of reference states T might not be available.

III. RESOURCE THEORIES OF CHANNELS
We now introduce resource theories of quantum channels, extending the framework described in the previous section. To take into account the various structures that may be imposed on channels, we employ the approach of Ref. [63] and consider a subset O all of quantum channels as the set of channels of interest. Each situation designates a set O ⊂ O all of free channels that does not contain any resource considered precious under the given setting. We also need to introduce the set of free operations that are accessible to manipulate channels. General channel transformations are described by quantum superchannels [86,87]. Superchannels are linear maps that map quantum channels to quantum channels, and they are physically realizable by sandwiching an input channel with a pre-processing channel and a post-processing channel, both of which can also be connected by a quantum memory. Letting O ′ all denote the set of output channels of interest, the set of relevant superchannels is specified as Then, one can consider a subset S ⊂ S all of superchannels as free superchannels, serving as free operations. The minimal requirement for free operations forces Θ ∈ S ⇒ Θ(M) ∈ O ′ , ∀M ∈ O. Analogous to the case of state theories, we also define the maximal set of free superchannels as the set of superchannels satisfying the above minimal condition and call it S max . We can analogously introduce resource quantifiers that do not increase under any set of free superchannels. The relevant channel resource measures that we need are the max-relative entropy measure [31,46,56,57], defined for every E ∈ O all as (7) and the min-relative entropy measure [56,57], defined as (8) where the optimization is restricted to every pure input state , without loss of generality [88].
Then, we can formalize the tasks of resource distillation and dilution. Resource distillation is a task that transforms a given channel E ∈ O all to a channel in the set T ⊂ O ′ all of reference channels using free superchannels S, while resource dilution transforms a reference channel to the desired channel E. The optimal performance of these tasks is characterized by the oneshot distillable resource and dilution cost, defined respectively as is the worst-case fidelity [88] (also called channel fidelity), and ℜ O refers to an arbitrary resource monotone, which we take as ℜ O = min,O in the above definition for simplicity. In our discussion below, we will make the natural assumption that the reference channel T is a pure target channel, in the sense that id ⊗ T ( ) is pure for every pure state . This covers general types of distillation processes encountered in physical resource theories, where T plays the role of a noiseless resource. If the input and output systems are identical, then the condition implies that T is unitary, while if the input and output systems have different dimensions, then it implies that T is an isometry. When the input space is trivial -which is the case, e.g., in the special case of resource theories of states -then T reduces to a pure-state preparation channel.

IV. YIELD-COST RELATIONS
Our first main result is a relation between the one-shot distillable resource and dilution cost, which we dub a yield-cost relation, applicable to (1) dynamical resource theories of channels, (2) any set of free channels, and (3) any set of free superchannels defined for the given free channels.

Theorem 1.
Suppose that for every channel T in the chosen reference set T, id ⊗ T ( ) is pure for every pure state , and T satisfies min,O (T ) = max,O (T ). Then, for all 1 ∈ [0, 1) and 2 ∈ [0, 1− 1 ], for every quantum channel E, and for every set S ⊆ S max of free superchannels, the following inequality holds where ( 1 , 2 ) is a function defined as otherwise.
For 1 + √ 2 < 1, each quantity can be tighter than the other for certain error regions. Indeed, direct calculation reveals that This gives a fundamental relation between distillable resource and dilution cost in the one-shot regime under any chosen set of free superchannels. In particular, when transformation errors are taken into account, the one-shot cost could be smaller than the one-shot yield. Our result establishes a quantitative trade-off relation between a potential yield-cost gap and the transformation inaccuracy.
The only assumption required for the result to hold is the choice of reference channels which satisfy min,O (T ) = max,O (T ). The collapse of the two measures to the same value is a common property of maximally resourceful states or channels [12,41,63], which are precisely the most appropriate references employed in distillation and dilution protocols in practice. We review examples of reference states and channels in several physical settings in Appendix B. We will also return to this condition in Section V to provide more intuition behind the assumption and give an operationally motivated understanding of it.
Theorem 1 is a direct consequence of the following lemma.
A natural question following these results is whether they smoothly connect to an asymptotic yield-cost relation. In asymptotic distillation (and analogously for dilution), the goal of the tasks is commonly to obtain as many copies of a fixed reference channel as possible from multiple copies of the given channel, in such a way that the transformation error vanishes in the limit of infinite copies. The figure of merit is then the ratio of the number of obtained copies of the reference channel to the number of used copies of the given channel. The following result provides a relation between the optimal rates for the asymptotic distillation and dilution in a more general setting, in which the errors do not necessarily approach zero in the asymptotic limit.
Theorem 3. Let E be an arbitrary input channel, and let T be some target reference channel for which id ⊗ T ( ) is pure for every pure state . Let be any rate of distillation such that there exists a sequence {Θ } of free superchannels with Also, let be any rate of dilution such that there exists a sequence {Θ } of free superchannels with Suppose that the following conditions are satisfied: (i) It holds that lim inf →∞ + < 1; (ii) The resource theory is closed under tensor products; that is, Then the following inequality holds Let us briefly discuss the assumptions of Theorem 3. Condition (i) simply means that the errors of distillation and dilution do not get too large at the same time; it is satisfied, for instance, when we take lim →∞ = 0 and lim inf →∞ < 1 or vice versa. Condition (ii) is a basic property obeyed by virtually all theories encountered in practice.
Although the result of Theorem 3 is appealing since it follows straightforwardly from our one-shot relation in Theorem 1, one can ask whether it is possible to derive asymptotic relations that have better dependence on quantities such as ∞ max,O (T ) and ∞ min,O (T ), as well as are free from the constraint on the target channel T , that is, the constraint that id ⊗ T ( ) is pure for every pure state . We discuss several variations on this idea in Appendix C, where we present other bounds that are potentially tighter and do not impose any condition on T but come with an additional condition on the sequences { } and { } of achievable errors or require the computation of more complicated regularized quantities. For our purposes, it will be sufficient to use the condition of Theorem 3, and indeed we will see that the reliance on the max-/min-relative entropies is not a problem in most theories of practical interest.
The result of Theorem 3 implies in particular the strong converse bounds given in Corollary 4 below, for the rates of resource manipulation tasks. Before stating this result, let us recall the definitions of the basic asymptotic quantities involved. We define the asymptotic distillable resource ∞ S as the largest achievable rate at which copies of T can be extracted from a given channel, and, analogously, the asymptotic resource cost ∞ S as the least rate at which copies of T are needed to produce a given channel. Imposing that the error in such transformations vanishes in the asymptotic limit, we have (20) As counterparts to the above asymptotic quantities, we define also the corresponding strong converse quantities as [17,89] with the interpretation of the strong converse distillable resource as the smallest rate at which the error is guaranteed to converge to one in the asymptotic limit, and that for the strong converse dilution cost as the largest rate at which the error is guaranteed to converge to one. The following equivalent expressions for the strong converse quantities hold: which can alternatively be understood as weakening the requirements imposed on the transformation error in Eq. (20). Here the interpretation is as follows: unlike the quantities in (20), the strong converse quantities no longer ask that the error converges to zero, but represent the best rates at which the error does not converge to one -they therefore provide a general threshold on achievable asymptotic transformations even when perfect conversion is not required. Note that ∞ S (E, T ) ≤˜ ∞ S (E, T ) and ∞ S (E, T ) ≥˜ ∞ S (E, T ) by definition.
Theorem 3 implies the following strong converse bounds.  Then the following inequalities hold: We remark here that the definitions of strong converse quantities and Corollary 4 imply the following fundamental inequality: Here, condition (i) is generally satisfied in physical theories, but condition (ii) imposes non-trivial requirements on the choice of a suitable target channel T . In particular, in addition to the collapse of the max-and min-relative entropy measures to the same value, it requires that the min-relative entropy measure is additive, i.e. min,O (T ⊗ ) = min,O (T ). Although this constitutes a seemingly restrictive condition, the existence of channels or states satisfying this requirement is actually a common property of important resource theories, obeyed, e.g., by quantum entanglement, quantum thermodynamics, magic, and quantum communication (cf. [58, Table 1]). Intuitively, the condition can be thought of as characterizing how well the given reference T serves as an intermediary channel in distillation and dilution protocols.
We stress that all the results in this section automatically provide bounds for state theories as special cases; when one is interested in a resource theory of quantum states with free states F and free operations O, the same results hold by replacing free superchannels S with free operations O and free channels O with free states F. Furthermore, in the case of state transformations, the generalized quantum Stein's lemma [15,90] implies that the asymptotic distillable resource coincides with its strong converse rate when the set of free states and the reference state satisfy some mild assumptions -see Appendix D for details. When the generalized quantum Stein's lemma holds, for the asymptotic distillation and dilution for an arbitrary state with respect to a reference state Φ under free operations O, we can improve the bound in Corollary 4 tõ This applies in particular to the resource theory of quantum entanglement, which solves an open problem posed in Ref. [23]. The implication of (25) is that the yield-cost relation still holds even when neither of the errors is required to vanish in the asymptotic limit, which gives a significant strengthening of the inequalities in (23).
We remark that another approach to asymptotic transformation rates of states and the consequent relations between asymptotic resource yield and cost in general resource theories was considered in Ref. [19].

V. OPERATIONAL ACCOUNT FOR REFERENCE STATES
We saw that the relation between min,O and max,O plays a major role in establishing the yield-cost relation in general resource theories. Here, we give operational insights into the properties of these quantifiers and in particular the condition min,O (T ) = max,O (T ) that we have encountered previously. By studying the interplay between these and other related measures, we will be able to quantify a number of smoothed resource monotones for target reference states in general resource theories. In this section, we restrict our attention to theories of quantum states, and we use F and O to specify the set of free states and free operations, respectively. In what follows, we also assume that F is a convex and closed set.
In addition to min,F and max,F , we also consider another type of robustness measure known as the standard robustness [80], defined as We also define two smooth robustness quantities as [82,90] , and two types of hypothesis-testing relative entropy measures [10,41,90]: where is the hypothesis-testing relative entropy [7,91,92]. We used aff(F) to denote the affine hull of F, which is the smallest affine subspace that contains F. The min-relative entropy is obtained as a special case of the hypothesis-testing relative entropy measure as min,F ( ) = =0 ,F ( ). We use this correspondence to define the affine min-relative entropy measure [41,63] as Then, the following ordering holds for an arbitrary state : where the first inequality follows because F ⊆ aff(F), the second inequality because min ( ) ≤ max ( ) for all states and , and the third inequality from the definitions of the generalized and standard robustness measures. Depending on the structure of F, some of these measures may exhibit drastic behavior. min,F and max,F take finite values for every state as long as F contains at least one full-rank state, which is satisfied by most of the relevant theories. However, ,F may diverge if F is reduced dimensional [41], meaning that F has zero volume in the set of all states and span(F) is not the whole space of self-adjoint operators. Examples of reduced-dimensional theories include the theories of coherence and thermal non-equilibrium. On the other hand, for full-dimensional theories, which are not reduced dimensional and include the theories of entanglement and magic as examples, ,F remains finite but min,aff (F) takes the value zero for every state . Therefore, ,F is usually a relevant resource quantifier in full-dimensional theories while min,aff (F) is relevant in reduced-dimensional theories [41,63].
Although these measures have been introduced in a rather abstract manner, they play crucial roles in the quantitative characterization of distillation and dilution -under certain assumptions on the target state Φ, the value of the one-shot resource yield O ( ) under the maximal set of free operations is directly related to ,F ( ) or ,aff (F) ( ), while the value of the resource cost O ( ) corresponds to the value of ,F ( ) or max,F ( ) [12,41]. A necessary requirement for a precise description of distillation or dilution to be possible was that, when evaluated on the reference state Φ, the resource measures all collapse to the same value; specifically, in full-dimensional theories one needs that while in reduced-dimensional theories one instead requires In some theories, such as entanglement or coherence, the existence of such states is natural: the maximally entangled or coherent states always satisfy the requirement. It is rather remarkable that maximally resourceful states satisfy min,F (Φ) = max,F (Φ) in any convex resource theory [41], but this is still not sufficient to guarantee that Eqs. (33)- (34) hold in general -we expect this to be a resource-dependent property which needs to be verified explicitly in each specific setting.
Here, we find that Eqs. (33)- (34) can have operational implications, which then help evaluate other resource measures introduced in this section. In particular, we provide an understanding of the conditions (33)- (34) in terms of a twirling-like free operation.
such that ★ is a state orthogonal to Φ, satisfying Tr , one can further take ★ to be a free state.
This free operation possesses a property similar to wellknown group twirling operations such as the isotropic twirling ∫ ⊗ * (·) † ⊗ * † d in entanglement theory [75], in that it maps states to the given reference state or its complement, while stabilizing a specific state Φ. Such twirling operations were found to be useful to evaluate several entanglement measures for states invariant under them [93]. In particular, the existence of such an operation allows us to obtain exact expressions for smoothed resource measures, which a priori require a non-trivial optimization over all states within an error .
, then for every ∈ [0, 1) it holds that A crucial property which makes this result possible is that optimal states in the optimization of the smoothed resource measures can always be taken of the form Φ = Φ + (1 − ) ★ , as obtained through the operation in Lemma 5. Such states share many useful properties with isotropic states of entanglement theory, and indeed several quantitative insights in the description of isotropic states in works such as Ref. [93] can be extended to general resource theories. In particular, the entropic resource quantifiers can be computed exactly for such states.
Proposition 7. Suppose a state Φ satisfies min,F (Φ) = ,F (Φ) =: , and let ★ be the state in (35). Then, and We present a more complete discussion of the quantitative properties of the isotropic-like states Φ in Appendix E, where we also show how the smoothed entropic measures can be computed for this class of states.
These exact evaluations of the resource measures allow us to employ the argument in Ref. [23] to obtain an alternative bound to that given in Theorem 1.

Theorem 8. If the chosen reference set obeys
This improves the bound in Theorem 1, as it gives a tighter upper bound for all regions of ( 1 , 2 ) with 1 + 2 < 1, as we prove in Appendix G. We also remark that this bound becomes essentially tight when O = O max , ∈ T, and either 1 or 2 is 0. This can be shown by using 1 O max,F ( ) when min,aff (F) ( ) = max,F ( ) (up to some floor or ceiling due to a discrete structure of T) [63], as well as explicitly evaluating these measures by Proposition 6.
So far we have employed the property of several resource measures evaluated for a reference state to show the existence of a twirling-like operation (35) and presented its applications. We now also investigate the opposite direction, asking whether the existence of a certain type of free operation provides insights into the relation between different resource measures. The following result addresses this question. Lemma 9. Let S(Φ) be a set of channels stabilizing Φ, defined as where CP is the set of completely positive maps. Then, if there exists a free operation Λ ∈ O max ∩ S(Φ), the equality holds. In addition, if Λ is completely free, i.e., id ⊗ Λ ∈ O max where id is the identity map on an arbitrary ancillary system, then Moreover, ifΛ can be taken as a free subchannel in (45), where cone( ) ≔ ≥ 0, ∈ .
Together with Lemma 5, we conclude a general operational correspondence between the collapse of resource measures and the existence of twirling-like free operations: we have that However, it is not always easy to find a channel Λ ∈ O max ∩ S(Φ) to apply Lemma 9. The following result helps to find such a channel by formally relating the group-theoretic twirling operations inspired by the original LOCC twirlings [74,75] to the form of (45) necessary to apply Lemma 9. This result allows us to find an appropriate free operation to ensure the condition required in Theorems 1 and 8. We indeed find that Lemma 9 and Proposition 10 are helpful to obtain new insights into specific resource theories, which we now demonstrate.

VI. APPLICATION: THEORY OF MAGIC
As we emphasized before, our results immediately hold in general types of quantum resources of both states and channels, and we direct the interested reader, e.g., to the recent Refs. [12,58,59,63,67] as well as to Appendix B for a discussion of how such general methods can be applied in specific theories such as quantum communication. Here, we discuss a non-trivial example in which the quantification of resource measures for many important states has still not been understood -the theory of magic (non-stabilizerness) [76,77]. Applying the results of our work, we show how they can be used to provide new quantitative insights and reveal broad and useful relations for this resource.
To realize scalable fault-tolerant quantum computation, it is essential to encode the whole computation in a higherdimensional space using an error-correcting code. Clifford gates particularly stand out as a subset of quantum gates that admits efficient fault-tolerant encoding on many error-correcting codes. However, to form a universal gate set, we also need to implement a non-Clifford gate. It is usually accomplished by the gate teleportation technique [94,95], which simulates the action of a non-Clifford gate by combining a Clifford operation and a magic state, which cannot be created solely by Clifford operations. Since magic states are hard to produce in a faulttolerant manner, they are precious resources under this setting, motivating us to consider the quantification and manipulation of them using a resource-theoretic formalism [76,77].
In particular, the optimal performance of magic state distillation and dilution has been a central question in the field, as it is the most resource-demanding part to realize fault-tolerant universal quantum computation. Our results establish a relation between the optimal performance of these two, under the assumption that the reference resource states satisfy the aforementioned conditions on resource measures. Here, we investigate these conditions for some well-known states that can be good candidates for reference states in distillation and dilution protocols. We find that our approach, particularly Lemma 9 and Proposition 10, provides new operational insights into the evaluation of resource measures, which may be of independent interest.
Stabilizer states are those that can be represented by a probabilistic mixture of eigenstates of Pauli operators. In the resource theory of magic, stabilizer states form the set F STAB of free states. Although it is often assumed that quantum computation is carried out in multi-qubit systems, higher-dimensional qudit systems also stand as potential candidates for a largescale quantum computing architecture [96,97]. Resource theories have been developed for both settings [76,77,98], and we will consider several standard magic states defined in these scenarios.
Let us start with multi-qubit systems. For single-qubit states, the farthest states from the set of stabilizer states are positioned at 1 √ 3 (±1, ±1, ±1) in Bloch coordinates, which are connected to each other by Clifford unitaries. We call one of them the face state, which is written as where , , are qubit Pauli operators. Then, Lemma 9 and Proposition 10 allow us to obtain the following relation: Proposition 11. The qubit face state satisfies for every integer ≥ 1.
Since the face state is a maximizer of min,F STAB , called a golden state [12,41], the case of = 1 recovers the result in Ref. [12]. Also, noting that the stabilizer extent introduced in Ref. [99] is identical to max,F STAB [36], Eq. (50) can be shown by using the additivity of min,F STAB ( ⊗ ) and max,F STAB ( ⊗ ), as well as the equivalence of min,F STAB ( ) and max,F STAB ( ) shown in Ref. [99]. Our result provides a different operational approach to this relation based on a strategy with a broad potential applicability.
We also get an analogous result for a special class of magic states that includes the state and the Toffoli state, which are usually used for gate teleportation protocols.

Proposition 12.
Let be a unitary in the third level of the Clifford hierarchy, and let | be a stabilizer state defined on a multi-qudit system with local dimension , where = 2 or is an odd prime. Then, the state | = | satisfies min,F STAB ( ⊗ ) = max,F STAB ( ⊗ ) for every integer ≥ 1.
This provides an alternative proof of the result in Ref. [99] and extends it to qudit systems, including the qudit generalization of the state [97]. Note that max,F STAB ( ⊗ ) = max,F STAB ( ) holds for every of up to three qubits [99], and indeed also for single-qubit mixed states [100].
Although min,F STAB and max,F STAB coincide for the above cases, one can actually show that for every single-qubit state, max,F STAB is strictly less than ,F STAB . The natural question is whether there exists a state for which the three quantities collapse to the same value, allowing us to use both Theorems 1 and 8. No such state has been previously known in the theory of multi-qubit magic, which prevented previously known bounds for one-shot distillation and dilution from being tight [12,41]. To show that a suitable choice does indeed exist, let us consider the three-qubit fiducial Hoggar state [77,[101][102][103][104] defined as The Hoggar state serves as one of the fiducial states from which group covariant SIC-POVMs can be generated [104]. We show that the three measures collapse for the Hoggar state.
We note that the value of ,F STAB (Hog) was reported in Ref. [77].
Let us now turn our attention to qutrit states. In the qutrit magic theory, two classes of states were identified to have the maximum sum negativity of the discrete Wigner function [76]. The first class is represented by the Strange state [76,98,105,106], defined as We can use Lemma 9 to show the collapse of the three measures for this state.
Our results complement the findings in Ref. [98], which considered min,F ′ and max,F ′ with respect to a larger set of free operators (not necessarily normalized quantum states) based on the negativity of the discrete Wigner function and found that min,F ′ ( ) = max,F ′ ( ) = log(5/3).
Another maximizer of the sum negativity is the Norrell state [76] defined as We can get a similar collapse for the measures, but to a different value.
These results may make one wonder whether there is a general characterization for when the three measures take the same value. Although we still do not have a definitive answer for this question, we can make the following observation. For a given state , Lemma 5 implies that if min,F STAB ( ) = max,F STAB ( ), we must have max,F STAB ( ) < ,F STAB ( ) unless there exists a free state acting on supp ½− −1 . This is equivalent to the condition that the weight resource measure [58,107], defined for the set F of free states as This condition can be explicitly verified to hold for the Strange state and Norrell state, for which we have seen the three measures to coincide. On the other hand, for the qutrit state [97], defined as Interestingly, numerical investigations suggest that, for most qutrit states, the weight measure for the complement is equal to zero, indicating that the non-zero gap between max,F STAB and ,F STAB is a generic feature shared by the majority of qutrit states. We leave a thorough investigation on the relationship between the weight resource measure and the robustness measures for future work.

VII. CONCLUSIONS
We established a quantitative relation between the one-shot distillable resource and resource cost for general quantum resource theories, including both state-based resources as well as dynamical resources of quantum channels. We also showed the corresponding bounds in the asymptotic regime and recovered the familiar relation between the distillable resource and resource cost in the form of strong converse bounds. We investigated the conditions that appear in the yield-cost relation and related it to a class of free operations that have similar properties to twirling operations. We employed such operations to obtain analytical expressions for several smoothed resource measures for general resource theories of states and tightened the yield-cost relation. We then applied our operational technique to evaluate resource measures for several standard resource states in the resource theory of magic, recovering previous results with different techniques and presenting new evaluations of measures for some magic states.
Outstanding questions include the extension of the results that were only shown for state theories in this work to channel theories, such as the two-sided strong converse bound (25) and the relation to the twirling-like operation. The difficulty in characterizing the manipulation of quantum channels, and in particular their asymptotic properties [60], makes such questions non-trivial to answer. Another interesting direction is to use the operational techniques introduced here to shed light on settings other than magic theory. On the other hand, it will also be beneficial to gain a deeper understanding of the structure of magic theory, for which additional operational insights might be helpful.

ACKNOWLEDGMENTS
We thank an anonymous referee of Ref. [23] for suggesting to improve a result in Ref. [23] by using a tighter triangle inequality of the purified distance from Ref. [108], which eventually allowed us to arrive at the bound in Let us first define the smoothed channel divergences [46,109]: which construct the following resource monotones [46,56,57]: We begin by showing some useful lemmas. The first two are channel extensions of the relations shown in Refs. [110,111] between the hypothesis-testing relative entropy and the maxrelative entropy of quantum states. They have an interpretation in the resource theory of asymmetric distinguishability as a one-shot yield-cost relation for bits of asymmetric distinguishability [55].

for every , (3): max-min inequality.
We note that alternative bounds with trace-distance and diamond-distance smoothing were shown in Refs. [55,112].
We also have the following alternative bound.

Lemma 17.
For all 1 , 2 ≥ 0 with 1 + 2 < 1, two arbitrary channels E, M satisfy Proof. The proof of [111,Theorem 4] established that for all 1 , 2 ≥ 0, 1 + 2 < 1 and all states , . The statement for the channels follows by employing the same argument in the proof of Lemma 16.
We also recall monotonicity properties under one-shot channel transformations involving smooth measures.
Lemma 18 (Theorems 1 and 3 in Ref. [63]). If there exists a free superchannel Θ ∈ S max such that (Θ(E), N) ≥ 1 − , then for every resource monotone ℜ O , In particular, Now let N be a channel such that id ⊗ N ( ) is pure for every pure state . If there exists a free superchannel Θ such that (Θ(E), N) ≥ 1 − , then We are now in a position to prove Theorem 1 and Lemma 2.
Proof of Theorem 1 and Lemma 2. Theorem 1 is obtained from Lemma 2 by maximizing the left-hand side over T 1 ∈ T and minimizing the right-hand side over T 2 ∈ T for fixed errors 1 , 2 . Thus, it suffices to show Lemma 2. Let T 1 be an arbitrary channel for which there exists Θ 1 ∈ S such that (Θ 1 (E), T 1 ) ≥ 1 − 1 , and let T 2 be an arbitrary channel for which there exists Θ 2 ∈ S such that (Θ 2 (T 2 ), E) ≥ 1 − 2 . Then, where in the first and the third inequalities we used Lemma 18, and in the second inequality we used Lemmas 16 and 17.

Theorem 3 can be shown similarly.
Proof of Theorem 3. Let us first observe that the regularized quantities ∞ min,O and ∞ max,O are both well defined. This follows since max,O and min,O are subadditive under assumption (ii) -as can be seen from their definitions using the additivity of max and min on tensor-product arguments -and hence the limit in the definition of the regularized quantities exists by Fekete's lemma. Now, by assumption (i), there exists a subsequence { } of indices such that lim →∞ + < 1. Then ≤ lim = lim We briefly review several examples of physical settings, as well as a reference channel T , that satisfy the conditions of Theorem 1. Examples discussed here are either state preparation channels with one-dimensional input or unitaries with input and output of the same dimension, ensuring that id ⊗ T ( ) is pure for every pure state . They further satisfy min,O (T ) = max,O (T ) and thus serve as appropriate references for distillation and dilution protocols that meet the conditions in Theorem 1.
Let us first consider state theories, where O is a set of channels preparing free states (denoted by F in Section II). The theory of bipartite entanglement [79], in which separable states serve as free states, takes a maximally entangled state |Φ + = =1 −1/2 | as a reference state and obeys min,O (Φ + ) = max,O (Φ + ) = log . In the case of the theory of coherence [25], where the free states are the diagonal states with respect to a given preferred basis, a maximally coherent state, i.e., a uniform superposition of the basis states | + = =1 −1/2 | , meets the condition min,O ( + ) = max,O ( + ) = log . The theory of thermal non-equilibrium [3] is defined by fixed temperature and Hamiltonian. The "work bit" represented by the eigenstate of the Hamiltonian with the highest energy is considered as a standard unit resource, and it also satisfies the conditions of Theorem 1. In fact, the above observation can be generalized to an arbitrary state theory; every theory with an arbitrary convex set F of free states is equipped with a golden state Φ gold [12] such that min,F (Φ gold ) = max,F (Φ gold ) [41]. See also Section VI in the main text for further discussion on the theory of magic states.
Many important dynamical resource theories are relevant to the setting of quantum communication. A central purpose of quantum communication is to transmit a quantum state from one party to another. In such a scenario, the identity channel connecting the two parties is considered as the most useful channel. It is then natural to take the -dimensional identity channel id as a reference in distillation and dilution protocols, and indeed, the identity channel satisfies the conditions in Theorem 1 under several coding schemes that are specified by different choices of O [58,63]. For instance, in the theory of no-signalling-assisted communication, we get min,O (id ) = max,O (id ) = 2 log , while in quantum communication assisted by LOCC, separable, or positive partial transpose codes we get min,O (id ) = max,O (id ) = log . A related setting is a framework that quantifies how much quantum memory a given channel can preserve [47]. This theory takes the set of entanglement-breaking channels as free channels, and the identity channel again serves as an appropriate reference channel [53]. In relation to Section VI, dynamical resource theories of magic [54,113] admit several unitary gates as reference channels satisfying the conditions in Theorem 1 [58].

Appendix C: Alternative asymptotic bounds
Besides Theorem 3, we can also show alternative asymptotic bounds, which are tighter but have additional conditions on the achievable errors. Another advantage of our alternative bounds is that they do not require the reference channel T to be pure, i.e., id ⊗ T ( ) does not need to be pure for every pure state .
Let us begin by presenting some useful lemmas. We first introduce the smoothed hypothesis-testing relative entropy measure as Then, we can relate this smoothed measure and the standard hypothesis-testing measure as follows. Note that for an arbitrary channel E ′ satisfying (E ′ , E) ≥ 1 − , an arbitrary positive semidefinite operator satisfying 0 ≤ ≤ ½, and an arbitrary state , where the first inequality is because of the optimization form of the trace distance 1 2 − 1 = max 0≤ ≤½ Tr[ ( − )] satisfied for arbitrary two states and [114], and the second inequality is because where we used the relation between trace distance and fidelity [115] in the first inequality and the definition of channel fidelity (10) (C6) where in the last line we used the max-min inequality. Taking − log on both sides concludes the proof.
The following result, which is a variant of Theorem 1 in Ref. [63], is also useful.

Lemma 20.
If there exists a free superchannel Θ ∈ S max such that (Θ(E), N) ≥ 1 − , then for an arbitrary resource monotone ℜ O , Proof. Let (E, N) ≔ 1 − (E, N) be the sine distance of quantum channels (also known as purified distance), which satisfies the triangle inequality for all channels E, N, T [88]. Noting that ℜ O is a resource monotone for all , we can use Lemma 18 to get Using the triangle inequality of the purified distance, we get (N ′′ , N) ≤ √ + √ for all channels N, N ′ , N ′′ that satisfy Remark. We can improve the triangle inequality (C9). Define the following "best-case" fidelity where the supremum in the first line is taken over every possible ancillary system , and the second equality is due to the data-processing inequality, ensuring that tracing out the ancillary system does not decrease the fidelity. The last form particularly allows for an efficient computation via the following semidefinite program, which follows from (C12) and the known semi-definite program for fidelity [116]: where L(H ) denotes the set of linear operators acting on the output Hilbert space H for the channels E and N. Then, we have the following inequality. Proof. Recall that the purified distance ( , ) ≔ 1 − ( , ) defined for arbitrary two states and satisfies the following triangle inequality [108] ( , ) ≤ ( , ) for every set of three states , , such that ( , ) 2 + ( , ) 2 ≤ 1. We can employ this to obtain Eq. (C14) is tighter than (C9) in general because max (E, N) ≤ 1 for every E and N. Although (C9) and Lemma 20 suffice to establish the forthcoming results, Lemma 21 may find use in other different settings.
We are ready to show the first alternative asymptotic bound, which is tighter than Theorem 3 and does not assume anything on T , while having less flexibility in the achievable errors.

Proposition 22. Let E be an arbitrary input channel and let T be some target reference channel. Let be any rate of distillation such that there exists a sequence {Θ } of free superchannels with
Also, let be any rate of dilution such that there exists a sequence {Θ } of free superchannels with Suppose that Then, the following inequality holds Proof. Let { } be a subsequence of indices such that ∞ ≔ lim →∞ < 1. Then, for every with 0 < < 1 − ∞ , we get (C22) The second inequality follows from Lemma 20, the third inequality from Lemmas 16-17 and the fact that + ( √ + √ ) 2 < 1 holds for sufficiently close to 0 and for sufficiently large , the fourth inequality from Lemma 18 noting that ,O is a resource monotone for a fixed , and the fifth inequality from Lemma Since this holds for every with 0 < < 1 − ∞ , we can further take lim →0 and use In particular, this gives a strong converse inequalitỹ In the case of state transformations with mild assumptions (c.f., Appendix D), the asymptotic equipartition propertỹ [90], and consequently we get as we discuss in Appendix D.
Finally, if we further impose a stronger condition to the achievable errors, we can obtain an even tighter bound.

Proposition 23. Let us consider the setting in which we instead have the condition lim →∞
Then, where the second and third lines follow from Lemma 20. Since lim →∞ = lim →∞ = 0, for arbitrary constant > 0, it is ensured that < and < for sufficiently large . Since ℜ O is nonincreasing with respect to , we can bound the last expression as Since this holds for every > 0, we can bound the expression in (C31) by taking the limit → 0 as resulting in the desired inequality in (C30).
This extends and complements similar relations known for state transformations in settings such as entanglement [14,18] and a general class of resources [16,19]. In particular, this reduces to the intuitive bound ≥ when ℜ Note, however, that the regularized resource measure may take 0 for all channels in some cases [117].

Appendix D: Strong converse property of distillable resource
Ref. [90] discussed a generalization of quantum Stein's lemma for resource theories satisfying mild assumptions. Using this, Ref. [15] characterized the asymptotic distillable entanglement under the set of non-entangling operations with the regularized relative entropy of entanglement. In fact, their argument shows more than that -the regularized relative entropy of entanglement also serves as a strong converse distillation rate.
Here, we extend this strong converse property to general resource theories by combining the results in Refs. [15] and [41] in the case when the generalized quantum Stein's lemma holds. This can then turn the one-sided strong converse inequality in Corollary 4 to a double-sided inequality, making both quantities -yield and cost -strong converse rates for each other.
Before stating the result, we recall the following characterization of the fidelity of distillation. where Then, we get the following relations.

Proposition 25. Let F be a set of free states which:
1. is convex and closed, 2. contains a full-rank state, 3. is closed under partial trace and composition of free states.
Also, let Φ be a state that satisfies Proof. Let O ( → ) be the fidelity of distillation from to under free operations O, defined as Lemma 24 and the assumed additivity of min,F gives where = min,F (Φ) = ,F (Φ). Note that F ( ; ) is a convex optimization program, and one can obtain its dual program by following standard techniques in convex optimization theory [118] (cf. [90]).
for all and . Let ∞ rel,F ( ) be the regularized relative entropy resource measure [15,16] where ( ) is the relative entropy defined for arbitrary two states and taking Tr[ log ] − Tr[ log ] if supp( ) ⊆ supp( ) and +∞ otherwise. Let us take = 1 ∞ rel,F ( ) + for some > 0. Then, for each , we can take = 1 ∞ rel,F ( ) + 2 to get Ref. [90,Prop. III.1] showed that the first term approaches 0 in the limit of → ∞ for every > 0. Therefore, combining it with (D6), we get that for all > 0, On the other hand, if we take = 1 ∞ rel,F ( ) − , the optimum for each needs to satisfy < because otherwise ) would diverge as → ∞. Therefore, The right-hand side approaches 1 for every > 0 [90] and thus leads to for every > 0. (D13) and (D15) imply .
(D16) Finally, combining this with Corollary 4, we get Noting that which concludes the proof.
We note that the fact that 1 ˜ ∞ max,O (E) is a strong converse rate for distillation in general resource theories of channels was previously shown in Ref. [58]. However, the relatioñ and in particular the fact that 1 ∞ rel,F ( ) constitutes an achievable rate of distillation under O max is a very non-trivial result established in Ref. [90], applicable only to resources of quantum states. It is an open question whether an extension of this result to channel theories can be obtained (cf. [60]). It would also be interesting to understand whether the double-sided strong converse bound described in this section can be shown without relying on the generalized quantum Stein's lemma of Ref. [90].

Appendix E: Proof of Lemma 5, Proposition 6 and Proposition 7
We first show Lemma 5.
Proof of Lemma 5. Suppose min,F (Φ) = ,F (Φ). By definition of the standard robustness and the closedness of the set F, there exist states , ★ ∈ F such that Also, by definition of min,F (Φ) and noting that min,F (Φ) = =0 ,F (Φ), one can confirm that an operator ★ = Π Φ satisfies F. Using this operator, define the following channel One can check that Λ ∈ O max as follows. For every ∈ F, we get The convexity of F and the form of (E1) implies that all states of the form The proof for the case of min,aff (F) (Φ) = max,F (Φ) goes analogously. The only difference is that the inequality in (E4) becomes an equality, and ★ does not necessarily coincide with Π Φ . By definition of the generalized robustness, there exist a free state ∈ F and some state ★ ∈ D such that Also, by definition of min,aff (F) (Φ) ≔ Using this operator, define the following channel It is now easy to check that Λ ∈ O max using (E8) and (E7). Next, we show that Tr[ ★ ★ ] = Tr[Φ ★ ] = 0. If we apply Λ to in (E1), we get where is a contractive measure under free operations, i.e., ( , ) ≥ (Λ( ), Λ( )), ∀ , for an arbitrary free channel Λ ∈ O max . Suppose ,F (Φ) = min,F (Φ) =: for some state Φ, and let ★ be the state that appears in (35). Also, let D andF be the sets of states defined as Then, for every ∈D, we can restrict the optimization in (E12) as On the other hand, if max,F (Φ) = min,aff (F) (Φ) =: , then every ∈D satisfies We are now ready to prove Propositions 6 and 7. We will in fact show a more general result, which immediately implies both of the Propositions and allows for the computation of the smoothed entropic measures for all isotropic-like states Φ . Proposition 27. Suppose a state Φ satisfies min,F (Φ) = ,F (Φ) =: , and let ★ be the state in (35). Then, every Also, let min and max be the minimum and maximum that satisfy cl (( , 1 − ), ( , 1 − )) ≥ 1 − , where cl ( , ) ≔ √ 2 is the fidelity for two classical distributions. Then, for all ∈ [0, 1).
We remark that one can obtain an analogous result for smoothing with different distance measures. In particular, the trace-distance smoothing leads to simple expressions with min = − and max = + .
Proof. Let us first consider the case when min,F (Φ) = ,F (Φ) = . Since (· ·) satisfies the data-processing inequality, we can use Lemma 26 to get whereF is the set of free states defined in (E13). Let Λ and ★ be the channel and operator in (35).
Let A be the set of ( , ) that satisfies (E22). Then, we can compute ,F (Φ ) as log min where we used Tr[ ★ Φ] = 1 and Tr[ ★ ★ ] = 0. Let us first consider the case = 1. The condition for A in this case turns to Then, (E23) can be further computed as On the other hand, suppose = 0 and 0 ≤ ≤ 1. Then, the condition for A takes the form From this, it is clear that for the case 0 < < 1, we must have = = 1, which makes the quantity in (E23) equal to 0. In the case of = 0, which forces = 1, 0 ≤ ≤ 1, the optimum in (E23) is achieved at = = 0, = 1, resulting in 0 also. The case of = 1 is included in (E25); we get by setting = 0.
To summarize, we showed To show the expression for max,F , note that where in the second line we used Lemma 26 because max (· ·) satisfies the data-processing inequality for all quantum channels. Consider again the map Λ in (35), which maps an arbitrary state to a state inD (c.f., (E13)), while stabilizing every state Φ ∈D and ∈F as Λ(Φ ) = Φ and Λ( ) = . Then, ≤ implies Λ( ) ≤ Λ( ) = , ∀ ∈F. We also have Let min and max be the minimum and maximum that satisfies (E30). Then, we can compute max,F (Φ ) as which concludes the proof for the expression of max,F . The smooth standard robustness can be computed similarly.
and ,F is contractive under every free operation E ∈ O max , as for all states and , where we set˜ as the optimal state realizing the standard robustness, and in the third line we used that (E (˜ ), E ( )) ≥ (˜ , ) ≥ 1 − . Thus, we can use Lemma 26 to compute where in the first line, we used the notation ≤ F ⇐⇒ − ∈ cone(F). When min ≤ 2 − , we immediately get ,F (Φ ) = 0, which is achieved at = . On the other hand, when min ≥ 2 − , we have > 1− 1− and > 1−2 1−2 , in which the minimization over and results in ,F (Φ ) = − log 1 min . These two can be combined as The proof for the case when min,aff (F) (Φ) = max,F (Φ) goes analogously, where we basically change the region of optimization over free states fromF to {˜ } defined in (E15). Since ( ) satisfies the data-processing inequality even if is not a positive operator [63], we can employ the same argument as that in Lemma 26 to get where we used that the map Λ in (35) transforms all ∈ aff(F) to˜ . We can then restrict the optimization to = ★ + (½− ★ ) with the condition (E22). Recall that A is the set of ( , ) that satisfies (E22). Then, we can compute ,aff (F) (Φ ) as log sup When = 1, the condition on A becomes (E24). Then, we evaluate this as On the other hand, when = 0 and 0 < < 1, we are constrained to = = 1, leading to value 0. When = 0, we have = 1, 0 ≤ ≤ 1, giving log 1 1−2 − . When = 1, we have 0 ≤ ≤ 1, = 1, giving .
Noting that the above argument gives the same conclusion for ,F as well, we reorganize the above form to reach As for max,F , we follow the same argument up to (E31) to get If min ≥ 2 − , we always have 2 − ≥ 1− 1−2 − and thus max,F (Φ ) = log min 2 − ≤ 1− 1−2 − and thus max,F (Φ ) = log 1− max 1−2 − . If min ≤ 2 − ≤ max , then max,F (Φ ) = 0, which is achieved at concluding the proof.
The generality of Lemma 26 has wide applicability beyond the above measures. As an example, it allows us to provide an exact evaluation for a measure based on the trace distance.
where in the first line we used Lemma 26, and in the third line we used that Tr[Φ ] = 0. The case for min,aff (F) (Φ) = max,F (Φ) can be shown analogously.
It is easy to see that every output of the group twirling operation is a symmetric state, because, for every ∈ , where in the third line we used the left and right invariance of the Haar measure.
The projection turns every state into the form where { } is probability distribution, and ( , ) is a quantum state acting on H ( ) ⊗ H ( ) . Also, the forms of (I3) and (I6) imply that P ( ) † = P ( † ) ∀ , ∀ ∈ , and hence Ξ • P ( ) = P • Ξ( ) for every state , where in the second line, we used the fact that Ξ maps every state to a symmetric state, which has the form (I4). Then, Since Ξ( ) is a symmetric state having a structure of (I4), and ( ) in (I9) acts only on H ( ) nontrivially, we must have ∫

(I11)
We now show that this group twirling operation satisfies the condition for the operation that appears in Lemma 9. By assumption, |Φ is an eigenvector of with eigenvalue for every ∈ . This implies that |Φ is in the invariant subspace of ( ) corresponding to the one-dimensional irreducible representation { } ∈ . We assign the label ★ for this representation. Then, every ( ) has the form

(I13)
The second term is a completely positive map. Also, the convexity of O and the assumption that · † ∈ O imply that Ξ ∈ O ⊆ O max . Thus, we can apply Lemma 9 to conclude the proof.
where P denotes all -qubit Pauli operators. Crucially, since any stabilizer state satisfies st ≤ 1, we can define the set Then min,F STAB ( ) ≥ min,F P ( ) for every state . Take then the Hoggar state, which can be written as and consider any state ∈ F P , which can always be written as (J9) Using the fact that ,F STAB (Hog) ≥ max,F STAB (Hog) ≥ min,F STAB (Hog) ≥ min,F P (Hog) concludes the proof of the first part of the result.
To show the second part of the result, note that the Hoggar state |Hog is the +1 eigenvector of all unitaries in the group generated by two unitaries with order 7 and order 12 [104], defined as 7 ≔ where = 2 /8 . Let us call this group and consider the representation { } ∈ that has the identical form to the element of the unitary group . As one can explicitly check, construct a corresponding element in { ( ) k→k ′′ } k ′′ =1 , which are distinct from each other for different 's. This shows that we must at least have k ′′ ≥ k ′ , but this violates the assumption that k ′ > k ′′ . This concludes the proof that k ′ takes the same value for all k ′ ≠ 0, leading to the second equality in (J14).