Quantifying resource in catalytic resource theory

# Quantifying resource in catalytic resource theory

Anurag Anshu111Center for Quantum Technologies, National University of Singapore, Singapore. a0109169@u.nus.edu   Min-Hsiu Hsieh222Centre for Quantum Software and Information, University of Technology Sydney, NSW 2007, Australia. Min-Hsiu.Hsieh@uts.edu.au   Rahul Jain333Center for Quantum Technologies, National University of Singapore and MajuLab, UMI 3654, Singapore. rahul@comp.nus.edu.sg
###### Abstract

We consider a general resource theory that allows the use of free resource as a catalyst. We show that the amount of ‘resource’ contained in a given state, in the asymptotic scenario, is equal to the regularized relative entropy of resource of that state, which then yields a straightforward operational meaning to this quantity. Such an answer has been long sought for in any resource theory since the usefulness of a state in information-processing tasks is directly related to the amount of resource the state possesses in the beginning. While we need to place a few assumptions in our resource theoretical framework, it is still general enough and includes quantum resource theory of entanglement, coherence, asymmetry, non-uniformity, purity, contextuality, stabilizer computation and the classical resource theory of randomness extraction as special cases. Since our resource theoretic framework includes entanglement theory, our result also implies that the amount of noise one has to inject locally in order to erase all entanglement contained in an entangled state is equal to the regularized relative entropy of entanglement, resolving an open question posted in [Groisman et al., Phys. Rev. A. 72: 032317, 2005]. On the way to prove the main result, we also quantify the amount of resource contained in a state in the one-shot setting (where one only has a single copy of the state), in terms of the smooth max-relative entropy. Our one-shot result employs a recently developed technique of convex-split lemma.

## 1 Introduction

A question that is commonly asked in Physics is how a certain property of a physical system can be used to achieve useful tasks, and how to quantify the amount of such a property in a meaningful way. Various areas in Physics have developed methodology to answer these questions relevant to their own areas; however, many of the approaches are difficult, sometimes even impossible, to employ outside of their respective fields. Quantum resource theory then emerges when quantum information theory is found to provide a unified platform for characterizing the resource [1, 2] because in a nutshell, they can all be viewed as interconversion of different system states with system-dependent constraints. Since then, individual resource theory has been able to characterize targeted properties in some information-processing tasks using entropic quantities defined on system states.

The core of a resource theory is built upon two main system-dependent requirements for the resources and allowed operations; namely, (i) the existence of a set of states that are free and those not in the set are expensive; and (ii) the allowed operations are those that map the set of free states to itself. Various resource theories have been developed in the past decade. The most notable example is the resource theory of entanglement [3], where the set of free states corresponds to the collection of separable states and the allowed free operations are the local quantum operations and classical communication (LOCC). These two classes of states and operations have attracted stand-alone interests besides resource theory [4, 5]. Under this resource framework, one can then ask the amount of valuable resource (cf. quantum entanglement) possessed by an entangled state relative to the set of free states (cf. separable states). This question motivates the scenario of injecting noise locally to the system in order to destroy the quantum entanglement, i.e., the randomness cost. A complete characterization of this question has remained open; though, gapped upper and lower bounds have been provided in the asymptotic i.i.d. setting in Ref. [6]. It is worthwhile to mention that investigation of a variant of the above setting, where the local noise is used to destroy the total correlation in an entangled state, relates the minimal randomness cost to the quantum mutual information [6]. This seminal result gives the first operational meaning to this entropic quantity and advances significantly our understanding of entanglement theory. Being able to answer the optimal randomness cost to bring entangled states to separable states thus bears equivalent significance, if not more important, since the existence of entanglement is believed to make quantum systems superior to their classical counterparts and the amount of entanglement is generally linked to its information-processing power [7]. Likewise, this crucial question of the amount of valuable resource possessed by a state relative to its free resource is then adhesive to every resource theory, be it quantum coherence, thermodynamics, etc.

One can view the above erasing framework as a restricted model of state transformation, where the final state belongs to the free resource [8, 9]. In the general model of state transformation, it has been demonstrated that not every state transformation is possible, and could become possible if a catalyst is involved [10]. Even if the transformation process is possible in the beginning, adding catalyst would likely make the process much more efficient. Therefore, individual resource theories have included catalyst in their formalism [11, 12].

We thus consider a general resource theory framework that allows free resource being used as a catalyst. We demonstrate that this framework includes previous major resource theories of (i) entanglement [3, 6, 13, 14], (ii) coherence [15, 16, 14, 17], (iii) thermodynamics [18, 19, 20, 21, 22, 23], (iv) non-uniformity [11], (v) purity [24], (vi) randomness extractors [25, 26], (vii) contextuality [27], (viii) asymmetry [28] and (ix) stabilizer computation [29] as special instances. Treating resource theory in a general framework enjoys the major advantage that the resource conversion tasks done for one particular resource can already be adapted to other resources almost trivially. Hence, it has started to attract more attention [8, 9]. We establish that the entropic quantity, the regularized relative entropy of resource [8], characterizes the amount of useful resource in a given state relative to its free resource in the asymptotic i.i.d. setting. In other words, if one has infinitely many copies of the same state, the randomness cost (per copy of the state) for erasing the resource contained in the given state and bringing it to the closest free state is equal to this entropic quantity. This yields a crucial operational meaning to this entropic quantity. Since the resource theory of quantum entanglement is a special case of our framework, our result directly provides the missing answer444We remark that an independent work for this answer can be found in Ref. [30] to the minimal randomness cost for erasing the amount of entanglement in an entangled state to a separable state, a question first posted in Ref. [6].

We also obtain matching upper and lower bounds on the randomness cost if one only has a single copy of a given state, i.e., the one-shot scenario. Our one-shot bounds are given in terms of smooth max-relative entropy; hence our result also provides a new operational meaning to this quantity in the resource theoretic framework. We emphasize that being able to obtain matching one-shot bounds in such a general resource theory framework is rare [8], and is due to the technical tool: the convex split lemma [31]. This again provides another excellent example that quantum information theory helps in the understanding of quantum physics.

The paper is organized as follows. Section 2 contains assumptions required for our catalytic resource framework. We then define our information-processing task of quantifying the amount of resource and present our main theorem. We also discuss several individual resource theories as special cases of our framework. Section 3 contains definition of entropic quantities and useful lemmas required in the proofs. The proofs of our main result are given in Sections 46. We conclude in Section 7.

## 2 Catalytic Resource Framework and Main Results

The question that we will answer in this general catalytic resource theory framework is how much resource is contained in a given resource state (relative to its free resource). This question is crucial, as this amount directly relates to the power the state possesses in achieving information-processing tasks.

We first present assumptions that we have to make in our resource theoretic framework. It is apparent that if no limit is set on the allowed operations and free resources, then it is almost impossible to obtain useful characterization, as also noted in Ref. [8]. Thus, we need a sufficient (yet small) number of assumptions to keep the theory interesting.

Our framework for both single-partite and multi-partite settings is discussed below, with the assumptions on free resources and allowed operations. Our assumptions capture the requirement we impose on an experimenter (or experimenters in the multi-partite setting) who wants to quantify a useful property of his/her physical system. For this, we start with the natural experimental framework, where the experimenter has decided upon the global Hilbert space and its tensor decomposition into several Hilbert spaces (for example, as ‘physically separated’ Hilbert spaces). Given this framework, we impose the conditions that are to be satisfied by the set of free resources and the allowed unitary operations. It is important to note that the conditions are only for a given framework; we do not restrict the choice of the global Hilbert space and its tensor decomposition by the experimenter.

For the notations and definitions used in this section, please refer to Section 3.

### 2.1 Single-partite case

As discussed above, an experimenter starts with a collection of quantum registers and performs operations on them. The joint state lives on a global Hilbert space . This Hilbert space possesses an ordered decomposition into systems . Given the Hilbert space and its ordered decomposition (the order of the decomposition may matter in a given physical scenario, hence we take it into account), we define the set of free states and allowed operations. The definition is analogous to the postulates given in Ref. [8].

###### Definition 1 (Single party resource theory).

Fix an integer . Given and registers where corresponds to , the set of free resources and free operations are as follows.

1. For every non-empty ordered set (where are increasing sequence of integers), let and satisfy the following properties. For brevity, we set

1. is a convex and closed set.

2. For any , .

• This assumption states that if two quantum states are free resources, then their tensor product is a free resource as well.

3. For any and , .

• This assumption states that if a quantum state on more than one registers is a free resource, then we obtain a free resource by partial trace over a subset of these registers.

4. It holds that , where is the dimension of register .

• Under this assumption, we require the maximally mixed quantum state to be a free resource on the last register.

2. The complete set of free resources is .

3. For a fixed L, the free operations are defined as the set of all unitaries that satisfy

1. .

2. if and only if .

4. The set of all free operations are .

We remark that these assumptions we required are natural and mild [8]. Above, we also observe that the set (for every L) forms a group.

Now we are in a position to formally define our task, that we call an -transformation of to .

Let , be an integer and fix a Hilbert space with decomposition , which is chosen by the experimenter. Let the register correspond to , register correspond to and register correspond to . An experimenter holds a quantum state . Using a state , she applies a unitary to obtain a joint quantum state :

 ΘMEJ=U(ρM⊗μEJ)U†.

It is required that there exists a such that , where the chosen distance measure is defined in Equation (1). The number of discarded qubits is .

In above task, no restriction is made on the transformation of the free resource state . But in many cases, it is desirable that this quantum state be returned close to its original form and hence act as a catalyst. For example, the framework of catalytic decoupling [32] studies such a scenario. Our achievability result shall belong to such class of transformations. Hence we have the following definition.

###### Definition 3.

Task 2 is said to be a -catalytic transformation of if and for some .

Task 2 is motivated by the work [6], which considered the problem of transforming a bipartite quantum state into a product state with the aid of shared randomness and local unitaries. It was shown that the number of bits of randomness required (in other words, the randomness cost) is . Task 2 is along the lines of the framework considered in above work, but with an additional freedom of allowing the use of additional free resources that can aid in the transformation of the desired quantum state. Moreover, when the register is classical (in a more precise sense to be discussed below), we can interpret to be the randomness cost of the protocol. In Definition 3, we have provided further restriction that the free resource (expect for the randomness used) be returned with small error.

We provide a near optimal characterization of the randomness cost of Task 2 in Theorem 5 below. Our achievability protocol requires a further assumption related to the last register (and its connection to rest of the registers), as this register serves as the source of ‘classical randomness’. This assumption is as follows.

###### Assumption 4.

Invoke the notation in Definition 1. Then there exists a canonical basis on (where is the dimension of ) such that

 F=conv{1ℓr∑jUjσU†j⊗|j⟩⟨j|Mr:σ∈F{1,2,…r−1},Uj∈U{1,2,…r−1}},

and the experimeter only performs an operation from the set

 {ℓr∑j=1Uj⊗|j⟩⟨j|Mr:Uj∈U{1,2,…r−1}}.

Moreover, suppose it holds that for all . Then implies for all .

Under this assumption, all the states in Task 2 are classical-quantum states, with register being classical and being diagonal in the canonical basis. Thus, the assumption allows us to interpret as the randomness cost of the protocol. We have the following theorem.

###### Theorem 5.

Fix , and a quantum state .

• Achievability: Suppose Assumption 4 holds. Then there exists an -catalytic transformation of to , where .

• Converse: For every - transformation of to , it holds that

 log|J|≥minσM∈FDεmax(ρM∥σM),

if Assumption 4 is true. Otherwise,

 log|J|≥12minσM∈FDεmax(ρM∥σM).

The proof of this theorem is given in Section 4. Now we consider the asymptotic i.i.d. properties of our results.

###### Definition 6.

Suppose Assumption 4 holds. We say that the asymptotic randomness rate of catalytic transformation of is , if for every , there exists an such that for all , there exists a -catalytic transformation of to . Define the following relative entropy of resource:

 E(ρ)=infσM∈FD(ρM∥σM)

and the regularized relative entropy of resource:

 E∞(ρ)=limn→∞1nE(ρ⊗nM).

Let denote the set of free resources for the register . Define the constant

 C(F):=limn→∞1ninfτ∈Fn∥logτ∥∞.

Using Theorem 5, we obtain the following.

###### Theorem 7.

Suppose the resource theory is such that . Then for a quantum state , the asymptotic randomness rate of catalytic transformation of is given by .

It completely characterizes the per copy randomness requirement in the asymptotic i.i.d. setting. The proof is given in Section 6. Moreover, we show that the number of qubits of catalytic free resource grows only polynomially in (Theorem 28).

### 2.2 Multi-partite case

Our formalism also extends to the multi-partite case. We consider the case of parties and introduce some minor modifications to Definition 1 and Assumption 4. Let be the global Hilbert space in the multiparty setting. We start with a decomposition of , where is a Hilbert space shared by all the parties.

###### Definition 8 (Multi-partite resource theory).

Fix an integer . Given

 H=H1⊗H2…⊗Hr

and registers , , , where corresponds to and is held by -th party, the set of free resources and free operations are as follows.

1. For every non empty ordered set

 L={s1,s2,…s|L|}⊆{1,2,…r}

let satisfy the following properties. For brevity, we let be the register corresponding to set L.

1. is a convex and closed set.

2. For any , .

3. For any and , .

2. The complete set of free resources is .

3. For a fixed L, the free operations are defined as the set of all unitaries that satisfy

1. .

2. if and only if .

4. The set of all free operations are .

In an analogous fashion to Task 2, we define the multi-partite task.

###### Task 9 (An (ε,log|J|,t)-transformation of ρM1M2…Mt to F).

Let , be an integer and fix a Hilbert space with decomposition , which is chosen by the experimenters. Let the register correspond to , register correspond to (with register held by -th party) and register correspond to . The experimenters share a quantum state . Using a state , they apply the unitary to obtain a joint quantum state :

 ΘM1M2…MtEJ1J2…Jt=U(ρM1M2…Mt⊗μEJ1J2…Jt)U†.

It is required that there exists a such that , where the chosen distant measure is defined in Equation (1).

We can similarly define a catalytic transformation in this framework.

###### Definition 10.

Task 9 is said to be a -catalytic transformation of if and for some .

We will also need the following assumption for achievability protocol.

###### Assumption 11.

Invoke the notation from Definition 8. There exists a canonical basis on (where the dimension of is independent of ) such that

 F=conv{1ℓrℓr∑k=1UkσU†k⊗|k⟩⟨k|M1r⊗|k⟩⟨k|M2r⊗…|k⟩⟨k|Mtr:Uk∈U{1,2,…r−1}}

and experimenter only performs an operation from the set

 {∑kUk⊗|k⟩⟨k|M1r⊗|k⟩⟨k|M2r⊗…|k⟩⟨k|Mtr:Uk∈U{1,2,…r−1}}.

For brevity, define

 Iℓr,t:=1ℓrℓr∑k=1|k⟩⟨k|M1r⊗|k⟩⟨k|M2r⊗…|k⟩⟨k|Mtr.

Moreover, suppose it holds that for all . Then

 σM11M21…Mt1,M12M22…Mt2,…M1rM2r…Mtr∈F

implies

 σM1jM2j…Mtj,M12M22…Mt2,…M1j−1M2j−1…Mtj−1,M11M21…Mt1,M1j+1M2j+1…Mtj+1,…M1rM2r…Mtr∈F

for all .

We are now in a position to prove the following theorem.

###### Theorem 12.

Fix , and a quantum state .

• Achievability: Suppose Assumption 11 holds. There exists an -catalytic transformation of to , where .

• Converse: For every -transformation of to , it holds that

 log|J|≥minσM1M2…Mt∈FDεmax(ρM1M2…Mt∥σM1M2…Mt),

if Assumption 11 is true.

The proof of this theorem follows similar to the proof of Theorem 5 and is given in Corollaries 26 and 27 in Section 5.

### 2.3 Known resource theories

To conclude this section, we show that our general resource framework includes at least the following individual resource theory.

1. Resource theory of entanglement [3, 6, 13, 14]: In this resource theory, the set of free states, , is the collection of separable states. The perfectly correlated state is a separable quantum state for all , and hence is a free resource. The formalism also extends to multi-partite entanglement, where the free resources are convex combination of product quantum states.

2. Resource theory of coherence [15, 16, 14, 17]: In this resource theory, the set of free states, , is the collection of diagonal states in a pre-determined basis. This is captured by Definition 1, where we choose a basis on each Hilbert space and take the free resources to be the set of diagonal quantum states. The properties such as being closed under partial trace and tensor product are easily seen to be satisfied. Moreover, maximally mixed state is diagonal in any basis, and hence belongs to the free resources.

3. Resource theory of asymmetry [28]: In this resource theory, the set of free resources are the states that are invariant under some group transformation. To construct for some , one fixes a group with a unitary representation in dimension . The set is the collection of states that satisfy . The resource theory can then be constructed on tensor product of such Hilbert spaces (see Section in [28]). To verify the partial trace condition (Item ), let be such that . Then

 σM1 = = TrM2(Ug⊗V†g′Vg′)σM1M2(U†g⊗I))=TrM2(Ug⊗I)σM1M2(U†g⊗I)) = UgσM1U†g.

Symmetry under permutation of registers of identical dimension can also be verified in similar way, as must be invariant under for all .

4. Resource theory of nonuniformality [11] and purity [24]: In this resource theory, the only free state is the completely mixed state. This is easily captured by Definition 1. This is equivalent to the formalism of randomness extractors [25, 26].

5. Resource theory of Thermodynamics [18, 19, 20, 21, 22, 23]: In this resource theory, the free states are Gibbs quantum states, that are states of the form , for an arbitrary Hamiltonian . To capture it in Definition 1, we assign a Gibbs state as the only free resources in the set (for all ). Thus, for a system with registers, the free resource is . The condition that is equivalent to the condition . Moreover, this also implies . Finally, the maximally mixed state is a Gibbs quantum state with Hamiltonian . This establishes consistency with Definition 1.

Theorem 7 applies as long as . For a Gibbs state , it holds that

 ∥logρβ(H)∥∞≤∥βH∥∞+logTr(e−βH)≤∥βH∥∞+logd,

where is the support size of the Gibbs state. Thus, Theorem 7 applies for bounded .

6. Resource theory of contextuality [27]: In this resource theory, the free resources are the set of conditional probability distributions that can be described ‘classically’ (a notion that is made more precise in [27]). All the conditional probability distributions in this theory satisfy a consistency condition (which can be viewed as an analogue of the non-signaling condition in the case of non-locality). It can be observed that the set of free resources satisfies the conditions in Definition 1, being convex and closed under permutations and tensor product. Uniform probability distribution (or the maximally mixed state) also belongs to the free resource.

7. Resource theory of stabilizer computation [29]: In this resource theory, the free resources are the convex hull of the set of all pure states that can be generated by the action of a Clifford unitary on a standard state (for example, ). Thus, maximally mixed state belongs to this set. This set is convex and closed. The free resources can be extended to the tensor product of registers along the lines as discussed in [28].

## 3 Preliminaries

Let be the Hilbert space associated to a register . Let denote the dimension of . Let be the set of all normalized quantum states acting on . For two subsets , let . Let be a unitary operator acting on .

The definitions below have been adapted from the references [33, 34, 35, 36, 37, 38, 39] For such that , its relative entropy is given by

 D(ρA∥σA):=Tr(ρAlogρA)−Tr(ρAlogσA),

and is equal to infinity otherwise. Similarly, the relative entropy variance of is defined as

 V(ρ∥σ):=Tr(ρ(logρ−logσ)2)−(D(ρ∥σ))2,

when .

The max-relative entropy of such that is

 Dmax(ρA∥σA):=inf{λ∈R:2λσA≥ρA}.

The smooth max-relative entropy is

 Dεmax(ρA∥σA):=supρ′A∈Bε(ρA)Dmax(ρ′A∥σA),

where the -ball, , is defined via the purified distance

 Pur(ρA,σA):=√1−F2(ρA,σA). (1)

In the above, is the fidelity between two states.

We will use the following facts.

###### Fact 13 (Triangle inequality for purified distance,  [39, 34]).

For quantum states ,

 Pur(ρA,σA)≤Pur(ρA,τA)+Pur(τA,σA).
###### Fact 14 (Uhlmann’s Theorem, [40]).

Let . Let be a purification of . There exists a purification of such that,

 F(|θ⟩⟨θ|AB,|ρ⟩⟨ρ|AB)=F(ρA,σA).
###### Fact 15.

Let and be an extension of . Then there exists an extension of such that

 F(ρAB,σAB)=F(ρA,σA).
###### Proof.

Introduce a register and consider a purification of . There exists a purification of such that , due to Uhlmann’s Theorem (Fact 14). Thus, is the desired extension due to the relation

 F(ρA,σA)≥F(ρAB,σAB)≥F(|ρ⟩⟨ρ|ABC,|σ⟩⟨σ|ABC)=F(ρA,σA).

###### Fact 16.

Let and be a classical-quantum extension of , that is , where . Then there exists a classical-quantum extension of such that

 F(ρAB,σAB)=F(ρA,σA),

and .

###### Proof.

From Fact 15, there exists an extension of such that . Now, consider the map defined as . Observe that . Define . It holds that is a classical-quantum extension of . Moreover,

 F(ρA,σA)≥F(ρAB,σ′′AB)≥F(ρAB,σ′AB)=F(ρA,σA).

Let be the projector onto the support of . Define the quantum state . Observe that is also a classical-quantum extension of and . Then

 F(ρAB,σAB) = Tr(√√ρABσAB√ρAB) = 1√Tr(σ′′BΠB)Tr(√√ρABΠBσ′′ABΠB√ρAB) = 1√Tr(σ′′BΠB)Tr(√√ρABσ′′AB√ρAB) = 1√Tr(σ′′BΠB)F(ρAB,σ′′AB) ≥ F(ρAB,σ′′AB).

This completes the proof. ∎

###### Fact 17.

Let be a bipartite quantum state. Let be the projector onto the support of . Then

 ΘAB⪯|B|ΘA⊗ΠB.
###### Fact 18.

Let be a classical quantum state with as the quantum part. Let be the projector onto the support of . Then

 ΘAB⪯ΘA⊗ΠB.
###### Fact 19 ([35, 36]).

Let and be an integer. Let be quantum states. Define . It holds that

###### Fact 20.

For the function and , it holds that .

###### Proof.

We have

 Φ(−x)=∫−x−∞e−t2/2√2πdt=∫∞0e−(−x−t)2/2√2πdt≤e−x2/2∫∞0e−(−t)2/2√2πdt=12e−x2/2.

Thus, , which completes the proof. ∎

###### Fact 21.

Let be a quantum state and be a collection of quantum maps. Define a series of quantum states recursively as . It holds that

 Pur(ρi,ρ1)≤(i−1)maxi⋅{Pur(Ei(ρ1),ρ1)}.
###### Proof.

Consider

 Pur(ρi,ρ1)=Pur(Ei(ρi−1),ρ1)≤Pur(Ei(ρi−1),Ei(ρ1))+Pur(Ei(ρ1),ρ1)≤Pur(ρi−1,ρ1)+Pur(Ei(ρ1),ρ1).

This completes the proof. ∎

###### Lemma 22 (Convex-split lemma [31]).

Let , and be quantum states, be an integer and . Consider the following quantum state

 τM1M2…Mn=1nn∑j=1ρMj⊗σM1⊗σM2…⊗σMj−1⊗σMj+1…⊗σMn

where and . Then,

 Pur(τM1M2…Mn,σM1⊗σM2…⊗σMn)≤ε+√2kn.

## 4 Proof of Theorem 5

### 4.1 Achievability

###### Theorem 23 (Achievability).

Fix and . Suppose Assumption 4 holds. There exists an -catalytic transformation of to , where .

###### Proof.

Let and let be the quantum state achieving the minimum in the definition of . Let be a random variable taking values uniformly in . Introduce registers . Let be the quantum state in which an experimenter uses as a catalyst.

The protocol is as follows. Experimenter introduces the quantum state , which belongs to due to Definition 1 (Item ). Controlled on the value in , the experimenter swaps register with . This operation belongs to from Assumption 4. Let the resulting global quantum state be . We have and

 τM1M2…Mn=1nn∑j=1ρMj⊗σM1⊗σM2…⊗σMj−1⊗σMj+1…⊗σMn.

Using convex-split lemma (Lemma 22), this allows us to conclude that

 Pur(τM1M2…Mn,σ⊗nM)≤ε+δ.

This completes the proof. ∎

Remark 1: The number of qubits used as a catalyst in above task is .

### 4.2 Converse Proof

In this section, we prove a converse bound for resource transformation, showing near optimality of Theorem 23. We will separately consider the cases where Assumption 4 holds and where it does not.

###### Theorem 24.

Fix an . For every - transformation of to , it holds that