Quantum coherence is a physical resource that is essential for various tasks in quantum computing (e.g. Deutsch-Jozsa algorithm [Hil16]), cryptography (e.g. quantum key distribution [CML16]), information processing (e.g. quantum state merging [SCR16], state redistribution [AJS18] and channel simulation [DFW18]), thermodynamics [LJR15] and metrology [FD11]. A series of efforts have been devoted to building a resource framework of coherence in recent years [Abe06, GS08, LM14, BCP14, SAP17], characterizing in particular the state transformations and operational uses of coherence in fundamental resource manipulation protocols [WY16, CSR16, SCR16, CH16, RFWA18, FWL18, ZLY18]. As in any physical resource theory, a central problem of the resource theory of quantum coherence is distillation: the process of extracting canonical units of coherence, named coherent bits, from a given quantum state using a choice of free operations.
The usual asymptotic approach to studying the problem in quantum information theory is to assume that there is an unbounded number of independent and identically distributed (i.i.d.) copies of a quantum state available and that the transformation error asymptotically goes to zero. The task of coherence distillation in the asymptotic scenario has been first investigated in [WY16] and has been recently completed in [Lam19]. In spite of their theoretical importance, the asymptotic assumptions become unphysical in reality due to our limited access to a finite number of copies of a given state, making it necessary to look at non-asymptotic regimes. The first step in this direction is to consider the one-shot setting that distills coherence from a single instance of the prepared state. Such a scenario has been investigated in [RFWA18] and has been mostly completed in [ZLY18] . These works estimate the one-shot distillable coherence under different free operations by their corresponding one-shot entropies. The one-shot entropies most accurately describe the operational quantity, yet they tend to be difficult to calculate for large systems, even in the i.i.d. case. This motivates us to further investigate the second order expansions of distillable coherence.
. These works estimate the one-shot distillable coherence under different free operations by their corresponding one-shot entropies. The one-shot entropies most accurately describe the operational quantity, yet they tend to be difficult to calculate for large systems, even in the i.i.d. case. This motivates us to further investigate the second order expansions of distillable coherence.
More precisely, take the coherence distillation under incoherent operations (IO) as an example. Let be the maximum number of coherent bits that can be distilled from the given state while the transformation error is within . The asymptotic result by Winter & Yang [WY16] together with its strong converse property [ZLY19] can be expressed as:
where is the quantum relative entropy and is the completely dephashing channel in a given basis. The coefficient of in (1), i.e. , is called the first order coefficient of distillable coherence. In a second order expansion, one seeks to further analyze the term , which usually turns out to be of the order . One of our key results here is to refine (1) to
where denotes the quantum information variance and
denotes the quantum information variance andin (2), i.e. , is called the second order coefficient of the distillable coherence. In general, an expansion into terms in , and such as (2) is called a second order expansion.
The significance of second order expansions is multifold. First, second order expansions of distillable coherence provide a useful approximation for finite block length , refining optimal rates that typically correspond to the first order coefficient in asymptotic expansions. Second, they determine the rate of convergence of the averaged distillable coherence to its first order coefficient. This is analogous to the relation between the Central Limit Theorem and the Berry-Esseen Theorem, as the latter determines the rate of convergence in the former. Finally, second order expansions of distillable coherence can be used to derive the strong converse property, an information-theoretic property that rules out a possible tradeoff between the transformation error and the distillable coherence of a protocol.
, refining optimal rates that typically correspond to the first order coefficient in asymptotic expansions. Second, they determine the rate of convergence of the averaged distillable coherence to its first order coefficient. This is analogous to the relation between the Central Limit Theorem and the Berry-Esseen Theorem, as the latter determines the rate of convergence in the former. Finally, second order expansions of distillable coherence can be used to derive the strong converse property, an information-theoretic property that rules out a possible tradeoff between the transformation error and the distillable coherence of a protocol.
The usual approach to deriving the second order expansion of an information task is to combine the one-shot entropy bounds on the information quantity and the second order expansion of the corresponding entropies (e.g. [Hay08, TH13, Li14, DL14, TBR16, FWTD19, WFT19]). However, as second order expansions have a strong dependence on the error parameter , the existing one-shot entropy bounds on distillable coherence [RFWA18, ZLY19] are insufficient to get a tight second order expansion. That is, the second order coefficients in the expansion of the one-shot entropy lower and upper bounds are often mismatched. To solve this, we introduce a variant of randomness extraction framework in the context of quantum coherence theory [YZCM15, HZ18] and build an exact connection of this task with coherence distillation. Such a connection provides us a new perspective to the study of distillation process. Finally, expanding a one-shot entropy lower bound on the extractable randomness and a one-shot entropy upper bound on the distillable coherence, we obtain the desired second order expansion as presented in (2).
The exact one-shot relation between randomness extraction and coherence distillation builds a bridge between two seemly different information tasks, providing new perspectives to the study of both problems. Moreover, our second order expansions initiate the first large block length analysis in quantum coherence theory, filling an important gap in the literature.
Outline and main contributions
The main contributions of this paper can be summarized as follows:
In Section 3, we propose a variant of randomness extraction framework in the context of quantum coherence theory, which is later proved to be closely related to the task of quantum coherence distillation.
In Section 4, we establish an exact relation between the task of randomness extraction and the task of quantum coherence distillation in the one-shot regime. More precisely, we show that the maximum number of secure randomness bits () extractable from a given state is equal to the maximum number of coherent bits () that can be distilled from the same state. That is, for any quantum state , error tolerance , it holds
where free operation class whose definitions can be found in Section 3.1.
In Section 5, we combine the derived one-shot relation between and , their known one-shot entropy bounds as well as the second order expansion of these entropies to get the second order expansion of our information tasks. That is, for any quantum state , error tolerance , free operation class , it holds
In Section 6, we provide an alternative proof of the strong converse property of quantum coherence distillation and incoherent randomness extraction by using the established second order expansions.
The argument of our results uses various technical statements (e.g. properties of smooth entropies), which are presented in Appendix A. Finally, a refined estimation to the one-shot distillable coherence by quantum hypothesis testing relative entropy is given in Appendix B, which provides an alternative proof of our second order expansions.
In this section we define several quantities and set the notation that will be used throughout this paper. We label different physical systems by capital Latin letters (e.g. ). We often use these labels as subscripts to guide the reader by indicating which system a mathematical object belongs to. We drop the subscripts when they are evident in the context of an expression (or if we are not talking about a specific system). The corresponding Hilbert spaces of these physical systems are denoted as respectively. The corresponding alphabet sets are denoted by the same letters in mathcal font (e.g. ). For example, where is the dimension of Hilbert space . Let be the computational basis on Hilbert space . The set of positive semidefinite operators on is denoted as . The set of quantum states, which are positive semidefinite operators with unit trace, on is denoted as . Denote the completely mixed state on as . The identity operator and the identity map are denoted as and respectively. A quantum operation is a completely positive trace-preserving (CPTP) map from to . All logarithms in this work are taken base two.
For any , the purified distance is defined in terms of the generalized quantum fidelity as with [Tom15]. For any and , their quantum hypothesis testing relative entropy is defined as . The smooth max-relative entropy is defined as . The second order expansions of quantum hypothesis testing relative entropy [TH13, Li14] and smooth max-relative entropy [TH13] are, respectively, given by
where is the quantum relative entropy, is the quantum information variance and is the inverse of the cumulative distribution function of a standard normal random variable.
3 Quantum coherence distillation and incoherent randomness extraction
In this section we first review the resource theory of quantum coherence and the operational task of quantum coherence distillation. We then introduce a variant of randomness extraction framework in the context of quantum coherence theory which plays a crucial role in the later discussions.
3.1 Resource theory of quantum coherence
The resource theory of coherence consists of the following ingredients [BCP14]: the set of free states and the set of free operations, that is, a set of quantum operations that do not generate coherence. The free states, so-called incoherent states, are the quantum states which are diagonal in a given reference orthonormal basis . We will use to denote the diagonal map (completely dephasing channel) in this basis. Then the set of incoherent states is denoted as . For convenience, we will also use the cone of diagonal positive semidefinite matrices, which is denoted as . The maximal resource state on is the maximally coherent state with dimension . Denote its density operator as . The resource theory of coherence is known not to admit a unique physically-motivated choice of allowed free operations [WY16, CG16, MS16, DVS16, SAP17]. The relevant choices of free operations that we will focus on are: maximally incoherent operations (MIO) [Abe06], defined to be all operations such that for every ; dephasing-covariant incoherent operations (DIO) [CG16, MS16], which are maps such that ; incoherent operations (IO) [BCP14], which admit a set of incoherent Kraus operators such that for all and ; the intersection of IO and DIO is denoted as [ZLY19]. Another two classes of free operations commonly studied are strictly incoherent operations (SIO) [WY16] and physically incoherent operations (PIO) [CG16]. We do not investigate further details of SIO and PIO, as it has been recently shown that quantum coherence is generically undistillable under these two classes [LRA19, Lam19]. Finally, the inclusion relations between free operation classes can be summarized as , , while IO and DIO are not contained by each other.
3.2 Framework of quantum coherence distillation
The task of coherence distillation aims to transform a given quantum state to a maximally coherent state such that the transformation error is within a given threshold and that the obtained maximally coherent state has dimension as large as possible. More formally, for any free operation class , any given state and error tolerance , the one-shot distillable coherence is defined as
3.3 Framework of incoherent randomness extraction
The task of incoherent randomness extraction aims to obtain as many random bits as possible at Alice’s laboratory that is secure from possible attackers such as Eve. A general incoherent randomness extraction protocol is characterized by a triplet , where is an incoherent operation in a certain class, is a completely dephasing channel and is a hash function. A detailed procedure of randomness extraction by is depicted in Figure 1. For any given quantum state held by Alice, we denote its purification as 111Note that are not necessarily orthogonal to each other.
where is the reference system held by Eve. Alice first performs a free operation on her part of the system. If she uses a quantum operation whose final state is always a specific incoherent state, say the maximally mixed state , the resulting conditional entropy equals , which increases unlimitedly as increases. To avoid such a trivial advantage for Alice, similar to the study of quantum key distribution [TLGR12, SP00, Hay06] and private capacity in quantum Shannon theory [LWZG09], we assume that the environment system of the free operation is also controlled by Eve. This is because it is not easy to exclude the possibility that Eve accesses a system that interacts with Alice’s operation. Hence Eve has control over two systems and in total. To cover such a worst scenario, we consider the Stinespring representation of , where is the initial pure state on the environment and is the isometry from to . 222Note that a free operation does not necessarily admit a free dilation [CG16]. Thus is not necessarily incoherent though is free. After the action of , the total output state is a pure state
Next, Alice performs an incoherent measurement, with respect to the computational basis, on her part of the state. The output state is then denoted as
Finally, a hash function is applied on her part of the system to extract the randomness that is secure from Eve. For any deterministic function , and any classical-quantum (CQ) state , denote . Then the output state in the final step is given by
To quantify the security of randomness in a quantum state with respect to the reference system , we employ the following security measure:
The one-shot extractable randomness under given free operation is defined as
The one-shot extractable randomness under free operation class is then defined as
Note that the identity map is always free in coherence theory. Thus is a valid incoherent randomness extraction protocol for any , which was studied in [TH13, Section III]. That is, Alice directly performs an incoherent measurement on her given state . In this case, the environment system reduces to trivial and system . It has been shown in [TH13, Theorem 8] that for any ,
where is the conditional min-entropy and is the dephased CQ state in the protocol.
Remark 1 On the one hand, the randomness extraction protocol without using incoherent operations (e.g. the one considered in [ZLY19, Figure 1.(b)]) is too restrictive, as such a framework does not make good use of free resources at hand. On the other hand, an extraction protocol that does not consider Eve’s attack on the free operation is too trivial because Alice can generate an arbitrary amount of randomness by using a free replacer channel . Hence, the setup in Figure 1 contributes to a reasonable randomness extraction framework in the context of quantum coherence theory.
Remark 2 The randomness extraction framework proposed here is also closely related to the one in [HZ18]. But we should note the following subtle differences: (i) the independence of the resulting randomness is quantified using the trace distance in [HZ18] instead of the purified distance we use in (12). Though the trace distance can give us the nice property of universal composability (see e.g. [Ren05]), the choice of purified distance in (12) is crucial for obtaining the exact relation between coherence distillation and randomness extraction in the next section, which then becomes a key ingredient to proving the second order expansions; (ii) while the paper [HZ18] has a discussion on the large block length regime, its analysis is focused on the exponential decreasing rate for the amount of the leaked information , but we will put more focus on the rate of extractable randomness in the one-shot and large block length regime with constant .
Remark 3 Note that performing incoherent unitary operations in the first step does not make any difference with the protocol by identity map . This justifies our consideration of general incoherent operations. More precisely, for any incoherent unitary , it holds with . To see this, recall that any incoherent unitary on can be written as with a permutation and phase factors [SAP17, Section II.A.2]. Then a direct calculation gives that , implying the equivalence of extraction protocols and . Thus follows by definition.
4 Relation between coherence distillation and randomness extraction
In this section we present an exact relation between coherence distillation and incoherent randomness extraction, the proof of which showcases a nice one-to-one correspondence between coherence distillation protocols and incoherent randomness extraction protocols.
We first present a technical result that will be used in the proof of the one-shot relation.
Proposition 1 (Main technical result)
For any quantum state , error tolerance and incoherent randomness extraction protocol such that , there exists a quantum operation such that and .
A detailed proof is given at the end of this section.
Theorem 2 (One-shot relation)
For any quantum state , error tolerance and free operation class , the following equation holds
Remark 4 Recall that there is an exact characterization of one-shot distillable coherence under MIO and DIO operations [RFWA18, Proposition 2]. Together with the above (16), we have the following relations
where the minimum is taken over all Hermitian operators on satisfying the conditions and is the least number such that the solution corresponds to the logarithm of an integer.
Remark 5 Note that a one-shot relation between distillable coherence and extractable randomness has appeared in [ZLY19, Equation (80)]. Unlike the precise equation in (16), the relation in [ZLY19] is given in the form of one-shot lower and upper bounds with unmatched error dependence and additional correction terms. However, the clean form in (16) plays a pivotal role in deriving the second order expansions where the error dependence matters.
[Proof of Theorem 2] We first show the direction Denote and suppose that this rate is achieved by a free operation such that . Consider a randomness extraction protocol . Note that is a purification of . By Uhlmann’s theorem [Uhl76] there exists an extension of , denoted as , such that . Then we have
where the second equality follows by , the second inequality follows by the data-processing inequality of purified distance, the third equality follows from the assumption of . Thus we know that is an achievable randomness extraction rate, which implies .
For the other direction, we denote and suppose that this rate is achieved by an extraction protocol with . Notice that applying the protocol on quantum state is the same as applying a protocol on with purification and reference system . By Proposition 1 there exists a quantum operation such that and . Since and , we have and this operation achieves the distillation rate . This implies and completes the proof.
[Proof of Proposition 1] Let be a quantum state that attains the minimum in
Let on be a purification of . Thus we have
Define the incoherent isometry from to as
We choose normalized vectors
We choose normalized vectorsand normalization factors such that
By Uhlmann’s theorem [Uhl76] there exists a unitary on such that
Take . We have
It remains to check . Note that admits a Kraus decomposition with operators . For any computational basis and any , we have
by direct calculation. Thus . For any computational basis and , we can first check that
Thus it holds
5 Second order analysis
In this section we discuss the second order expansions of distillable coherence and extractable randomness. The technical statement is presented as follows.
Theorem 3 (Second order expansion)
For any quantum state , error tolerance and free operation class , the following second order expansion holds
where denotes the inverse of the cumulative distribution function of a standard normal random variable.
Remark 6 Comparing the second order expansion of in [TH13, Corollary 16] and the result above, we can conclude that a general incoherent randomness extraction protocol has no advantage over the protocol in the sense that they lead to the same first order asymptotics [HZ18] and the second order asymptotics of extractable randomness.
Remark 7 The distillable coherence under MIO/DIO/IO/DIIO not only have the same first order asymptotics as observed in [WY16, RFWA18, ZLY19] but also have the same second order asymptotics, indicating that they are equivalently powerful for coherence distillation in the large block length regime.
Remark 8 For any quantum state written in the computational basis, we can assign it to a bipartite maximally correlated state . The second order expansion of distillable entanglement of under local operations and classical communication (LOCC) is also given by [FWTD19, Proposition 10]. Together with the result in Theorem 3, the coincidence of these second order expansions leads to a new evidence to the long-standing conjecture (see e.g. [SAP17, Section II.D]) that any incoherent operation acting on a state is equivalent to a LOCC operation acting on the associated maximally correlated state .
[Proof of Theorem 3] The converse part is a direct consequence of a known result in [RFWA18, Proposition 2]. That is, for any quantum state it holds , where is the least number such that the solution on the r.h.s. corresponds to the logarithm of an integer. Choosing a feasible solution as , we have . Given the i.i.d. state , we have
where and the first inequality follows by definition. By definition we also have . Hence, given the i.i.d. state , we have
Here we used a fact that in the expansion, which follows as is continuously differentiable around . Finally the proof is completed by noticing that and . These two equations are calculated in Lemma 5.
6 Strong converse property
The direct part of quantum coherence distillation states that for any rate below the optimal rate, there is a corresponding distillation protocol that accomplishes the task successfully. More precisely, if we denote the transformation error in the protocol for uses of the underlying resource by , then for any rate below the optimal rate there exists a protocol, whose transformation error vanishes in the asymptotic limit . Such rates are called achievable, and the optimal rate is defined as the supremum over all achievable rates. In contrast, the converse part states that for any distillation protocol with a rate above the optimal rate, the error does not vanish asymptotically, that is, it is bounded away from 0 in the asymptotic limit . This is usually called weak converse. In principle, it leaves open the possibility of a trade-off between error and rate of a protocol. However, the strong converse property rules out such a possibility, stating that for any distillation protocol with a rate above the optimal rate, the corresponding transformation error incurred in the protocol converges to one. In other words, such protocols become worse with increasing block length , and eventually fail with certainty in the asymptotic limit.
The strong converse property of coherence distillation has been pointed out by [ZLY19, Theorem 16]. In this part we give an alternative proof and showcase a standard argument how a second order result automatically implies the strong converse property. For simplicity, we denote and . For any achievable rate , we have . By Theorem 3, we have