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Resource Theory of Coherence — Beyond States Khaled Ben Dana,1, ∗ Marı́a Garcı́a Dı́az,2, † Mohamed Mejatty,1, ‡ and Andreas Winter2, 3, § arXiv:1704.03710v3 [quant-ph] 25 May 2017 1 Laboratory of Advanced Materials and Quantum Phenomena, Department of Physics, Faculty of Sciences of Tunis, University Tunis El Manar, Tunis 2092, Tunisia 2 Departament de Fı́sica: Grup d’Informació Quàntica, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain 3 ICREA—Institució Catalana de Recerca i Estudis Avançats, Pg. Lluı́s Companys, 23, ES-08001 Barcelona, Spain (Dated: 22 May 2017) We generalize the recently proposed resource theory of coherence (or superposition) [Baumgratz, Cramer & Plenio, Phys. Rev. Lett. 113:140401; Winter & Yang, Phys. Rev. Lett. 116:120404] to the setting where not only the free (“incoherent”) resources, but also the objects manipulated, are quantum operations rather than states. In particular, we discuss an information theoretic notion of coherence capacity of a quantum channel, and prove a single-letter formula for it in the case of unitaries. Then we move to the coherence cost of simulating a channel, and prove achievability results for unitaries and general channels acting on a d-dimensional system; we show that a maximally coherent state of rank d is always sufficient as a resource if incoherent operations are allowed, and rank d2 for “strictly incoherent” operations. We also show lower bounds on the simulation cost of channels that allow us to conclude that there exists bound coherence in operations, i.e. maps with non-zero cost of implementing them but zero coherence capacity; this is in contrast to states, which do not exhibit bound coherence. I. INTRODUCTION Since its discovery to our days, quantum mechanics has provided a mathematical framework for the construction of physical theories. Indeterminism, interference, uncertainty, superposition and entanglement are concepts of quantum mechanics that distinguish it from classical physics, and which have become resources in quantum information processing. Quantum resource theories aim at capturing the essence of these traits and quantifying them. Recently, quantum resource theories have been formulated in different areas of physics such as the resource theory of athermality in thermodynamics [1–6] and the resource theory of asymmetry [7, 8]. Furthermore, general structural frameworks of quantum resource theories have been proposed [9]. Resource theories using concepts of quantum mechanics have been developed since the appearance of the theory of entanglement [10–12]. Very recently, Baumgratz et al. [13], following earlier work by Åberg [14], have made quantum coherence itself, i.e. the concept of superposition |ψi = X ψi |ii, i the subject of a resource theory; see also [15]. ∗ † ‡ § [email protected] [email protected] [email protected] [email protected] The present paper is concerned with this resource theory of coherence, and here we briefly recall its fundamental definitions, as well as some important coherence measures; for a comprehensive review, see [16]. Let {|ii : i = 0, . . . , d − 1} be a particular fixed basis of the d-dimensional Hilbert space H; then all density matrices inPthis basis are “incoherent”, i.e. those of the form d−1 δ̃ = i=0 δi |iihi|. We denote by ∆ ⊂ S(H) the set of such incoherent quantum states. The definition of coherence monotones requires the identification of operations that are incoherent. These map the set of incoherent states to itself. More precisely, such a completely positive and trace preserving (cptp) map is specified by a set of Kraus operators {Kα } satP isfying α Kα† Kα = 11 and Kα ∆Kα† ⊂ ∆ for all α. A Kraus operator with this property is called incoherent; we call it strictly incoherent if both K and K † are incoherent [17, 18]. We distinguish two classes of incoherent operations (IO): (i) Incoherent completely positive and trace preserving quantum P operations (non-selective maps) T , which act as T (ρ) = α Kα ρKα† (note that this formulation implies the loss of information about the measurement outcome); (ii) quantum operations for which measurement outcomes are retained, given by ρα = p1α Kα ρKα† occurring with probability pα = Tr Kα ρKα† . The latter can be modelled as a nonselective operation by explicitly introducing a new register to hold the (incoherent) measurement result: X Te(ρ) = Kα ρKα† ⊗ |αihα|. α Here, we have made use of the convention that when composing systems, the incoherent states in the tensor 2 product space are precisely the tensor products of incoherent states and their probabilistic mixtures (convex combinations) [14]. An operation is called strictly incoherent (SIO), if it can be written with strictly incoherent Kraus operators. We define also the maximally coherent state on a dPd−1 dimensional system by |Ψd i = √1d i=0 |ii, from which every state in dimension d can be prepared [13, 17, 19]. Note that the definition of maximally coherent state is independent of a specific measure for the coherence [13, 20]. A list of desirable conditions for any coherence measure, i.e. a functional from states to non-negative real numbers, was also presented [13]: 1. C(ρ) = 0 for all ρ ∈ ∆; 2. Monotonicity under non-selective incoherent maps: C(ρ) ≥ C(T (ρ)); 3. (Strong) monotonicity under selective incoherent P maps: C(ρ) ≥ α pα C(ρα ); P P 4. Convexity: i pi C(ρi ) ≥ C( i pi ρi ). The first two are definitely required to speak of a coherence measure, the third is sometimes demanded axiomatically, but often is really more of a convenience, and convexity should be thought of as nice if present but not absolutely necessary. Among the most important examples are the following three measures: For pure states ϕ = |ϕihϕ|, the entropy of coherence is defined as C(ϕ) := S ∆(ϕ) , (1) where ∆ is thePdephasing (i.e. coherence-destroying) map ∆(ρ) = i |iihi|ρ|iihi|; for mixed states, it is extended by the convex hull construction to the coherence of formation [13, 14] X X Cf (ρ) := min pi C(ψi ) s.t. ρ = p i ψi . (2) i i Finally, the relative entropy of coherence [13, 14] Cr (ρ) := min D(ρkσ) = S ∆(ρ) − S(ρ), σ∈∆ (3) with the relative entropy D(ρkσ) = Tr ρ(log ρ − log σ). Both Cr and Cf satisfy all properties 1 through 4 above. In the present paper, we expand our view from states as coherent resources to operations, showing how to extract pure state coherence from a given operation (section II), and how to implement operations using coherent states as a resource (section III). We briefly discuss the case of qubit unitaries as an example (section IV), and conclude in section V, where we observe that in operations, there is bound coherence, something that doesn’t exist for states. Most generally, we propose a definition for the rate of conversion between two channels using only incoherent operations, and present many open questions about this concept. II. COHERENCE GENERATING CAPACITY AND COHERENCE POWER OF TRANSFORMATIONS The free operations in our resource theory are the incoherent ones (IO), which means that in some sense, any other CPTP map represents a resource. How to measure it? Or better: How to assess its resource character? In the present section, we are focusing on how much pure state coherence can be created asymptotically, using a given operation T : A −→ B a large number of times, when incoherent operations are for free. The most general protocol to generate coherence must use the resource T and incoherent operations according to some predetermined algorithm, in some order. We may assume that the channels T are invoked one at a time; and we can integrate all incoherent operations in between one use of T and the next into one incoherent operation, since IO is closed under composition. Thus, a mathematical description of the most general protocol is the following: One starts by preparing an incoherent state ρ0 on A ⊗ A0 , then lets act T , followed by an incoherent transformation I1 : B ⊗ A0 −→ A ⊗ A1 , resulting in the state ρ1 = I1 (T ⊗ id)ρ0 . Iterating, given the state ρt on A ⊗ At obtained after the action of t realizations of T and suitable incoherent operations, we let T act and the incoherent transformation It+1 : B ⊗ At −→ A ⊗ At+1 , resulting in the state ρt+1 = It+1 (T ⊗ id)ρt . At the end of n iterations, we have a state ρn on A ⊗ An , and we call the above procedure a coherence generation protocol of rate R and error , if |An | = 2nR and the ren duced state ρA n = TrA ρn has high fidelity with the maximally coherent state, n hΨ2nR |ρA n |Ψ2nR i ≥ 1 − . The maximum number R such that there exist coherence generating protocols for all n, with error going to zero and rates converging to R, is called the coherence generating capacity of T , and denoted Cgen (T ). Theorem 1 For a general CPTP map T : A −→ B, Cgen (T ) ≥ sup Cr (T ⊗ id)ϕ − C(ϕ), (4) |ϕi∈A⊗C where the supremum over all auxiliary systems C and pure states |ϕi ∈ A ⊗ C. Furthermore, Cgen (T ) ≤ sup Cr (T ⊗ id)ρ − Cr (ρ), (5) ρ on A⊗C where now the supremum is over mixed states ρ on A ⊗ C. If T is an isometry, i.e. T (ρ) = V ρV † for an isometry V : A ,→ B, the lower bound is an equality, and can be simplified: Cgen (V ·V † ) = sup C (V ⊗ 11)ϕ(V ⊗ 11)† − C(ϕ) |ϕi∈A⊗C = max C V ϕV † − C(ϕ). |ϕi∈A (6) 3 This result, the main one of the present section, should be compared to the formula, similar in spirit, for the entangling power of a bipartite unitary [21, 22]. Furthermore, the above formulas for the coherence generating capacity are related to the coherence power (w.r.t. the relative entropy measure) Pr (T ) = max Cr T (ρ) − Cr (ρ), (7) ρ on A investigated by Garcı́a Dı́az et al. [23] and Bu et al. [24]. Let us also introduce the same maximization restricted to pure input states, Per (T ) = max Cr T (ϕ) − C(ϕ). (8) |ϕi∈A Note that the only difference to our formulas is that we allow an ancilla system C of arbitrary dimension. If we consider, for a general CPTP map T , the extension T ⊗ idk and Pr(k) (T ) ⊗dnC (ρ)−nδe r copies of ρ to Ψ2 , incurring an error of 2 in trace norm in each repetition. At the end, we have (k − 1)n(R − 2δ) + nC(ϕ) copies of Ψ2 , up to trace distance O(k 2 ), using the channel a k−1 total of kn + nR0 times, i.e. the rate is ≥ (R − 2δ) k+R , 0 which can be made arbitrarily close to R by choosing δ small enough and k large enough (which in turn can be effected by sufficiently small ). For the upper bound, Eq. (5), consider a generic protocol using the channel n times, starting from ρ0 (incoherent) and generating ρ1 , . . . , ρn step by step along the way, such that ρn has fidelity ≥ 1 − with Ψ⊗nR . By the 2 asymptotic continuity of C [17, Lemma 12], C (ρn ) ≥ r r p nR − 2δn − 2, with δ = (2 − ), so we can bound nR − 2δn − 2 ≤ Cr (ρn ) = Cr (ρt+1 ) − Cr (ρt ) t=0 ≤ Per(k) (T ) := Per (T ⊗ idk ), (9) := Pr (T ⊗ idk ), n−1 X n−1 X Cr (T ⊗ id)ρt − Cr (ρt ), t=1 then we have † Cgen (V ·V ) = sup Per(k) (V k † † ·V ) = Per (V ·V ), (10) and in general sup Per(k) (T ) ≤ Cgen (T ) ≤ sup Pr(k) (T ). (11) k k Proof We start with the lower bound, Eq. (4): For a given ancilla C and |ϕi ∈ A ⊗ C, let R = Cr (T ⊗ id)ϕ − C(ϕ). For any , δ > 0, we can choose, by the results of [17], a sufficiently large n such that ⊗bnC(ϕ)+nδc Ψ2 IO 7−→≈ ϕ⊗n , IO ⊗dnCr (ρ)−nδe ρ⊗n 7−→≈ Ψ2 , with ρ = (T ⊗id)ϕ, and where ≈ refers to approximation of the target state up to in trace norm. We only have to prove something when R > 0, which can only arise if T is not incoherent, meaning that there exists an initial state |0i mapped to a coherent resource σ = T (|0ih0|), i.e. Cr (σ) > 0. In the following, assume R > 2δ. Now, we may assume that n is large enough so that with R0 = C(ϕ)+δ Cr (σ) + δ, IO ⊗bnC(ϕ)+nδc σ ⊗bnR0 c 7−→≈ Ψ2 where we have used the factthat ρ0 is incoherent and that ρt+1 = It+1 (T ⊗ id)ρt , with an incoherent operation It+1 , which can only decrease relative entropy of coherence. However, each term on the right hand sum is of the form Cr (T ⊗ id)ρ − Cr (ρ) for a suitable ancilla C and a state ρ on A ⊗ C. Thus, dividing by n and letting n → ∞, → 0 shows that R ≤ supρ on A⊗C Cr (T ⊗ id)ρ − Cr (ρ). For an isometric channel T (ρ) = V ρV † , note that the initial state ρ0 in a general protocol is without loss of generality pure, and that T maps pure states to pure states. The incoherent operations It map pure states to ensembles of pure states, so that following the same converse reasoning as above, we end up upper bounding R by an average of expressions Cr (T ⊗id)ρ −Cr (ρ), with pure states ρ, i.e. Eq. (6), since we also have Cgen (T ) ≥ C (T ⊗ id)ϕ − C(ϕ) from the other direction. The fact that no ancilla system is needed, is an elementary calculation. Indeed, for a pure state |ϕi ∈ A ⊗ C, C (T ⊗ id)ϕ) − C(ϕ) = S (∆ ◦ T ⊗ ∆)ϕ) − S (∆ ⊗ ∆)ϕ = S (∆ ◦ T ⊗ id)ρ) − S (∆ ⊗ id)ρ , with ρ = (id ⊗ ∆)ϕ X = pi ϕi ⊗ |iihi| , . The protocol consists of the following steps: 0: Use bnR0 c instances of T to create as many copies ⊗bnC(ϕ)+nδc of σ, and convert them into Ψ2 (up to trace norm ). 1-k (repeat): Convert first bnC(ϕ) + nδc of the already created copies of Ψ2 into n copies of ϕ; then apply T to each of them to obtain ρ = (T ⊗ id)ϕ; and convert the n i = X pi S ∆(T (ϕi )) − S ∆(ϕi ) i ≤ max S ∆(T (ϕi )) − S ∆(ϕi ) , |ϕi∈A and we are done. t u Remark We do not know, at this point, whether the suprema over the ancillary systems in the upper and 4 lower bounds in Eq. (11) are necessary in general, i.e. it (k) (k) might be that Pr (T ) = Pr and/or Per (T ) = Per ; note that the latter is the case for unitaries, even though it seems unlikely in general, cf. [21, 22]. We do know, however, that Pr is convex and non-increasing under composition of the channel with incoherent operations [23, Cor. 1 and 2]. It should be appreciated that even the calculation of Pr (T ) appears to be a hard problem. The investigation of further questions, such as the additivity of Pr , Per or Cgen , may depend on making progress on that problem. Proof For X ∈ {r, f }, any ancilla system C and any state ρ, CX (T1 ⊗ T2 ⊗ idC )ρ − CX (ρ) = CX (T1 ⊗ T2 ⊗ idC )ρ − CX (T1 ⊗ idB2 ⊗ idC )ρ + CX (T1 ⊗ idB2 ⊗ idC )ρ − CX (ρ) = CX (T2 ⊗ idA1 C )σ − CX (σ) + CX (T1 ⊗ idB2 C )ρ − CX (ρ) Remark The same reasoning as in the proof of Theorem 1, replacing Cr with Cf , shows that Cgen (T ) ≤ sup Cf (T ⊗ id)ρ − Cf (ρ) ρ on A⊗C (12) = sup Pf (T ⊗ idk ), where we have introduced the state σ = (T1 ⊗ idB2 ⊗ idC )ρ. By taking the supremum of the left hand side over all ancillas C and all states ρ, we obtain PX (T1 ⊗ T2 ) ≤ PX (T1 ) + PX (T2 ); since the opposite inequality is trivial, using tensor product ancillas and tensor product input states, and employing the additivity of Cr and Cf [17], we have proved the equality. In the case of isometries, Eq. (6) in Theorem 1 already er (Ti ) = Per (Ti ) = Cgen (Ti ). For the tensor prodshows P uct, we again have trivially Per (T1 ⊗ T2 ) ≥ Per (T1 ) + Per (T2 ), by using tensor product input states. To get the opposite inequality, we proceed as above: for any pure state |ϕi ∈ A1 ⊗ A2 , C (T1 ⊗ T2 )ϕ − C(ϕ) = C (T1 ⊗ T2 )ϕ − C (T1 ⊗ idB2 )ϕ + C (T1 ⊗ idB2 )ϕ − C(ϕ) = C (idA1 ⊗ T2 )ψ − C(ψ) + C (T1 ⊗ idB2 )ϕ − C(ϕ) er (T2 ) + P er (T1 ) ≤P k with the coherence of formation power, given by Pf (T ) := maxρ Cf T (ρ) − Cf (ρ). Despite the fact that Cf (ρ) ≥ Cr (ρ), since the upper bound is given by a difference of two coherence measures, it might be that for certain channels, the bound (12) is better than (5), and vice versa for others. Since the supremum over k of the coherence powers of T ⊗idk play such an important role in our bounds, we introduce notation for them, Pr (T ) := sup Pr (T ⊗ idk ), (13) k Pf (T ) := sup Pf (T ⊗ idk ), (14) k er (T ) := sup Per (T ⊗ idk ), P (15) k and call them the complete coherence powers with respect to relative entropy of coherence and coherence of formation, respectively. For these parameters, and the coherence generation capacity of isometries, we note the following additivity formulas. Proposition 2 For any CPTP maps T1 : A1 −→ B1 and T2 : A2 −→ B2 , Pr (T1 ⊗ T2 ) = Pr (T1 ) + Pr (T2 ), Pf (T1 ⊗ T2 ) = Pf (T1 ) + Pf (T2 ). Furthermore, for isometries Ti (ρ) = Vi ρVi† , er (Ti ) = Per (Ti ) = Cgen (Ti ) P and Per (T1 ⊗ T2 ) = Per (T1 ) + Per (T2 ). In other words, the coherence generating capacity of isometries is additive. ≤ PX (T2 ) + PX (T1 ), = Per (T2 ) + Per (T1 ), with the (pure) state ψ = (T1 ⊗ idB2 )ϕ, and we are done. t u III. IMPLEMENTATION OF CHANNELS: COHERENCE COST OF SIMULATION We have seen that a CPTP map can be a resource for coherence because one can use it to generate coherence from scratch, in the form of maximally coherent qubit states. True to the resource paradigm, we have to ask immediately the opposite question: Is it possible to create the resource using pure coherent states and only incoherent operations? Here we show that the answer is generally yes, and we define the asymptotic coherence cost Csim (T ) as the minimum rate of pure state coherence necessary to implement many independent instances of T using only incoherent operations otherwise. We start by recalling the implementation of an arPd−1 bitrary unitary operation U = ij=0 Uij |iihj| by means of an incoherent operation with Kraus operators 5 {Kα }, and using the maximally coherent state |Ψd i = Pd−1 √1 i=0 |ii as resource [25, Lemma 2]. Since it is imd portant for us that the implementation is in fact by means of a strictly incoherent operation, and for selfcontainedness, we include the proof. Proposition 3 (Chitambar/Hsieh [25]) Consuming one Pd−1 copy of Ψd as a resource, any unitary U = i,j=0 Uij |iihj| acting on Cd can be simulated using strictly incoherent operations. Proof We essentially follow [25]; see also [13] for the qubit case. Let the Kraus operators have the following form: Kα = d−1 X Uij |iihj| ⊗ |αihi+α mod d|. The unitaries Ujk can be written as Ujk = Z j X k , with the phase and cyclic shift unitaries 1 0 1 1 0 ω Z= , , X = .. .. .. . . . 1 0 ω d−1 2πi where ω = e d is the d-th root of unity. This scheme can be reduced to a destructive (hence incoherent) POVM on A ⊗ B 0 with outcomes jkα, followed by the application of the incoherent(!) Ujk . In detail, the probability of getting outcome jk is Pr{jk | σ} = Tr Φ(jk) (Kα ⊗ 11)σ(Kα ⊗ 11)† , where σ is a state on A ⊗ B 0 and the i,j=0 It is easy to see that these Kraus operators satisfy Kα ∆Kα† ⊂ ∆, and that they are indeed strictly incoherent, for all α = 0, 1, . . . , d − 1. Furthermore, with the notation q = i + α mod d, d−1 X Kα† Kα = d−1 X d−1 X |Φ(jk) i = (11 ⊗ Z j X k )|Φd i are the Bell states. We can define the POVM elements Mjkα = (Kα ⊗ 11)† Φ(jk) (Kα ⊗ 11), so that X Tr (T ⊗ id)σ Φ(jk) = Tr(Kα ⊗ 11)σ(Kα ⊗ 11)† Φ(jk) α ∗ Uij Ukl |jqihiα||kαihlp| = α=0 i,j,k,l=0 α=0 = d−1 X " ∗ Uij Ukl δi,k |jqihlp| ! Uij Uil∗ |jqihlq| δj,l |jqihlq| = 11. α=0 j,l=0 Now, let |φi = Pd−1 k=0 (Kα† ⊗ 11)Φ(jk) (Kα ⊗ 11) α d−1 d−1 X X d−1 X d−1 X !# X = Tr σ = Tr α=0 i,j,l=0 = Tr σ(Kα† ⊗ 11)Φ(jk) (Kα ⊗ 11) α d−1 X α=0 i,j,k,l=0 = X φk |ki, then 1 Kα (|φi ⊗ |Ψd i) = √ U |φi ⊗ |αi. d Thus, under this incoherent operation, and tracing out the ancilla afterwards, the system will be in the desired t u state U |φihφ|U † with certainty. Now we pass to the general case of CPTP maps, which extends the above result for unitaries, with two different protocols. Theorem 4 Any CPTP map T : A −→ B can be implemented by incoherent operations, using a maximally coherent resource state Ψd , where d = |B|. P Proof Let T (ρ) = α Kα ρKα† be a Kraus decomposition of T , with Kraus operators Kα : A −→ B. The idea of the simulation is to use teleportation of the output of T , which involves a maximally entangled state ΦD on B 0 ⊗B 00 , a Bell-measurement on system B ⊗B 0 with outcomes jk ∈ {0, 1, . . . , d − 1}2 , and unitaries Ujk on B 00 . X σMjkα = Tr σMjk , α P with Mjk = α Mjkα . This leads to a new equivalent scheme in which, given a state ρ on A and a maximally entangled state on B 0 ⊗ B 00 , we can apply the measurement Mjk on A⊗B 0 with outcomes jk, and unitaries Ujk acting on B 00 . Formally, let us define the Kraus operators of the protocol by letting 0 00 B Ljkα := [hΦ(jk) |(Kα ⊗ 11)]AB ⊗ Ujk . (16) It can be checked readily that they satisfy the normalization condition X † X Ljkα Ljkα = Mjkα ⊗ 11 = 11 ⊗ 11. jkα jkα Applying the Kraus operators Ljkα on all the system we get Ljkα |φiA |Ψd iB 0 B 00 = hΦ(jk) |(Kα |φi)(11 ⊗ Ujk )|Ψd i = hΦ(jk) |Kα |φi|Ψ(jk) i 1 = Kα |φi. d P P Hence, jkα Ljkα (ρ ⊗ Φd )L†jkα = α Kα ρKα† = T (ρ), and the proof is complete. t u 6 Theorem 5 Any CPTP map T : A −→ B can be implemented by strictly incoherent operations and a maximally coherent state Ψd , where d ≤ |A||B|. Proof The channel T is, first of all, a convex combination of extremal CPTP maps Tλ , each P of which has at most |A| Kraus operators [26]: T = λ pλ Tλ . Clearly, we only have to prove the claim for the Tλ . Because of the bound on the Kraus operators, each Tλ has a unitary dilation Uλ : A ⊗ B → B ⊗ A, such that Tλ (ρ) = TrA Uλ (ρ ⊗ |0ih0|B )Uλ† , with a fixed incoherent state |0i ∈ B. As the Uλ act on a space of dimension d = |A||B|, we can invoke the simulation according to Proposition 3. t u Remark Comparing Theorems 4 and 5, we note that the latter always consumes more resources, but it is guaranteed to be implemented by strictly incoherent operations, a much narrower class than the incoherent operations. We leave it as an open question whether the resource consumption of Theorem 4 can be achieved by strictly incoherent operations, or whether there is a performance gap between incoherent and strictly incoherent operations. Note that these two classes, incoherent operations and strictly incoherent operations, are distinct as sets of CPTP maps, although it is known that they induce the same possible state transformations of a given state into a target state for qubits [27] and for pure states in arbitrary dimension [28] (correcting the earlier erroneous proof of the claim by [19]; the SIO part of the pure state transformations is due to [17]). However, for the distillation of pure coherence at rate Cr (ρ) [17], IO are needed and it remains unknown whether SIO can attain the same rate. Crucially, any destructive measurement, i.e. any POVM followed by an incoherent state preparation, is IO, but the only measurements allowed under SIO are of diagonal, i.e. incoherent, observables. The results so far are about the resources required for the exact implementation of a single instance of a channel, in the worst case. It is intuitively clear that some channels are easier to implement in the sense that fewer resources are needed; e.g. for the identity or any incoherent channel, no coherent resource is required. In the spirit of the previous section, we are interested in the minimum resources required to implement many independent instances of T . Definition 6 An n-block incoherent simulation of a channel T : A −→ B with error and coherent resource Ψd (on space D) is an incoherent operation I : An ⊗ D −→ B n , such that T 0 (ρ) := I(ρ ⊗ Ψd ) satisfies ≥ kT 0 − T ⊗n k = sup |φi∈An ⊗C k(T 0 ⊗ idC )φ − (T ⊗n ⊗ idC )φk1 . Here, C is an arbitrary ancilla system; the error criterion of the simulation is known as diamond norm [29] or completely bounded trace norm [30], see also [31]. The rate of the simulation is n1 log d, and the simulation cost of T , denoted Csim (T ), is the smallest R such that there exist n-block incoherent simulations with error going to 0 and rate going to R as n → ∞. The best general bounds we have on the simulation cost are contained in the following theorem. Theorem 7 For any CPTP map T : A −→ B, Cgen (T ) ≤ Csim (T ) ≤ log |B|. (17) Furthermore, Csim (T ) ≥ max sup Pr (T ⊗ idk ), sup Pf (T ⊗ idk ) , k k (18) where we recall the definitions of the relative entropy coher ence power, Pr (T ) = maxρ Cr T (ρ) − Cr (ρ), and of the coherence of formation power, Pf (T ) = maxρ Cf T (ρ) − Cf (ρ). Proof We start with Eq. (17): The upper bound is a direct consequence of Theorem 4. The lower bound follows from the fact that T is implemented using maximally coherent states at rate R = Csim (T ) and incoherent operations. Generation of entanglement on the other hand uses T and some more incoherent operations. Since incoherent operations cannot increase the amount of entanglement, the overall process of simulation and generation cannot result in a rate of coherence of more than R. Regarding Eq. (18), the idea is that for > 0 and n large enough, since the simulation implements a CPTP map T 0 that is within diamond norm from T ⊗n , us⊗n(R+) ing incoherent operations and Ψ2 as a resource. Applying the simulation to the state ρ⊗n , results in ⊗n (T 0 ⊗idkn )ρ⊗n ≈ (T ⊗idk )ρ , hence we have an overall incoherent operation ⊗n(R+) Ψ2 IO ⊗ ρ⊗n 7−→ (T 0 ⊗ idkn )ρ⊗n . By monotonicity of CX (X ∈ {r, f }) under IO, and CX (Ψ2 ) = 1, this means n(R + ) ≥ CX (T 0 ⊗ idkn )ρ⊗n − CX (ρ⊗n ), where we have used additivity of Cr and Cf [17]. Since this holds for all ρ, we obtain n(R + ) ≥ PX (T 0 ⊗ idkn ) ≥ PX (T ⊗n ⊗ idkn ) − nκX − 4 ≥ nPX (T ⊗ idk ) − nκX − 2, invoking in the second line Lemma 8 below, with κr = 4 log |B| and κf = log |B|+log k, and in the third a tensor 7 power test state. Since can be made arbitrarily small, and n as well as k arbitrarily large, the claim follows. t u Here we state the technical lemma required in the proof of Theorem 7. Lemma 8 The relative entropy coherence power and the coherence of formation power are asymptotically continuous with respect to the diamond norm metric on channels. To be precise, for T1 , T2 : A −→ B with 21 kT1 − T2 k ≤ , Pr (T1 ⊗ idk ) − Pr (T2 ⊗ idk ) ≤ 4 log |B| + 2g(), (19) Pf (T1 ⊗ idk ) − Pf (T2 ⊗ idk ) ≤ (log |B|+log k) + g(), (20) where g(x) = (1+x)h2 x 1+x = (1+x) log(1+x)−x log x. Proof For the first bound, observe Pr (T1 ⊗ idk ) − Pr (T2 ⊗ idk ) ≤ max Cr (T1 ⊗ idk )ρ − Cr (T2 ⊗ idk )ρ AC ρ = max S(BC)(∆T1 ⊗∆)ρ − S(BC)(T1 ⊗idk )ρ in fact close to a MIO operation, the coherence generating capacity Cgen (T ) is also close to 0, we do not know at the moment whether the same holds for the simulation cost Csim (T ). IV. QUBIT UNITARIES In this section, we want to have a closer look at qubit unitaries, for which we would like to find the coherence generating capacity and simulation cost. To start our analysis, we note that a general 2 × 2unitary has four real parameters, but we can transform unitaries into each other at no cost by preceding or following them by incoherent unitaries, i.e. combinations iα of the bit flip σx and diagonal (phase) unitaries e 0 . This implies an equivalence relation among 0 eiβ qubit unitaries up to incoherent unitaries. A unique representative of each equivalence class is given by U = U (θ) = c −s , s c (21) ρAC −S(BC)(∆T2 ⊗∆)ρ + S(BC)(T2 ⊗idk )ρ = max S(B|C)(∆T ρAC 1 ⊗∆)ρ − S(B|C)(T1 ⊗idk )ρ −S(B|C)(∆T2 ⊗∆)ρ + S(B|C)(T2 ⊗idk )ρ ≤ max S(B|C)(∆T ⊗∆)ρ − S(B|C)(∆T ⊗∆)ρ ρAC 1 where c = cos θ and s = sin θ and with 0 ≤ θ ≤ that c ≥ s ≥ 0. π 4, so One can calculate Cgen (U (θ)), using the formula from Theorem 1. Clearly, by choosing the test state ϕ to be 2 + S(B|C)(T1 ⊗idk )ρ − S(B|C)(T2 ⊗idk )ρ ≤ 2 2 log |B| + g() , ρAC ≤ (log |B| + log k) + g(), where the last line comes directly from the asymptotic continuity of the coherence of formation [17, Lemma 15]. We close the proof expressing our belief that it is possible to prove a version of Eq. (20) where k does not appear on the right hand side. t u Remark As a consequence, while for a channel T that is close to an incoherent operation (in diamond norm), or 0.8 coherence gain where in the first line we insert the same variable ρAC to maximise Pr (Tj ⊗ idk ) and notice that in that case, the term Cr (ρ) cancels; then in the second line, we use the definition of the relative entropy of coherence and in the third we use chain rule S(BC) = S(B|C) + S(C) for the entropy, allowing us to cancel matching S(C) terms; in the fourth line we invoke the triangle inequality, and finally the Alicki-Fannes bound for the conditional entropy [32] in the form given in [33, Lemma 2]. For the second bound, we start very similarly: Pf (T1 ⊗ idk ) − Pf (T2 ⊗ idk ) ≤ max Cf (T1 ⊗ idk )ρ − Cf (T2 ⊗ idk )ρ 1 0.6 0.4 0.2 0 0 0.05 0.1 0.15 0.2 0.25 θ/π FIG. 1. Plot of Cgen (U (θ)) = Per (U (θ)) as a function of θ ∈ [0, π4 ] (solid blue line), and comparison with h2 (cos2 θ) (dashed red line), which is the coherence generated by an incoherent input state. In particular, for θ ≈ 0, the ratio between the two functions is unbounded. The angle θ is plotted as a fraction of π. pure incoherent, Cgen (U (θ)) = Per (U (θ)) ≥ h2 (c2 ) = −c2 log c2 − s2 log s2 , (22) 8 with h2 (x) = −x log x − (1 − x) log(1 − x) the binary entropy. Perhaps surprisingly, however, this is in general not the optimal state [34, Cor. 5] (see also [23]), meaning that Per (U (θ)) is attained at a coherent test state ϕ, although no closed form expression seems to be known. In fact, simple manipulations show that we only need to optimise C U (θ)ϕU (θ)† −C(ϕ) over states |ϕi = U (α)|0i = cos α|0i + sin α|1i, 0 ≤ α ≤ π (i.e. no phases are necessary). The function to optimise becomes h2 cos2 (α+θ) −h2 cos2 α . Its critical points satisfy the transcendental equation sin(2α + 2θ) ln tan2 (α + θ) = sin(2α) ln tan2 α, (23) which can be solved numerically. Fig. 1 shows that Cgen (U (θ)) = Per (U (θ)) > h2 (cos2 θ) for across the whole interval, except at the endpoints θ = 0, π4 ; in Fig. 2 we plot the optimal α for U (θ). 0.1 0.08 * α /π 0.06 0.04 0.02 0 0 0.05 0.1 0.15 0.2 0.25 θ/π FIG. 2. The optimal value of α attaining Cgen (U (θ)) = Per (U (θ)) = h2 cos2 (α + θ) − h2 cos2 α , as a function of π θ ∈ [0, 4 ]. It is nonzero except at the endpoints θ = 0, π4 . Both angles, θ and α∗ , are plotted as fractions of π. for a general state |ψi. Without loss of generality, the incoherent Kraus operators achieving the transformation have the following general form: K = λ U (θ) ⊗ |0ihr| + R ⊗ |0ihr⊥ | , (24) where |r⊥ i = s0 |0i − c0 |1i is the vector orthogonal to |ri. We now need to find the form of R such that K is incoherent. For that, we impose incoherence of K when tracing out the ancillary part: h0|A K|0iA =: T0 and h1|A K|1iA =: T1 , where T0 and T1 must be 2dimensional incoherent operators. We then obtain that 0 R = s0 T0 − c0 T1 and λU = c0 T T1 . The latter 0 + s condi 1 0 0 −1 tion enforces that either T0 ∝ and T1 ∝ 0 1 1 0 1 −1 0 0 or viceversa; or T0 ∝ and T1 ∝ or vicev0 0 1 1 ersa. From these possibilities, we get 4 possible R matrices, which define 4 different Kraus operators Ki defined, according to Eq. (24), by Ri matrices as follows: s0 0 c c0 −s sc0 , R1 = 0 0 s sc0 c sc0 c0 0 −c s0 s sc0 R2 = , 0 0 −s sc0 −c sc0 c0 0 −c s0 −s sc0 , R3 = 0 0 −s sc0 c sc0 s0 0 c c0 s sc0 R4 = , 0 0 s sc0 −c sc0 and the general incoherent Kraus operator is K = λi Ki P (i = 1, 2, 3, 4). Finally, after imposing i |λi |2 Ki† Ki = 11, we obtain the following conditions on Ri and λi : X |λi |2 = 1, i X |λi |2 Ri† Ri = 11, i On the other hand, regarding the implementation of these unitary channels, all we can say for the moment is that Csim (U (θ)) ≤ 1, because we can implement each instance of the qubit unitary using a qubit maximally coherent state Ψ2 . It is perhaps natural to expect that one could get away with a smaller amount of coherence, but it turns out that with a two-dimensional resource state this is impossible. Proposition 9 The only qubit coherent resource state |ri ∈ C2 that permits the implementation of U (θ), 0 < θ ≤ π4 , is the maximally coherent state. Furthermore, any two-qubit incoherent operation I such that I(ρ⊗|rihr|) = U (θ)ρU (θ)† ⊗σ for general ρ, is such that the state σ left behind in the ancilla is necessarily incoherent. Proof We want to know for which state |ri = c0 |0i+s0 |1i IO the transformation |ψi|ri −−→ (U (θ)|ψi)|0i is possible, X |λi |2 Ri = 0. i It can be verified that these conditions are only fulfilled when |c0 | = |s0 | = √12 , i.e. |ri is maximally coherent. If the incoherent implementation of the unitary, instead of mapping two qubits (input and resource state) to one (output), but to two (output plus residual resource), i.e. I(ρ ⊗ |rihr|) = U (θ)ρU (θ)† ⊗ σ, then first of all σ has to be the same irrespective of the state ρ. Otherwise we would be able, by measuring σ, to learn some information about ρ without disturbing it. Now consider a pure incoherent input state ρ = |0ih0|, and note that the desired output state U (θ)|ψi has nontrivial coherence. But now observe that I takes in a state of coherence rank 2 [17], and produces a product of a pure state of coherence rank 2 with another state. Since the coherence rank cannot increase, even under individual 9 Kraus operators [17], it must be the case that σ is incoherent. t u This result might suggest an irreversibility between simulation and coherence generation for these unitaries, but we point out that it does not preclude the possibility of simulations using a higher rank, yet less coherent, resource state (cf. [35], where the analogue is demonstrated for LOCC implementation of bipartite unitaries using entangled resources); or of a simulation of many instances of U (θ) at a cost lower than 1 per unitary. V. CONCLUSION AND OUTLOOK We have shown that using a maximally coherent state and strictly incoherent operations, we can implement any unitary on a system (using Ψ for a d-dimensional unitary), and via the Stinespring dilation any CPTP map (using Ψd2 for a channel acting on a d-dimensional system). By teleportation, we prove that for incoherent operations and a d-dimensional maximally coherent state any noisy channel can be implemented. Vice versa, every incoherent operation gives rise to some capacity of generating pure coherence by using it asymptotically many times, and we have given capacity bounds in general, and a single-letter formula for the case of unitaries. We also found the additive upper bounds Pr (T ) and Pf (T ) on the coherence generation capacity Cgen (T ), even though we do not know whether these numbers are efficiently computable due to the presence of the extension ⊗idk , nor whether these extensions are even necessary. It is open at the moment whether the coherence generation capacity Cgen (T ) itself is additive for general tensor product channels, and er (T ) = likewise the lower bound given in Theorem 1, P e supk Pr (T ⊗ idk ); at least for isometric channels they are. The coherence generating capacity is never larger than the simulation cost, but in general these two numbers will be different. As an extreme case, consider any CPTP map T that is not incoherent, but is a so-called “maximally incoherent operation” (MIO, cf. [16]), meaning that T (ρ) ∈ ∆ for all ρ ∈ ∆. This class was considered in [9], and it makes coherence theory asymptotically reversible [17], all states ρ being equivalent to Cr (ρ) maximally coherent qubit states. Such maps exist even in qubits, cf. [36] correcting [27, Thm. 21]. We expect the simulation cost of any such T to be positive, Csim (T ) > 0. At the same time, Cgen (T ) = 0 by Theorem 1, because the relative entropy of coherence is a MIO monotone, and the tensor product of MIO transformations is MIO. To obtain an example, we can take any MIO channel for which there exists a state ρ such that Cf T (ρ) > Cf (ρ), since by Theorem 7, Csim (T ) is lower bounded by the difference of the two. (As an aside, we note that this cannot be realised in qubits, because for qubits, any state transformation possible under MIO is already possible under IO, for which Cf is a monotone [27, 37].) Concretely, consider the following states on a 2d-dimensional system A, which could be called coherent flower states, since their corresponding maximally correlated states (cf. [17, 38]) are the well-known flower states [39]. We write them as 2 × 2-block matrices, 1 11 U ρd = , (25) 2d U † 11 where U is the d-dimensional discrete Fourier transform matrix. Via the correspondence between Cr and the relative entropy of entanglement, and between Cf and the entanglement of formation, respectively, of the associated state, we know that Cr (ρd ) = 1 and Cf (ρd ) = 1 + 12 log d [39]. By the results of [9], however, for every > 0 and sufficiently large n, there exists a MIO transformation T (n) : (C2 )⊗n(1+) −→ An with ⊗n(1+) ρ(n) := T (n) Ψ2 ≈ ρ⊗n (26) d . By the asymptotic continuity of Cf [17], we have Cf (ρ(n) ) ≥ n 1 + 21 log d − n log(2d) − g(), while of ⊗n(1+) course the preimage ρ0 = Ψ2 has Cf (ρ0 ) ≤ n + n, so for small enough and n large enough, we have a gap: n Csim T (n) ≥ log d − n(2 + log d) − g() > 0, 2 (27) invoking Theorem 7. Thus, while states in the resource theory of coherence cannot exhibit bound resource — indeed, it was observed in [17] that vanishing distillable coherence, Cr (ρ) = 0, implies vanishing coherence cost Cf (ρ) = 0 —, operations can have bound coherence, and the gap can be large on the scale of the logarithm of the channel dimension. We observe that this effect can only occur for maximally incoherent operations, which are precisely the ones with Cgen (T ) = 0. We argued already that MIO channels have zero coherence generating capacity; in the other direction, if T is not MIO, it means that there exists an incoherent state ρ such that T (ρ) has coherence, and this can be distilled at rate Cr (T (ρ)) [17]. As MIO are closed under forming tensor products, it also follows that Cgen does not exhibit superactivation. This example and the subsequent considerations raise the question of how our theory would change if we considered all MIO transformations as free operations. By definition, the above example – by virtue of being MIO – has zero MIO-simulation cost, so there is no bound coherence any more. It may still be the case that there is in general a difference between MIO-simulation cost and MIO-coherence generating capacity, but deciding this possibility is beyond the scope of the present investigation. We only note that Theorem 1 gives us a singleletter formula for the MIO-coherence generating capacity, namely MIO Cgen (T ) = Pr (T ) = sup Cr (T ⊗id)ρ −Cr (ρ), (28) ρ on A⊗C 10 the complete relative entropy coherence power of T . The supremum is over all auxiliary systems C and mixed states ρ on A ⊗ C. Indeed, the upper bound of Eq. (5) still applies, because Cr is MIO monotone, which is all we needed in the proof of Theorem 1. For the lower bound, that it is attainable follows the same idea as the proof of Eq. (4), only that we can now use an arbitrary mixed state ρ in the argument, since its coherence cost under MIO equals Cr (ρ) [9, 17]. extend some of the various coherence monotones that have been studied for states to CPTP maps. ACKNOWLEDGMENTS One of the most exciting possibilities presented by the point of view of channels as coherence resources is the transformation of channels into channels by means of preceding and post-processing a given one by incoherent operations to obtain a different one. The fundamental question one can ask here is how efficiently, i.e. at which rate R(T1 → T2 ) one can transform asymptotically many instances of T1 into instances of T2 , with asymptotically vanishing diamond norm error. To make nontrivial statements about these rates, one would need to The present project was started during two visits of KBD to the Quantum Information Group at the Universitat Autònoma de Barcelona. The authors thank Ludovico Lami, Swapan Rana, Alexander Streltsov, Michalis Skotiniotis, and Mark Wilde for interesting discussions and comments on the coherence of channels. MGD is supported by a doctoral studies fellowship of the Fundación “la Caixa”. 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