Coherence Manipulation with Dephasing-Covariant Operations

PHYSICAL REVIEW RESEARCH 2, 013109 (2020)

Coherence manipulation with dephasing-covariant operations

Bartosz Regula ,1,2,* Varun Narasimhachar,1,2,† Francesco Buscemi,3,‡ and Mile Gu1,2,4,§ 1School of Physical and Mathematical Sciences, Nanyang Technological University, 637371 Singapore 2Complexity Institute, Nanyang Technological University, 637335 Singapore 3Graduate School of Informatics, Nagoya University, Chikusa-ku, 464-8601 Nagoya, Japan 4Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore

(Received 13 August 2019; published 31 January 2020)

We characterize the operational capabilities of quantum channels which can neither create nor detect quantum coherence vis-à-vis efficiently manipulating coherence as a resource. We study the class of dephasing-covariant operations (DIO), unable to detect the coherence of any input state, as well as introduce an operationally motivated class of channels ρ-DIO, which is tailored to a specific input state. We first show that pure-state transformations under DIO are completely governed by majorization, establishing necessary and sufficient conditions for such transformations and adding to the list of operational paradigms where majorization plays a central role. We then show that ρ-DIO are strictly more powerful: although they cannot detect the coherence of the input state ρ, the operations ρ-DIO can distill more coherence than DIO. However, the advantage disappears in the task of coherence dilution as well as generally in the asymptotic limit, where both sets of operations achieve the same rates in all transformations.

DOI: 10.1103/PhysRevResearch.2.013109

I. INTRODUCTION measurements, making them cheap and easy to implement in the resource-theoretic setting; the strictly incoherent opera- Quantum coherence, or superposition, is an intrinsic tions (SIO) [4,11] allow for a similar implementation with feature of quantum mechanics that underlies the advantages incoherent ancillae, but require arbitrary measurements. Re- enabled by quantum information processing and quantum grettably, these operations were found to be too limited in technologies [1]. The resource theory of quantum coher- their operational capabilities [13,25,26], suggesting that any ence [1–4] has found extensive use in the characterization nontrivial resource theory of coherence would require a larger of our ability to manipulate coherence efficiently within a set of allowed maps in order to be practically useful. Common rigorous theoretical framework, wherein the properties of choices of such larger sets, however, are often too permissive a resource are investigated under a suitable set of allowed and lack a physical justification [7]. (“free”) operations which reflect the constraints placed on The question of “freeness” of operations within the re- the manipulation of the given resource [5,6]. Despite many source theory of coherence has also been approached from promising developments in the establishment of a comprehen- a different perspective, by characterizing the nonclassicality sive description of coherence, the physical constraints gov- of the channels themselves—first in a very general man- erning its manipulation are unsettled [1,7,8], and one of the ner [27] and later with explicit applications to coherence most important outstanding questions in the resource theory theory [11,14,22]. These works suggest that the most impor- of quantum coherence is to understand the exact properties tant feature distinguishing truly nonclassical channels is their of the different operational paradigms under which coherence ability to detect the coherence present in a system and use it to can be investigated [4,7–26]. manipulate the state, and so any class of channels which is to Many proposed types of free operations stem from mean- be considered free must necessarily be incapable of coherence ingful physical considerations regarding their implementabil- detection. Although the study of maps based on various types ity: for instance, the physically incoherent operations [7] only of coherence nondetecting properties has attracted significant require the use of incoherent ancillary systems and incoherent attention in recent years [7,8,11,14,17–19,21,22], the limits of their power remain undiscovered. In this work, we explore the limitations imposed on co- *[email protected] herence manipulation by the inability of the free operations †[email protected] to create and detect coherence. We first study the class of ‡[email protected] dephasing-covariant incoherent operations (DIO) [7,8], con- §[email protected] stituting the largest class of operations that do not detect the coherence of any input state. We establish a complete descrip- Published by the American Physical Society under the terms of the tion of pure-state transformations under these operations by Creative Commons Attribution 4.0 International license. Further relating them with the theory of majorization, also revealing distribution of this work must maintain attribution to the author(s) an operational connection between DIO and other classes of and the published article’s title, journal citation, and DOI. free operations. To investigate the ultimate operational limits

2643-1564/2020/2(1)/013109(10) 013109-1 Published by the American Physical Society REGULA, NARASIMHACHAR, BUSCEMI, AND GU PHYSICAL REVIEW RESEARCH 2, 013109 (2020) of coherence nondetecting channels, we then introduce the is inherently free and cannot be used in such an experimental class of operations ρ-DIO, which are tailored to a specific in- protocol. put state and extend the class DIO. In order to understand and However, consider now a scenario in which the input compare their strength, we quantify the capabilities of ρ-DIO coherent state ρ of a protocol is known: the operations in the fundamental operational tasks of coherence distillation which cannot detect the coherence of the input state are then and dilution. We show in particular that such maps satisfy precisely those which satisfy ◦ (ρ) = ◦ (ρ) for this a curious property: even though ρ-DIO cannot detect the choice of ρ, and indeed it is not necessary to impose dephas- coherence of the state ρ, they can still distill more coherence ing covariance for all inputs if one is concerned with acting from ρ than the class DIO; however, this advantage disappears on ρ specifically. This point of view motivates us to define in the one-shot task of dilution as well as generally in the the class of ρ–dephasing-covariant incoherent operations (ρ- asymptotic limit, where DIO match the capabilities of ρ-DIO DIO), which we take to be the operations which commute with in all state transformations. Our results give insight into the the dephasing channel for a given input state ρ, and which precise limits on the operational power of free operations therefore incorporate the ultimate limitations caused by the which do not use coherence in manipulating coherence as a inability to detect or use the coherence of a particular input resource. state. It is clear that a ρ-DIO map can, in principle, create or detect coherence when acting on an input state other than ρ. II. DIO AND ρ-DIO However, the definition of ρ-DIO is justified whenever one We study quantum coherence as a basis-dependent con- deals with an explicit protocol which transforms a fixed input {| }d cept. We will therefore fix an orthonormal basis i i=1 which state to some desired output. Two such protocols form the we deem incoherent, and use I to denote the set of all foundations of the manipulation of coherence as a quantum re- states diagonal (incoherent) in this basis. We will use (·) = source: these are the tasks of coherence distillation [4,18,24], | |·|| i i i i i to denote the completely dephasing channel which aims to convert a given input state to a maximally in this basis. coherent state, and coherence dilution [4,17], which performs As discussed previously, it is natural to impose two condi- the opposite transformation of a maximally coherent input tions for a set of operations to be free: one is its inability to state to some desired state. The definition of ρ-DIO then mo- create coherence, in the sense that σ ∈ I ⇒ (σ ) ∈ I, and tivates the following question: can the operational capabilities the other is its inability to detect coherence, in the sense that of DIO be surpassed by operations which nevertheless do not measurement statistics under any incoherent measurement detect the coherence of the input state ρ? To address this ques- after an operation should remain the same regardless of tion, we first describe the transformations achievable under whether or not the input state possessed any coherence; pre- DIO and later investigate whether ρ-DIO can outperform the cisely, i|(ρ)|i=i|((ρ))|i for all i. Imposing only the former. first constraint leads to the class of maximally incoherent operations (MIO) [2], which can exhibit undesirable properties such as being able to increase the number of levels of a pure III. PURE-STATE TRANSFORMATIONS UNDER DIO state which are in superposition [12]. The class of channels which satisfies both constraints for all input states is precisely Although a fundamental and operationally meaningful DIO, equivalently defined to commute with the dephasing choice of operations, the class DIO remains relatively un- channel: ◦ (ρ) = ◦ (ρ) ∀ρ. These operations have explored and few of its properties are known. Other sets of previously been considered in various contexts [7,8,14,27], operations are better understood: in particular, it is known and indeed they admit several interpretations. The maps DIO that the transformations of pure states under the classes of can be regarded as inherently classical [22,27], as any classi- incoherent operations (IO) [3] and strictly incoherent opera- cal (incoherent) observer is unable to distinguish (ρ) from tions (SIO) [4,11] are governed by majorization theory, in a ◦ (ρ), and hence is unable to determine whether the manner similar to the manipulation of pure-state entanglement coherence of ρ has been employed in the process. The latter under local operations and classical communication [29]. |ψ= point shows that DIO can also be understood as the operations Precisely, one has that a pure-state transformation ψ | →|φ= φ | which do not use coherence [11,14], as the properties of the i i i i i i is achievable under IO or SIO ψ ≺ φ k |ψ |2 output system accessible to a classical observer are indepen- if and only if ( ) ( )[12,30,31], i.e., if i=1 i k |φ |2 ∀ ∈{ ,..., } dent of the coherence of the input. i=1 i k 1 d , where we assume that the coeffi- The ability to detect coherence is of particular importance cients of the states are arranged so that |ψ1| ··· |ψd |.Our in practical setups relying on quantum coherence, such as first contribution is to extend this relation to the class DIO. interferometric experiments [11,22,28]. A general interfer- Theorem 1. The deterministic pure-state transformation ometric protocol can be understood as consisting of three ψ → φ is possible under DIO if and only if (ψ) ≺ (φ). separate parts: first, a state in superposition is created; second, We refer to Appendix A for the full proof of the theorem. path-dependent phases are encoded in the state with suitable This establishes DIO as another class of operations in which unitary operations; and third, the information about the paths pure-state transformations are fully governed by majorization is extracted in a measurement. It is then explicit that the theory, and reveals an operational equivalence between DIO, ability to create (in the first step) and detect (in the last step) IO, and SIO in manipulating pure states. The equivalence is coherence is crucial for any such setup to work, and indeed nontrivial: the class DIO is incomparable with IO [12], and any operation which can neither create nor detect coherence there exist coherence monotones which can increase under

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DIO despite always decreasing under the action of SIO and distinguishing between two quantum states ρ and σ by a mea- { , − } ε ρ σ IO [32]. surement M 1 M , with DH ( ) exactly quantifying the The theorem immediately lets us apply a plethora of results smallest probability of incorrectly accepting the hypothesis to coherence manipulation under DIO. For instance, the recent of possessing state ρ as true (Tr Mσ ) while constraining the investigation of moderate-deviation interconversion rates un- probability of incorrectly accepting the hypothesis of being der majorization in Refs. [33,34] allows one to precisely char- in possession of state σ as true [Tr(1 − M)ρ] to be, at most, ε ε ρ||σ acterize DIO transformations beyond the single-shot regime; . We remark that DH ( ) is efficiently computable as a similarly, a recent investigation of quantum coherence fluctu- semidefinite program [44]. ation relations [35] relies purely on the theory of majorization, We relate the hypothesis testing relative entropy with dis- and our result immediately establishes that the results can tillation in the following. be directly applied to describe the fluctuations and battery- Theorem 2. The ε-error one-shot distillable coherence un- assisted transformations under DIO. der ρ-DIO for any input state ρ is given by The result can also be extended to so-called heralded ,ε ε C(1) (ρ) = D (ρ (ρ)) , (2) probabilistic transformations, where a state |ψ is transformed d,ρ−DIO H log to one of the states {|φ } with a corresponding probability p , = x j j where x log : log 2 . and the final state is identified unambiguously by a classical This formally establishes a property of the class of op- φ ⊗ flag register; formally, the output states take the form j erations ρ−DIO that one might intuitively expect: the more | jj|. One can similarly show the following. distinguishable a state ρ is from its dephased version (ρ), Proposition 1. There exists a DIO effecting the transforma- the more coherence we can extract from it using ρ-DIO. This ψ → φ ⊗| | tion j p j j j j if and only if is indeed very natural in the framework of quantum coherence, as it gives explicit operational meaning to the information ψ ≺ φ . ( ) p j ( j ) (1) contained in the off-diagonal elements of the density matrix j ρ. It is instructive to compare Eq. (2) with the expression ε We refer the reader to the appendices for details. This again for one-shot distillable coherence under DIO [18], where DH establishes an equivalence between DIO, IO, and SIO in such additionally has to be optimized over a set of operators and transformations, extending earlier partial results [21]. does not enjoy an exact interpretation in this context. Of particular importance will be the case ε = 0, that is, IV. COHERENCE MANIPULATION WITH ρ-DIO exact deterministic distillation of coherence. The result then reduces to The existence of a ρ-DIO transformation between states , 1 ρ and σ is equivalent to the existence of a quantum channel C(1) 0 (ρ) = D0 (ρ (ρ)) = log , d,ρ−DIO H log ρ such that (ρ) = σ and ((ρ)) = ((σ )). This has Tr ρ ( ) strong connections with the concept of relative majoriza- (3) ρ tion and the distinguishability between the states and where ρ is the projection onto the support of ρ. In particular, (ρ)[36–38], and may at first sight suggest that majorization combining the results of Theorems 1 and 2, we have the ρ will also play a role in -DIO transformations, making them following. no more powerful than DIO. We will show that this is, in fact, |ψ= ψ | Corollary 1. A pure state i i i can be determin- not the case. To investigate this problem, we now focus on the istically transformed to | m under DIO iff fundamental tasks of distillation and dilution. 2 1 max |ψi| , (4) i A. Distillation m while the transformation is possible under ψ-DIO iff The ε-error one-shot distillable coherence under the class ρ 1 -DIO is defined to be the largest size of the maximally ψ| ψ |ψ= |ψ |4 . √1 ( ) i (5) coherent state | m= |i reachable to within an error m i m i ε under a single ρ−DIO transformation; formally, we have A detailed derivation can be found in the appendices. (1),ε ρ C ,ρ− ( ) The above allows us to easily construct examples of states d DIO such that even though |ψ→| m is impossible under DIO, = ρ , − ε , ψ− : log max m max F ( ( ) m ) 1 the transformation can be achieved by DIO. Consider, ∈ρ−DIO |ψ = 5 , 3 , 3 T √ √ for example, the state : 8 16 16 ,forwhichit where F (ρ,σ) = ρ σ 2 is the fidelity and the logarithm 1 can be verified that ( 2 ) (ψ), which means the trans- is base 2. Our first result exactly characterizes this quantity in |ψ→| ε formation 2 is impossible by DIO (and, in fact, terms of the hypothesis testing relative entropy D , defined |ψ |4 = H by all MIO [18]). However, we easily compute i i as [39–41] 59 < 1 (1),0 ψ = 128 2 ,soCd,ψ−DIO( ) 1 and hence one coherence bit 2 ε ρ||σ DH ( ) can be exactly distilled. This explicitly shows an operational advantage provided by the operations ρ-DIO over DIO in :=−log min{Tr Mσ | 0 M 1, 1 − Tr Mρ ε}. state transformations and, in particular, in coherence distilla- This quantity finds use in the fundamental task of quan- tion. Such an advantage is rather surprising: to any classical tum hypothesis testing [42,43], where one is interested in observer, the distillation protocol applied to the state ρ is

013109-3 REGULA, NARASIMHACHAR, BUSCEMI, AND GU PHYSICAL REVIEW RESEARCH 2, 013109 (2020) indistinguishable from a classical operation, yet it can distill Interestingly, comparing the above with the results ob- more coherence than DIO or even the powerful class MIO. tained previously for DIO [17], we have that Furthermore, due to the majorization condition of Theorem 1, (1),ε (1),ε C , (ρ) = C , − (ρ); (8) it is easy to see that the exact distillation of coherence under c DIO c m DIO DIO cannot be enhanced by the use of a catalyst—that is, that is, the operations m-DIO provide no advantage over ψ ⊗ φ → m ⊗ φ is possible under DIO if and only if ψ → DIO whatsoever. Combining this with the fact that the m is possible—which then shows that ρ-DIO has strictly asymptotic coherence cost under DIO is exactly given by larger capabilities than even catalysis-assisted DIO. D(ρ (ρ)) [19], we similarly have that However, now consider the many-copy scenario in which ∞ ρ = 1 (1),ε ρ⊗n = ρ ρ . we can perform joint quantum operations on the compos- Cd, −DIO( ) lim lim Cc, −DIO( ) D( ( )) ρ⊗n m ε→0 n→∞ n m ite system . In the asymptotic limit of independent and (9) identically distributed (i.i.d.) systems, one can then define the We can also note that the zero-error coherence cost under distillable coherence under ρ-DIO as m-DIO (or DIO) is exactly given by logR(ρ) + 1, and ∞ ρ = 1 (1),ε ρ⊗n . noticing the multiplicativity of R + 1, one can see that Cd,ρ−DIO( ) lim lim C ,ρ⊗n− ( ) (6) ε→0 n→∞ n d DIO the asymptotic zero-error cost of coherence dilution will be A simple application of Theorem 1 together with the quan- simply given by log[R(ρ) + 1]. tum Stein’s lemma [45,46] reveals that we have, in fact, ∞ ρ = ρ ρ Cd,ρ−DIO( ) D( ( )), that is, the relative entropy of C. General transformations and monotones D ρ ρ coherence ( ( )) characterizes the asymptotic rate of co- When discussing asymptotic state transformations, one is herence distillation under ρ-DIO. But it is already known that ρ → σ ∞ ρ = ρ ρ in particular interested in the largest rate R( ) at which under DIO, we also have Cd,DIO( ) D( ( )) [18,19], ρ σ ρ copies of can be transformed to copies of under the given which means that -DIO do not perform any better than class of operations. Our results can be used to show that the DIO in the asymptotic limit. Taking into consideration the ρ ρ rate of any such transformation under -DIO is completely operational gap between the operations DIO and -DIO in characterized by the relative entropy between the states and single-shot transformations, the asymptotic result can be quite their diagonals. surprising since it effectively shows that the advantage pro- ρ → ω → ρ To see this, notice first that any transformation vided by -DIO over DIO will be relatively minor and will σ such that the operation taking ρ to ω is ρ-DIO and the disappear completely at the asymptotic level. operation taking ω to σ is ω-DIO results in an overall protocol Finally, one can define the zero-error distillable coherence ρ → σ , which is ρ-DIO. This allows us to employ maxi- as mally coherent states m as an intermediary in coherence ∞, 1 , ⊗ ⊗m 0 ρ = (1) 0 ρ n . transformations. Using the fact that m =|++| ,we C ,ρ− ( ) lim C ,ρ⊗n− ( ) (7) 2 d DIO n→∞ n d DIO ∞ ρ can interpret the distillable coherence Cd,ρ−DIO( )asthe 0 ρ ρ ρ →|++| ρ Noting the additivity of DH ( ( )), from Eq. (3) we imme- rate R( ) under -DIO, and the coherence cost ∞,0 ∞ ρ =− ρ C , − (σ )astherate1/R(|+ +| → σ ) under |+ +|- diately get that Cd,ρ−DIO( ) log Tr ρ ( ). c m DIO DIO; a straightforward argument in analogy with Ref. [47] then shows the following. B. Dilution Corollary 2. For any states ρ and σ , the maximal rate of the Consider the transformation of a maximally coherent state asymptotic transformation ρ → σ under ρ-DIO operations is m into a general state ρ,usinga m-DIO protocol. The one- given by shot coherence cost is given by D(ρ (ρ)) (1),ε R(ρ → σ ) = . (10) C , − (ρ) D(σ (σ )) c m DIO As this is also true for DIO [19], asymptotically, ρ-DIO := log min m max F (( m ),ρ) 1 − ε . ∈ m−DIO provide no advantage whatsoever over DIO in any state transformation. To characterize this quantity, we will consider the coherence Furthermore, one can obtain various useful sufficient con- monotone based on the max-relative entropy between ρ and ditions for the transformations in the one-shot setting. For ρ ( )[12], given by instance, we show that the monotone R can also be used to characterize state transformations under ρ-DIO which go R(ρ):= min{λ | ρ (1 + λ)(ρ)}. beyond coherence distillation and dilution. It is easy to verify that R((ρ)) R(ρ) for any ρ-DIO Proposition 2. If R(σ ) + 1 1/ Tr ρ (ρ), then there operation . Using this quantity, we have the following. exists a ρ-DIO map such that (ρ) = σ . Theorem 3. The ε-error one-shot coherence cost under m- In the particularly simple case of single-qubit transforma- DIO is given by tions, we furthermore establish an equivalence of ρ-DIO and ,ε DIO. C(1) (ρ) c, m−DIO Proposition 3. For any single-qubit states ρ and σ ,the transformation ρ → σ is possible under ρ-DIO if and only = logmin{R(ω) + 1 | ω ∈ D, F (ρ,ω) 1 − ε}, if it is possible under DIO, which holds if and only if [12] ρ σ ρ σ where D denotes the set of all density matrices. R ( ) R ( ) and 1 1 .

013109-4 COHERENCE MANIPULATION WITH … PHYSICAL REVIEW RESEARCH 2, 013109 (2020) ρ = | |ρ| | restriction to a specific input. The results provide insight into Here, 1 i, j i j . We note from Ref. [12] that single-qubit DIO transformations have been shown to be the structure of the ultimate physical constraints on coherence equivalent to both MIO and SIO transformations, and our manipulation with free operations, and establish connections result thus extends this equivalence to ρ-DIO also. This does in the operational description of quantum coherence by quan- not hold beyond dimension 2, as we have demonstrated in titatively relating coherence manipulation to the distinguisha- Sec. IV A a transformation from a qutrit to a qubit system bility between a state ρ and its dephased version (ρ). achievable with ρ-DIO, but impossible under MIO and DIO. Note added. Recently, Wang and Wilde [52]aswellas Necessary conditions for single-shot ρ-DIO transforma- Buscemi et al. [53] studied the distinguishability of pairs of tions can be characterized by monotones under this class, states in an operational setting and independently obtained i.e., functions which obey the property that if there exists results which overlap with parts of this manuscript. Note, a transformation ρ → σ under ρ-DIO, then f (ρ) f (σ ). however, that although our characterization of ρ-DIO relies Some DIO monotones discussed, e.g., in Ref. [12], will in precisely on the distinguishability between ρ and (ρ), the fact also be ρ-DIO monotones—this includes R or the tasks of coherence distillation and dilution studied herein relative entropy D(ρ (ρ)). Indeed, any divergence which are different from the operational framework of Ref. [52] satisfies the data-processing inequality will form a ρ-DIO concerned with manipulating distinguishability. monotone. Importantly, this includes Rényi relative entropies 1 α 1−α α ρ ρ = ρ ρ D ( ( )) α−1 log Tr ( ) , but only in the range ACKNOWLEDGMENTS α ∈ [0, 2] [48], which contrasts with the set DIO for which all α give a valid monotone [12]. For a pure state, the Rényi We acknowledge discussions with Ludovico Lami. This relative entropies reduce to Dα (ψ (ψ)) = S2−α (ψ)[12], work was supported by the National Research Founda- γ γ ψ = ψ tion of Singapore Fellowship No. NRF-NRFF2016-02, the where S ( ) 1−γ log ( ) γ are the Rényi entropies. National Research Foundation and L’Agence Nationale de This shows, in particular, that p norms of the squared moduli of the coefficients of a pure state are ψ-DIO monotones for p la Recherche joint Project No. NRF2017-NRFANR004 Van- in the range p ∈ [0, 1] and reverse monotones for p ∈ [1, 2]. QuTe, the program for FRIAS-Nagoya IAR Joint Project An outstanding question is whether such monotones form a Group, and the Japan Society for the Promotion of Science complete set, in the sense that the inequality f (ρ) f (σ )for (JSPS) KAKENHI Grant No. 19H04066. each monotone f implies that there exists a ρ-DIO transformation taking ρ to σ. Although a complete set of infinitely APPENDIX A: PURE-STATE TRANSFORMATIONS many monotones can be defined [49–51], it is unclear if UNDER DIO there exists a finite set of conditions fully characterizing Let {|x} = ... ⊂ Hin and {|y} = ... ⊂ Hout be the in- transformations under ρ-DIO. x 1 din y 1 dout coherent bases on the input and output Hilbert spaces, respec- tively. V. DISCUSSION Definition 1. DIO: A channel E : L(Hin ) → L(Hout )isa In this work, we tackled the fundamental question of how dephasing- covariant incoherent operation (DIO) if ◦ E = to efficiently manipulate the resource of quantum coherence E ◦ , where (·) is the dephasing channel with respect to the under operations which do not use coherence and thus, to a incoherent basis on the corresponding system. classical observer, appear classical. We studied this question We now cast the DIO property of a channel in terms of its E · = n · † under two classes of channels: DIO, respecting dephasing co- Kraus operator representations. Let ( ) i=1 Ki( )Ki ;we variance for any input state, and ρ-DIO, tailored to a specific do not have to worry about the value of n. The equivalence of input. We first shed light on the operational power of DIO ◦ E and E ◦ can be translated to the equality of their Choi and explicitly characterized pure-state transformations under operators: this class, revealing a relation between DIO and majorization theory and thereby connecting DIO to several other classes id ⊗ [ ◦ E] |x1x1x2x2| of free operations. To push the characterization of coherence x1,x2 manipulation under coherence nondetecting operations to its very limit, we introduced the class of operations ρ-DIO and = id ⊗ [E ◦ ] |x x x x | investigated the advantages that this extension provides. We 1 1 2 2 x ,x showed, in particular, that even though the coherence of the 1 2 ρ ⇒ | |⊗ | | | | †| | input state is not detected, -DIO allow for transformations x1 x2 ( y y Ki x1 x2 Ki y y ) impossible under DIO: they can distill more coherence than i,x1,x2,y DIO in the one-shot setting. Despite ρ-DIO constituting a = | |⊗ | | † . significant relaxation of the constraints of DIO, the increased x x (Ki x x Ki ) (A1) capabilities of such channels are limited to nonasymptotic i,x regimes—we showed the advantages to disappear completely This leads to the following handy properties of any Kraus at the asymptotic level, where both sets of operations achieve operator representation of a DIO: the same performance in all transformations. This sug- Observation 1. Define the vectors K(y, x) ∈ V ≡ Cn, gests that the simpler class ρ-DIO closely approximates the dindout in number as follows: performance of all DIO and no significant operational advantage can be obtained by tailoring the coherence nondetecting K(y, x):= (y|K1|x, y|K2|x,...,y|Kn|x). (A2)

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Also deﬁne Sy|x :=K(y, x), K(y, x), where ·, · is the stan- be bistochastic. A non-negative real matrix is sub- *stochastic dard Hermitian inner product on Cn. Then, the following con- if another non-negative matrix can be added to it to make it ditions together capture the DIO property of the completely *stochastic (“*” can stand for “column-”, “row-”, “bi-”, or the E · = n · † positive (CP) map ( ) i=1 Ki( )Ki : absence of any of these qualiﬁers). (1) Diagonal (i.e., incoherent) input produces diagonal For the purposes of proving the above theorem, we will use , , , = δ output on average: K(y x) K(y1 x) Sy|x yy1 . the following properties of majorization (see, e.g., [54]): (2) Diagonal- free input produces diagonal- free output on (1) For normalized probability vectors u and v, the relation average: K(y, x), K(y, x )=S | δ . u ≺ v holds if and only if there exists a bistochastic matrix T 1 y x xx1 dout = ∈, such that u = T v. (3) Trace is preserved: y=1 Sy|x 1 for all x {1, 2 ...,d }; in other words, the matrix S with components (2) For non-negative vectors u and v, if there exists a sub- in = Syx := Sy|x is column stochastic (justifying the “conditional” bistochastic matrix T such that u T v, then v is said to notation). weakly majorize u; weak majorization between normalized It is important to bear in mind that each of these conditions probability vectors implies nonweak majorization (i.e., the involves summing over all the n Kraus operators. In particular, previous condition). if the “on-average” condition (1) was tightened to apply to Proof. Since the overall output is the pure state φ,the each Kraus operator separately while removing condition (2) output of each individual Kraus operator must necessarily be altogether, the resulting conditions would characterize the proportional to φ. In other words, there exists a normalized class of incoherent operations (IO). Likewise, if both (1) and cˆ ∈ Cn such that

(2) were tightened to apply to each Kraus operator, we would din have strictly incoherent operations (SIO). , μ = ν Ki(y x1 ) x1 ci y (A5) For convenience, we also deﬁne the normalized vectors x =1 κ , = √1 , κ , , κ , =δ 1 ˆ (y x): K(y x), leading to ˆ (y x) ˆ (y1 x) yy1 Sy|x ∀ , , κ , , κ , =δ = i y. Multiplying both sides by Ki(y x) and summing over i, and ˆ (y x) ˆ (y x1) xx1 . In the cases where Sy|x 0, just deﬁne κˆ (y, x) to be some unit vector orthogonal to the rest, din n n μ , , = , ν suitably expanding the space. x1 Ki(y x)Ki(y x1 ) Ki(y x)ci y

x1=1 i=1 i=1

1. Deterministic pure- to- pure state conversion din ⇒ μ δ = , , ν Now consider the problem of determining the conditions x1 Sy|x xx1 K(y x) cˆ y = under which there is a DIO deterministically mapping a given x1 1 pure state ψ to another, φ. We shall prove the following ⇒ μx Sy|x =κˆ (y, x), cˆνy theorem. Say the DIO given by K(y, x) achieves the desired transformation. Then, dout dout 2 2 2 2 |ψ= μ | ⇒ px ≡|μx| =|μx| Sy|x = |κˆ (y, x), cˆ| |νy| Theorem 4. A given initial state x x x can be y= y= mapped deterministically to a given target state |φ=νy|y 1 1 by a DIO if and only if the majorization relation dout ≡ |κ , , |2 . p ≺ q (A3) ˆ(y x) cˆ qy (A6) y=1 2 2 holds, where px :=|μx| and qy :=|νy| . Remark 1. Note on majorization. For a vector v on a The second line above follows from condition (2); the fol- finite- dimensional real vector space, define v↓ as the vector lowing line by dividing throughout by Sy|x and applying the κ , whose components are the components of v arranged in non definition of ˆ (y x); in the last line, we just sum the previous increasing order. For example, (2, −5, 2, 4)↓ = (4, 2, 2, −5). line’s expressions over y and use the stochasticity of S. =|κ , , |2 For a pair of vectors (u, v), the majorization relation u ≺ v Now define Tx|y : ˆ (y x) cˆ . The normalization of {κ , }din (“v majorizes u”) is then defined as the conjunction of the cˆ and orthonormality of ˆ(y x) x=1 (for each y) and {κ , }dout following conditions: ˆ (y x) y=1 (for each x) together imply that T is sub- bis- tochastic. This implies that p is weakly majorized by q; ↓ ↓ u1 v1 ; normalization of the distributions implies (nonweak) ma- ↓ + ↓ ↓ + ↓ jorization. Incidentally, the same normalization arguments u1 u2 v1 v2 ; also imply that n = din suffices. . Thus, majorization of the input coherence distribution by . the output is necessary for the existence of a DIO determinis- ↓ + ↓ ···+ ↓ ↓ + ↓ ···+ ↓, tically mapping ψ → φ. Since this condition is already known u1 u2 ud v1 v2 vd (A4) to be sufficient for the existence of such an SIO, its sufficiency where d is the larger of the dimensionalities of u and v (we for the existence of such a DIO follows. append the shorter vector with a suitable number of trailing zeros). A real matrix with non-negative entries is column stochas- 2. Probabilistic pure- to- pure state conversion tic (row stochastic) if each of its columns (rows) adds up to A possible definition of probabilistic conversion of a given 1; a matrix that is both row and column stochastic is said to state |ψ to an ensemble {(η j, |φ j)} under a class of op-

013109-6 COHERENCE MANIPULATION WITH … PHYSICAL REVIEW RESEARCH 2, 013109 (2020) erations is one where a channel ≡{Kj} belonging to the From the properties of the majorization relation, we have ψ † = η φ φ class satisfies Kj Kj j j (possibly with some of the j ↓ weak≺ η ↓. mutual duplicates). This definition is not easily amenable to p j jq j (A14) the treatment of the previous section, owing to the lack of individual Kraus operator-based constraints in DIO. This is Summing over j yields in contrast with IO and SIO, whose definitions constrain each Kraus operator. Extension of our results to the pure- state- to- ↓ weak≺ η ↓, p j jq j (A15) mixed- state case is hindered by this obstacle. j j Nevertheless, we can say something useful about heralded probabilistic conversion from a pure state |ψ to an ensemble which strengthens under normalization considerations to non- { η , |φ } φ = ≺ ↓ ( j j ) . This entails that a conversion to j be heralded weak majorization. But, p j p j j p j . Therefore, by a correlated “flag” system whose state unambiguously ↓ identifies j. In other words, we require a DIO to map ψ to p ≺ η q . (A16) η σ ⊗ φ σ j j j j j j, with the “flag states” j on the first subsystem j unambiguously distinguishable. We might as well set these to some mutually orthogonal | j j| without loss of generality. Again, the converse follows from the corresponding result To keep the game fair, we shall require | j to constitute the about SIO. axiomatic incoherent basis for the flag system. Proposition 4. There exists a DIO effecting the transforma- ψ → η | |⊗φ APPENDIX B: ρ-DIO TRANSFORMATIONS tion j j j j j if and only if 1. Distillation ≺ η ↓, p jq j (A7) Consider the rate of distillation of coherence under the j operations ρ-DIO, i.e., channels such that ◦ (ρ) = ◦ (ρ) for some fixed ρ. We will use A, B=Tr(A†B)for j =|ν j|2 |φ = ν j| where p is as before and qy : y for j y y y . the Hilbert-Schmidt inner product. Proof. Without loss of generality, we can decompose the Theorem 5. The one-shot distillable coherence under ρ- requisite DIO using Kraus operators of the form DIO for any input state ρ is given by (1),ε ε Kj,m = Kj,m( j, y; x)| j⊗|yx|, (A8) ρ = ρ ρ , Cd,ρ−DIO( ) DH ( ( )) log x,y = x with where x log : log 2 . Proof. The ε-error one-shot rate of distillation can be ψ † = η | |⊗φ . Kj,m Kj,m j j j j (A9) expressed as m (1),ε ρ = { ∈ | ρ, − ε}, Cd,ρ−DIO( ): log max m N Fρ−DIO( m) 1 Adapting the notation of the previous section, the DIO condi- (B1) tions can be cast as where Fρ− is the achievable fidelity of distillation, i.e., , , , = δ δ DIO (1) K( j y; x) K( j1 y1; x) S j,y|x jj1 yy1 . , , , = , | δ ρ, = ρ , , (2) K( j y; x) K( j y; x1 ) S j y x xx1 . Fρ−DIO( m): max F ( ( ) m ) (B2) = ∈ρ−DIO (3) j,y S j,y|x 1 for all x. In fact, considering the form (A8) allows us to strengthen with m the m-dimensional maximally coherent state. condition (2) above to 1 d! † T · = π · Defining the twirling ( ) d! i=1 U (i) Uπ(i), where , , , = δ , each π(i) is a permutation of the basis vectors, we have that K( j y; x) K( j y; x1 ) j S j,y|x xx1 (A10) where u, v := u , v , for u, v in the abstract vector Fρ− (ρ,m) = max F (T ◦ (ρ), T( )) j m j m j m DIO ∈ρ− m space V defined above. The rest of our proof to Theorem 1 DIO can be applied as such, but with all such vectors restricted to = max F (T ◦ (ρ), m ), (B3) ∈ρ−DIO the subspace corresponding to a specific j. This leads to that is, it suffices to optimize over twirled maps of the form = |κ , , j|2 j, px S j,y|x ˆ ( j y; x) c qy (A11) = T ◦ . Due to permutation invariance, the output of y y such a map must satisfy i|(·)|i=π(i)|(·)|π(i)∀i and | · | =π | · |π ∀ = π where i ( ) j (i) ( ) ( j) i j, where is an arbitrary permutation; imposing these constraints, we can write any |ψ= j | ⊗|φ , Kj,m cm j j (A12) such map as 2 − with η = |c j | . Thus, 1 m j m m (Q) =X, Q m +1 − X, Q , (B4) m − 1 weak j p j ≡ px S j,y|x ≺ η jq . (A13) for some operator X. The complete positivity of is equiv- y x alent to the condition 0 X 1, and to further impose that

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∈ ρ-DIO, we need that This simplifies in particular for the case of a pure state |ψ= ψ | i i i : 1 = ◦ (ρ) (1),0 ψ = ψ| ψ |ψ−1 m Cd,ψ−DIO( ) log ( ) = ◦ ρ − ( ) = log (Tr (ψ)2 ) 1 − ⎢ ⎥ 1 m ⎢ −1⎥ =X,(ρ) m + (1 −X,(ρ)) , (B5) ⎢ ⎥ m − 1 ⎣ 4 ⎦ = log |ψi| . (B12) , ρ = 1 i which is satisfied if and only if X ( ) m . Altogether, we have Explicitly, we have that 4 1 Fρ−DIO(ρ,m) ψ → ⇐⇒ |ψ | . (B13) m i m i 1 = max X,ρ 0 X 1, X,(ρ)= . (B6) m |ψ = 5 , 3 , 3 T Notice that the state : 8 16 16 from the main text gives an explicit example of a case where coherence Although originally defined for m ∈ N, we extend the defini- monotones under IO and DIO can increase in the ρ-DIO trans- tion of Fρ− to any m 1asinEq.(B6). The one-shot rate DIO formation: specifically, consider the monotone C (|ψ) = of distillation is then 2 d |ψ↓|2 ψ↓ |ψ i=2 i , where i denote coefficients of arranged in ,ε C(1) (ρ) nonincreasing order by magnitude [31,55]. For this monotone, d,ρ−DIO |ψ |ψ = 3 |+ = 1 with as above, we have C2( ) 8 but C2( ) 2 . = log max{m ∈ R | Fρ−DIO(ρ,m) 1 − ε}log Finally, we remark the additivity of − log Tr ρ (ρ)inthe = − { , ρ | ,ρ − ε, } sense that log min X ( ) X 1 0 X 1 log ⊗n ⊗n ⊗n = ε ρ ρ , − log ρ⊗n ,(ρ )=−log ρ ,(ρ) DH ( ( )) log (B7) =−n log ρ ,(ρ), (B14) where Dε is the hypothesis testing relative entropy. H which, in particular, gives the asymptotic zero-error distillable By considering the asymptotic scenario, we then have coherence as ∞, ∞ 1 ,ε ⊗ 0 ρ = (1) ρ n C ,ρ− (ρ) =−log ρ ,(ρ). (B15) Cd,ρ−DIO( ) lim lim C ,ρ⊗n− ( ) d DIO ε→0 n→∞ n d DIO = ρ ρ , D( ( )) (B8) 2. Dilution by quantum Stein’s lemma [45]. Theorem 6. The one-shot coherence cost under m-DIO Recall from Ref. [18]thatforDIO,wehave operations is given by (1), C ,ψ (ρ) = log min{R(ω) + 1|ω ∈ D, F (ρ,ω) 1 − }. min Dε (ρ X ) c (1),ε H C , (ρ) = X = (X ) (B9) Proof. Using a twirling argument similar to the distillation d DIO Tr X = 1 log case, we can, without loss of generality, limit ourselves to operations of the form = ◦ T, which take the form and (Q) = m, QX +1 − m, QZ (B16) ∞ ρ = ρ ρ , Cd,DIO( ) D( ( )) (B10) for some operators X, Z. The complete positivity and trace preservation of impose that X, Z are valid quantum states. which shows that although ρ-DIO can have a larger rate of To impose that ∈ m-DIO, we have that one-shot distillation than DIO, asymptotically the operations 1 m − 1 have the same capabilities in distillation. (X ) = ◦ ( ) = ◦ ( ) = X + Z. m m m m Exact distillation (B17)

Now consider the case of zero-error distillation, that is, ex- Noticing that ( m ) = X, this means that the set of states act transformations ρ → m under ρ−DIO. From the above, ω such that ω = ( m )forsome m-DIO protocol is we have that the rate of zero-error distillation is given by precisely the set of states for which there exists a state σ such 1 ω + m−1 σ = ω that m m ( ). It is easy to see that this is only 1 σ = ω (1),0 ρ = 0 ρ ρ = , possible if ( ) ( ). Deﬁning the function Cd,ρ−DIO( ) DH ( ( )) log log ρ ,(ρ) ω + λσ g(ω):= min λ ∈ I,σ∈ D,(σ ) = (ω) , (B11) 1 + λ where ρ is the projection onto the support of ρ. (B18)

013109-8 COHERENCE MANIPULATION WITH … PHYSICAL REVIEW RESEARCH 2, 013109 (2020) it is not difficult to show that, in fact, g(ω) = R(ω)[12]. We any λ R(ω) + 1, there exists a state σ such that then have that the one-shot coherence cost under m-DIO is ω + (λ − 1)σ = λ(ω). By assumption, there exists, in σ ∈ (1),ε ρ particular, D, which satisfies Cc, −DIO( ) m 1 1 = logmin{m ∈ R | ω ∈ D, F (ρ,ω) ω + − 1 σ = (ω). (B24) ρ ,(ρ) ρ ,(ρ) 1 − ε, R(ω) m − 1} With this choice of σ , define the map = logmin{R(ω) + 1 | ω ∈ D, F (ρ,ω) 1 − ε}, (X ) = ρ , Xω +1 − ρ , Xσ. (B25) (B19) This map is clearly completely positive and trace preserving as required. (CPTP), and we have ρ = ω, Exact dilution ( ) ρ = , ρ ω + − , ρ σ Zero-error dilution can be characterized straightforwardly, ( ( )) ρ ( ) [1 ρ ( ) ] as the required coherence cost is simply = , ρ ω + 1 − σ ρ ( ) , ρ 1 (1),0 ρ ( ) C , − (ρ) =log[R(ρ) + 1]. (B20) c m DIO = (ω), (B26) Note that as required. −1/2 −1/2 R(ρ) = (ρ) ρ(ρ) ∞ − 1, (B21) The above condition is, in general, not necessary for ρ- DIO transformations, and indeed, in general, R(ρ) + 1 > which makes this quantity easy to express and compute. 1 , which means that even the trivial transformation ρ ρ ,(ρ) In particular, noting that for every state we have ρ → ρ rank((ρ))(ρ)[56], this straightforwardly shows that the might not satisfy the condition of the proposition. When the input state is a qubit, however, the transforma- maximally coherent state m can be transformed into any other state with rank (ρ) m, and it acts as a “golden tions can be exactly characterized (see, also, [57,58]). We will, in particular, establish an equivalence between ρ-DIO unit” under the operations m-DIO. For a pure state, we have exactly [12] and DIO in such transformations. Proposition 6. For any single-qubit states ρ and σ ,the R(ψ) = rank (ψ) − 1, (B22) transformation ρ → σ is possible under ρ-DIO if and only if it is possible under DIO. which further simplifies the characterization. Proof. Clearly, if a DIO transformation ρ → σ exists, then We note the additivity of log[R(ρ) + 1], immediately so does a ρ-DIO transformation by the inclusion DIO ⊆ ρ− establishing that the asymptotic zero-error coherence cost DIO. By Theorem 30 in Ref. [12], the DIO transformation is under m-DIO is given by ρ σ ρ σ possible if and only if R ( ) R ( ) and 1 1 . ∞,0 ρ C , − (ρ) = log[R(ρ) + 1]. (B23) Since R is trivially a -DIO monotone, as discussed earlier, it c m DIO · ρ suffices to show that 1 is a -DIO monotone, which will mean that the existence of a ρ-DIO transformation implies the 3. General transformations existence of a DIO transformation. We can generalize the approach of coherence dilution ρ − ρ To see that this is indeed true, note that ( ) 1 is to give a simple sufficient condition for transformations to clearly a ρ-DIO monotone due to the contractivity of the general states. For any ω ∈ D, we have the following: trace distance under CPTP maps. But for a single-qubit state 1 Proposition 5. If R(ω) + 1 , then there exists ρ ρ ρ ,(ρ) 00 01 ρ = ∗ ,wehave ρ − (ρ) = 2|ρ |= ρ − ρ ρ = ω ρ ρ 1 01 1 a -DIO map such that ( ) . 01 11 ω = ρ ρ Proof. Recalling that R ( ) 1, which means that 1 is also a -DIO monotone for all min {λ | ω (λ + 1)(ω)}, it is easy to see that for single-qubit states.

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