Coherence of operations and interferometry
CCoherence of operations and interferometry
Michele Masini,
1, 2, ∗ Thomas Theurer, † and Martin B. Plenio ‡ Laboratoire d’Information Quantique, Universit´e Libre de Bruxelles, 1050 Bruxelles, Belgium Institute of Theoretical Physics and IQST, Universit¨at Ulm,Albert-Einstein-Allee 11, D-89069 Ulm, Germany (Dated: February 10, 2021)Quantum coherence is one of the key features that fuels applications for which quantum mechanicsexceeds the power of classical physics. This explains the considerable efforts that were undertakento quantify coherence via quantum resource theories. An application of the resulting framework toconcrete technological tasks is however largely missing. Here, we address this problem and connectthe ability of an operation to detect or create coherence to the performance of interferometricexperiments.
I. INTRODUCTION
The emergence of quantum technologies that outper-form their classical counterparts [1–4] led to the insightthat properties in which quantum mechanics departsfrom classical physics are not only of foundational inter-est, but also of practical relevance. Consequently, theseproperties are now considered resources that can lead tooperational advantages. To understand precisely whichquantum property is responsible for what advantage andhow to employ them optimally is the motivation for thedevelopment of various quantum resource theories [5–23].Until recently, the focus was mainly on the quantificationof resources present in quantum states, which is achievedwith the help of static resource theories. The value ofoperations was determined indirectly by, e.g., their re-source generation capacity [24–28], which describes howmuch static resources they can create, or their resourcecost [29–31], i.e., the amount of static resources that isneeded to simulate them.Only recently, the direct quantification of resourcespresent in quantum operations started with the devel-opment of dynamical resource theories [32–46]. Com-pared to static resource theories, they show several ad-vantages [33]. Firstly, quantum technologies intend to ac-complish tasks that cannot be carried out with classicaldevices, and such devices are described by quantum op-erations. From a conceptual point of view, it seems thusmore natural to quantify the value of operations directlywithout a detour through states. Secondly, dynamical re-source theories are a unifying concept and include staticresource theories as a special case as the preparation ofa state is a specific quantum operation. Thirdly, theyoften allow for an operational treatment of subselection,i.e., the ability to subselect in one operation but not inanother can be naturally reflected in the framework. Fi-nally, dynamical resource theories can be used to quantifyproperties of quantum operations that cannot be cap- ∗ [email protected] † [email protected] ‡ [email protected] tured by the indirect methods mentioned above. One ofthese properties is the ability to detect coherence in thesense that its presence makes a difference in measure-ment statistics [47, 48], which is a necessary prerequisiteto its exploitation. It is therefore equally important toinvestigate how well an operation can create and detectcoherence, which was quantified both theoretically andexperimentally in Refs. [33, 49]. | (cid:105) | (cid:105)| (cid:105) | (cid:105) sample 1 sample 0 L.S. 0 L.S. 1
Beam splitter 1 Beam splitter 2
D. 0 D. 1FIG. 1. Mach-Zehnder interferometer. Two light sources L.S.0 and L.S. 1 illuminate the two input ports of a beam splitter.Its two output modes experience different phases, before theyreach a second beam splitter, after which they are measuredby the detectors D. 0 and D. 1.
Whilst coherence is undoubtedly central to the depar-ture from classical physics and underlies other quantumresources such as entanglement, it is important to inves-tigate its relevance in concrete applications [50–52]. InRef. [53], the authors established a direct connection be-tween static coherence [19, 54] and interferometry (seealso Refs. [55, 56]). Here, we go one step further and ap-ply dynamical resource theories to interferometers (seeFig. 1), which allows us to give an operational meaningto both the ability to detect and to create coherence.After a quick introduction of the relevant resource the-ories in Sec. II A, we describe in Sec. II B the multi- a r X i v : . [ qu a n t - ph ] F e b path interferometers [53] that we are going to analyze.In Sec. III, we then present our main results and con-struct families of measures that have a direct operationalinterpretation: they are not only proper quantifiers ofresourcefulness, but also determine the advantage thatdynamical coherence grants in concrete interferometricsetups. We then conclude in Sec. IV. II. DEFINITIONS
In the following, we recall some definitions and resultsthat will be used in the remainder of this article.
A. Resource theoretical setting
Firstly, we shortly review the dynamical resource the-ories that we will utilize throughout the article, but referto the original publication [33] for full details. Channelresource theories are defined by two ingredients, the setof free channels and the set of free superchannels . Whilea quantum channel is a linear map from quantum statesto quantum states, a superchannel S is a linear map fromquantum channels C to quantum channels C (cid:48) that has aphysical realization C (cid:48) = S ( C ) := D ( C ⊗ id) E , (1)where id is the identity channel and D and E are quantumchannels [57]. We will use the notation C B ← A when weneed to specify that a quantum channel has an inputsystem A and an output system B or simply write C A when input and output system are the same.In this work, we employ one resource theory to quan-tify the ability of an operation to detect coherence andanother one to describe its ability to create coherence.Firstly, for every finite dimensional system under consid-eration, we fix an incohererent basis, i.e., an orthonormalbasis {| i (cid:105)} i =0 ,...,M − and define total dephasing with re-spect to it as the operation ∆ that acts on every state σ as ∆( σ ) := (cid:88) i | i (cid:105)(cid:104) i | σ | i (cid:105)(cid:104) i | . (2)A quantum state ρ is called incoherent if it is diagonal inthe incoherent basis, i.e., if ∆ ρ = ρ .A measurement described by a positive operator-valued measure (POVM) with elements P ( n ) ≥ (cid:80) n P ( n ) = cannot detect coherence if its outcomestatistics solely depend on the populations of the statesto which it is applied. This is exactly the case if it isdiagonal in the incoherent basis [33], i.e., of the form P ( n ) = (cid:88) i P ( n ) i | i (cid:105)(cid:104) i | , (3)where {| i (cid:105)} is the incoherent basis. We will denote the setof all incoherent POVMs by P I . To extend this definition to arbitrary instruments, we must handle subselectionconsistently: for a quantum instrument X that allowsus to apply subselection according to a variable x , i.e.,with probability p x = tr( X x ( ρ )), we obtain an output ρ x = X x ( ρ ), we define a corresponding channel˜ X ( ρ ) = (cid:88) x X x ( ρ ) ⊗ | x (cid:105)(cid:104) x | . Formally, we thus store the outcome x in the incoherentbasis of an auxiliary system, from which we can extractit at a later point using an incoherent POVM. This al-lows us to reduce our analysis to channels and has theadditional advantage that we treat subselection in an op-erational manner. A channel is then unable to detectcoherence if it maps incoherent POVMs to incoherentPOVMs in the sense that the populations of its outputare independent of the coherences of its input. In our re-source theory that describes the detection of coherence,the set of free channels is therefore given by all operationsthat satisfy [33, 58, 59]∆ C = ∆ C ∆ . (4)We denote this set with DI .To describe the ability to create coherence, we choosethe maximally incoherent operations ( MIO ) [8, 58, 60]as free. Treating subselection as above, these are all chan-nels C that satisfy C ∆ = ∆ C ∆ . (5)As the name suggests, this is the maximal set of channelsthat maps incoherent states to incoherent states, i.e., thatcannot create coherence, which is why it is consideredfree when we investigate the creation of coherence as aresource.As free superchannels, we chose the ones with a decom-position as in Eq. (1) where both D and E are free chan-nels (elements of MIO in the creation incoherent settingor elements of DI in the detection incoherent setting).Thanks to the previously defined sets, we are now ableto quantify the ability of an operation to detect or cre-ate coherence. In a channel resource theoretical setting,a resource measure M ( C ) is a functional from quantumchannels to real numbers that satisfies the following prop-erties. • Nullity : If C is a free operation, then M ( C ) = 0. • Non-negativity : M ( C ) is non-negative. • Monotonicity : M ( C ) is monotonic under free su-peroperations, i.e., M ( S ( C )) ≤ M ( C ) for all freesuperchannels S . This condition is equivalent tothe simultaneous satisfaction of three simpler con-ditions, namely monotonicity under left and rightcomposition and monotonicity under tensor prod-ucts [33, Prop. 12]. If we denote with F the set offree channels the three conditions read M ( C ) ≥ M ( FC ) ∀C ∀F ∈ F , (6) M ( C ) ≥ M ( CF ) ∀C ∀F ∈ F , (7) M ( C ) ≥ M ( C ⊗ id Z ) ∀C ∀ Z. (8)Moreover, one often examines two additional propertiesthat are convenient but not necessary to consider M ( C )a proper resource measure [37, 61], namely • Faithfulness : M ( C ) = 0 exactly if C is a free chan-nel. • Convexity : For all t with 0 ≤ t ≤ C α , C β , M ( t C α +(1 − t ) C β ) ≤ tM ( C α )+(1 − t ) M ( C β ) . (9)We will use the term convex measure for functionalsthat are convex and qualify as resource measures. B. Multi-path interferometer
Next we introduce the idealized multi-path interfer-ometer [53] that we will investigate in this paper. Toanalyze the role of dynamical coherence in interferome-try, we begin with an incoherent state τ , as depicted inFig. 2. We then apply a quantum channel C to it, which sample i sample 0sample 1sample M − C C τ incoherent incoherent FIG. 2. Sketch of a multi-path interferometer. The quantumchannel C distributes an incoherent state τ to M differentpaths that can lead to different phases via, e.g., interactionswith different samples. The paths are recombined by a secondchannel C , and its output is incoherently measured. distributes τ to M different paths of the interferometerrepresented by the orthonormal basis | (cid:105) , . . . , | M − (cid:105) .Samples in the paths or different path lengths lead torelative phases which we describe by the application ofthe unitary channel Λ (cid:126)φ defined asΛ (cid:126)φ ( σ ) := (cid:88) i,j e i ( φ i − φ j ) | i (cid:105)(cid:104) i | σ | j (cid:105)(cid:104) j | , (10)where (cid:126)φ refers to the phases { φ i } . Afterwards, a channel C recombines the paths before a final incoherent mea-surement. The channel C is thus the generalization of beam splitter 1 in the Mach-Zehnder interferometer rep-resented in Fig. 1, C generalizes beam splitter 2, andthe incoherent measurement the two detectors. The goalof interferometry is now to deduce information about therelative phases from the measurement outcome. To makeour setting non-trivial, we assume from here on that thereexists at least one pair k, l such that φ k (cid:54) = φ l .Intuitively, the ability of C to generate coherence andof C to detect it will then affect how sensitive the mea-surement outcome is to relative phases: if C cannot cre-ate coherence, the state Λ (cid:126)φ C ( τ ) is independent of (cid:126)φ . Onthe other hand, if C cannot detect coherence as definedin Sec. II A, the final outcome of the incoherent measure-ment will be independent of the relative phases too. Inthe following, we will make these intuitions rigorous. III. MAIN RESULTS
In this section, we investigate the connection betweena channel’s ability to detect or create coherence and itsusefulness in interferometry in detail. To this end, weconsider the setup described in Sec. II B and depictedin Fig. 2. Since the detection and creation of coherenceare two different resources, we will treat them separately,beginning with the former.
A. Detecting coherence and interferometry
To investigate which role the detection of coherenceplays in interferometry, we analyze a setting that is bestdescribed in terms of a game between two parties, Aliceand Bob (see Fig. 3): Bob prepares a quantum state ρ and sends it to Alice. Alice applies to this state with Alice Bob incoherentBob ρ λ id + µ Λ (cid:126)φ E C
FIG. 3. Schematic representation of the game played by Aliceand Bob that is used to describe the role of the detection ofcoherence in interferometry. probability µ a channel Λ (cid:126)φ introduced in Eq. (10), andotherwise, with probability λ = 1 − µ , she leaves it un-changed. She then sends the state back to Bob. His taskis to guess if Alice applied Λ (cid:126)φ or not. To do this, he isallowed to first apply an arbitrary detection incoherentoperation E to the state he retrieved, followed by a fixedchannel C , and an incoherent measurement of his choice.Based on its outcome, he then announces his guess. As-suming that Bob knows λ and the phases (cid:126)φ , and that heuses the optimal state ρ , the best pre-processing E , andthe optimal incoherent measurement, his probability ofguessing correctly is given by [33, Prop. 17], p max λ,(cid:126)φ ( C ) = 12 + 12 max E∈DI (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ CE (cid:16) λ − µ Λ (cid:126)φ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (11)where ||G|| denotes the induced trace norm of the oper-ation G . Here and in the following, we always implicitlyassume that the in- and output dimensions of operationsand states that we connect fit, i.e., the state ρ that Bobsends to Alice is an element of the Hilbert space on whichΛ (cid:126)φ acts, and the maximization is understood to run overall detection incoherent channels E with in- and outputspaces determined by the ones of C and Λ (cid:126)φ .If Bob had no access to C , his measurement outcomewould not depend on whether Alice applied Λ (cid:126)φ or not,because the combination of E and the incoherent mea-surement alone is not sensitive to the changes that Λ (cid:126)φ induces. His best strategy would thus be to bet purelybased on his knowledge of λ . The increase in his prob-ability of guessing correctly that C provides is thereforegiven by the functionals M λ,(cid:126)φ ( C ) := max E∈DI (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ CE (cid:16) λ − µ Λ (cid:126)φ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − | λ − µ | , (12)which we will call pre-processed improvements . Thesefunctionals define a family of convex measures in the de-tection incoherent setting, which is the content of thefollowing Theorem. Its proof can be found in App. E,where we also provide the other proofs of the results pre-sented in the main text. Theorem 1.
The functionals M λ,(cid:126)φ ( C ) are convex mea-sures in the detection incoherent setting for all λ ∈ [0 , and for all (cid:126)φ ∈ R M . We show in App. A that without the optimal pre-processing E , the functionals in Eq. (12) are in generalnot proper measures in the detection incoherent setting.On the other hand, adding a free post-processing afterthe channel C does not increase Bob’s chances of success.This is a direct consequence of the fact that a detectionincoherent operation cannot convert an incoherent mea-surement to a coherent one, and we can thus absorb anyfree post-processing into the incoherent measurement.Together with the fact that M λ,(cid:126)φ ( C ) = M λ,(cid:126)φ ( C ⊗ id),which we show in the proof of Thm. 1, this implies thatreplacing the optimal free pre-processing E by a free pre-and post-processing as well as a memory channel, i.e., EC → ˜ E ( C ⊗ id) G , ˜ E , G ∈ DI , does not increase Bob’s chances of success.In discrimination games similar to the one that we dis-cuss here, it is frequently the case that the correct usageof auxiliary systems increases the chances of success [62,Chap 3.3]. Therefore, one might assume that it is bene-ficial for Bob to prepare a correlated state of a composedsystem AZ and hand only a part of it, i.e., A , to Alice(see Fig. 4 for the adapted protocol). This is however nottrue. Alice Bob incoherentBob ρ AZ λ id A + µ Λ A(cid:126)φ
E C id Z FIG. 4. Sketch of a potential method with which Bob mightincrease his chances in the guessing game represented inFig. 3. He prepares a correlated state and hands only a sub-system to Alice.
Theorem 2.
An auxiliary system does not increase thepre-processed improvements, i.e., for (cid:126) ˜ φ ( AZ ) such that Λ (cid:126) ˜ φ ( AZ ) := Λ A(cid:126)φ ⊗ id Z , it holds that M λ,(cid:126) ˜ φ ( AZ ) ( C ) = M λ,(cid:126)φ ( C ) ∀C ∀ λ ∀ (cid:126)φ ∀ Z. These results imply that the pre-processed improve-ments describe the maximal usefulness of an operation’sability to detect coherence in our guessing games: if wehave access to C and are allowed to combine it with arbi-trary operations that cannot detect coherence, but withnone that can, the pre-preocessed improvements quantifythe advantage that C grants. The pre-processed improve-ments are thus resource measures with a clear operationalinterpretation in terms of the games.Moreover, these guessing games are directly connectedto our interferometric setup: Bob’s preparation of the ar-bitrary ρ corresponds to a choice of the incoherent τ aswell as C in Fig. 2, and C represents the combination of E and C . Since we are only interested in C ( τ ), but notin C alone, we can always choose a pair C and τ suchthat C does not detect coherence ( C ( τ ) = ρ tr( τ ), where ρ is our optimal state). The pre-processed improvements M λ,(cid:126)φ ( C ) therefore describe the maximal usefulness of anoperation’s ability to detect coherence in a concrete in-terferometric task, i.e., deciding if, e.g., a set of sampleswas present or not. Since the pre-processed improve-ments are also valid resource measures in the detectionincoherent setting, we showed one of the main results ofthis paper: the ability to detect coherence is a resourcein interferometry.As we pointed out in Sec. II A, another desirable prop-erty for measures is faithfulness. In our operational set-ting, it would ensure that every non-free channel is atleast a little helpful for the interferometric task thatwe intend to accomplish. Faithfulness of the function-als M λ,(cid:126)φ ( C ) depends however on λ , which is the contentof the following Theorem. Theorem 3.
For all (cid:126)φ ∈ R M with at least two differentcomponents, the functionals M λ,(cid:126)φ ( C ) are faithful if andonly if λ = . In App. C, we discuss in more detail why the function-als M λ,(cid:126)φ ( C ) are only faithful for λ = .Now that we connected the pre-processed improve-ments with interferometry, a natural question to ask ishow to evaluate them numerically. This is for examplerelevant if we want to decide if one operation outperformsanother one in our discrimination games. Since the pre-processed improvements are defined via the optimizationsin Eq. (12) and the induced trace norm includes an ad-ditional optimization over states, evaluating M λ,(cid:126)φ ( C ) ishowever not straightforward. In App. B, we propose amethod based on semidefinite programming that in ad-dition leads to an optimal state and pre-processing. B. Creating coherence and interferometry
Here, we analyse a setting that connects the creationof coherence to interferometry. As in Sec. III A, we in-troduce it as a game between Alice and Bob (see Fig. 5).This time, Bob is provided with a fixed creating oper-ation C that he applies to an incoherent state of hischoice. Before sending this state to Alice, Bob is al-lowed to further apply an arbitrary creation incoherentoperation D onto C ( τ ). Alice receives the state DC ( τ ),applies again with probability µ a fixed operation Λ (cid:126)φ ,and sends the resulting state back to Bob. Finally, Bobperforms a generic measurement on the state he retrievedand guesses whether Alice applied Λ (cid:126)φ or not. Alice BobPOVMBob τ ∈ I C D λ id + µ Λ (cid:126)φ FIG. 5. Sketch of the game played by Alice and Bob thatis used to describe the role of the creation of coherence ininterferometry.
In this case, assuming that Bob prepares the best in-coherent state τ , applies the best post-processing D , andthe best final measurement, his probability of guessingcorrectly whether Alice applied Λ (cid:126)φ or not is given by [62] p max λ,(cid:126)φ ( C ) = 12 + 12 max D∈MIO (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) λ − µ Λ (cid:126)φ (cid:17) DC ∆ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (13)Analogously to the previous setting, we define the func-tionals N λ,(cid:126)φ ( C ) := max D∈MIO (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) λ − µ Λ (cid:126)φ (cid:17) DC ∆ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − | λ − µ | . (14) and call them post-processed improvements . The con-nection to our interferometric setup is again straightforward: the operation C in Fig. 2 is represented bythe joint action of C and D , and C and the incoherentmeasurement afterwards form the general POVM. Anal-ogously to Thm. 1, we further find Theorem 4.
The functionals N λ,(cid:126)φ ( C ) are convex mea-sures in the creation incoherent setting for all λ ∈ [0 , and for all (cid:126)φ ∈ R M . The post-processed improvements therefore quantifythe ability of an operation C to create coherence and, us-ing similar arguments as in the previous section, have theoperational interpretation that they describe the advan-tage that C ’s ability to create coherence grants in a con-crete interferometric setup. We thus established anothermain result, namely that the ability to create coherenceis a relevant resource in interferometry too.Concerning faithfulness, we have the following result. Theorem 5.
The functionals N ,(cid:126)φ ( C ) are faithful for all (cid:126)φ ∈ R M with at least two different components. This ensures again that the ability to create coherencecontributes in some interferometric tasks, namely in theones described by our games with λ = µ = . IV. CONCLUSIONS
In this work, we introduced families of dynamical re-source measures that allowed us to establish a connec-tion between an operation’s ability to detect or createcoherence and the performance of interferometric exper-iments. This shows that the abstract resource theoriesdefined in Ref. [33] have an operational meaning. Our re-sults concerning the ability of operations to create coher-ence should be compared to the static results of Ref. [53],where it was shown that every visibility functional thatsatisfies some meaningful properties can be used to de-fine a coherence measure that is strongly monotonic [12]under strictly incoherent operations [54]. One obtainsthese static coherence measures from the visibilities viaoptimizations over all measurements. Similar to our case,such optimizations are necessary to ensure that the co-herence is used ideally and not only present. Here, wetook a more direct approach that did not rely on visi-bilities. This allowed us to define measures that are notrestricted to strictly incoherent operations, but hold forthe larger class of maximally incoherent operations in-stead. An interesting question is whether one could alsodefine dynamical resource measures based on visibility,and whether this would lead back to strictly incoherentoperations.Whilst we provided a method to compute the pre-processed improvements and showed that they are notfaithful for λ (cid:54) = µ , it is an open question whether there ex-ist analogous results for the post-processed cases. More-over, one could combine the two resource theories we ap-plied and consider a fixed operation used for the creationof coherence and one for its detection. Potentially, theresulting success probabilities could then be expressed asproducts of two measures. Another interesting idea is toremove, e.g., the optimal pre-processing in our measure,and require that Alice applies an optimal Λ (cid:126)φ instead.Whilst we investigated this approach too, we were notable to prove monotonicity (for input dimensions of C greater than three).In our investigations, we considered the application offixed phases that are known to Bob. Whilst this is cer-tainly a relevant scenario, e.g., if one checks whether aknown sample is present or not, one often uses interfer-ometers to gather information about unknown phases.It is then an open question whether one could use, e.g.,Fisher information, to construct measures in these sce-narios [53, 63, 64]. In conclusion, the investigation ofthe technological relevance of dynamical coherence is farfrom being completed, but our proof of principle showsthat these theories might help to better understand andthus exploit quantum properties. ACKNOWLEDGMENTS
We thank Mirko Rossini, Dario Egloff, and LudovicoLami for discussions and feedback.
Appendix A: The need for an optimal pre-processing
As mentioned in Sec. III, in this Appendix, we show thenecessity of the optimal pre-processing in the definitionsof M λ,(cid:126)φ ( C ) (see Eq. (12)). To this end, we assume thatBob cannot apply the optimal pre-processing prior to theoperation C . The analogues of M λ,(cid:126)φ ( C ) are then L λ,(cid:126)φ ( C ) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ C (cid:16) λ − µ Λ (cid:126)φ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − | λ − µ | . (A1)Via the construction of an explicit counterexample, wenow show that the L λ,(cid:126)φ ( C ) are not monotonic in general.To this end, we provide Bob with a detecting operationof the form ˜ C = C A ⊗ id B , where C A / ∈ DI and dim( A ) =dim( B ). In addition, we choose λ = µ = and Λ (cid:126)φ :=id A ⊗ Λ B(cid:126)φ (cid:48) . The resulting guessing game for Alice and Bobis represented in Fig. 6, and it is straightforward to verifythat L (cid:126)φ ( ˜ C ) = 0, because ˜ C detects in subspace where Λ (cid:126)φ encodes no information. We now define a superchannel S acting as S ( C ) = CC , (A2)where C is a SWAP channel that exchanges the systems Alice Bob incoherentincoherentBob ρ AB id A C A (cid:54)∈ DI (cid:16) id + Λ (cid:126)φ (cid:48) (cid:17) B id B FIG. 6. Schematic representation of the guessing game de-scribed in the main text. A and B , i.e., C (cid:88) i,j,k,l ρ ij,kl | ij (cid:105)(cid:104) kl | AB = (cid:88) i,j,k,l ρ ji,lk | ij (cid:105)(cid:104) kl | AB . (A3)The superchannel S is free since C is free (it only rela-bels Hilbert spaces). Applying S to ˜ C , we obtain the sit-uation represented in Fig. 7, from where we deduce that L (cid:126)φ ( S ( ˜ C )) can be different from zero for specific choicesof (cid:126)φ (cid:48) and C A , because now we detect in the subspacein which information is encoded. We conclude that the Alice Bob incoherentincoherentBob ρ AB id A C A (cid:16) id + Λ (cid:126)φ (cid:48) (cid:17) B id B FIG. 7. Addition of a SWAP operation to the game in Fig. 6,which shows that L λ,(cid:126)φ ( C ) is not monotonic. functionals L λ,(cid:126)φ ( C ) are in general not measures in thedetection incoherent setting. Appendix B: Evaluation of the pre-processedimprovements
Here we show how one can evaluate the pre-processedimprovements numerically and how this leads to an op-timal state and pre-processing. We begin with two Lem-mas.
Lemma 6.
The two sets X := (cid:40) X AB = id A ⊗ C B ← A (cid:88) i,j ρ i,j | ii (cid:105)(cid:104) jj | AA (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C CPTP, ρ = (cid:88) i,j ρ i,j | i (cid:105)(cid:104) j | quantum state (cid:41) , Y := { Y AB | Y ≥ , Tr B Y = ∆( σ A ) , σ A quantum state } are equal.Proof. First we show that every X ∈ X is also an elementof Y . Assume X ∈ X . From this follows X ≥ B X = (cid:88) i,j ρ i,j | i (cid:105)(cid:104) j | A Tr ( C [ | i (cid:105)(cid:104) j | A ]) = (cid:88) i ρ i,i | i (cid:105)(cid:104) i | A is a diagonal state since ρ is a state by assumption.To prove the reverse direction, we assume that Y ∈ Y .From this assumption follows that Y is a quantum state.Every purification Z ABC = | Ψ (cid:105)(cid:104) Ψ | of Y , i.e., Tr C Z = Y , is by assumption also a purification of a ∆( σ A ) = (cid:80) i σ i | i (cid:105)(cid:104) i | A . Let us define the set S = { i : σ i (cid:54) = 0 } and˜ A = span {| i (cid:105) ∈ S } . Using the isometric freedom in pu-rifications we find | Ψ (cid:105) = (cid:88) i √ σ i | i (cid:105) A ⊗ V BC ← ˜ A | i (cid:105) ˜ A where V is an isometry, which we use to define the map˜ C ( τ ) = Tr C (cid:20) V BC ← ˜ A τ (cid:16) V CB ← ˜ A (cid:17) † (cid:21) that is CPTP by construction. Then Y = Tr C (cid:34)(cid:88) i,j √ σ i σ j | i (cid:105)(cid:104) j | A ⊗ V BC ← ˜ A | i (cid:105)(cid:104) j | ˜ A (cid:16) V CB ← ˜ A (cid:17) † (cid:35) = (cid:88) i,j √ σ i σ j | i (cid:105)(cid:104) j | A ⊗ Tr C (cid:20) V BC ← ˜ A | i (cid:105)(cid:104) j | ˜ A (cid:16) V CB ← ˜ A (cid:17) † (cid:21) =id A ⊗ ˜ C B ← ˜ A (cid:88) i,j √ σ i σ j | ii (cid:105)(cid:104) jj | A ˜ A . (B1) Now, we introduce the map Π defined by Kraus operators K := (cid:88) i ∈ S | i (cid:105) ˜ A (cid:104) i | A , L j := | ψ (cid:105) ˜ A (cid:104) j | A ∀ j (cid:54)∈ S, (B2)where | ψ (cid:105) ˜ A is a normalized quantum state.It is easy to check that K † K + (cid:80) j (cid:54)∈ S L † j L j = A and thatthe map C B ← A := ˜ C B ← ˜ A Π ˜ A ← A is such that Y = id A ⊗ C B ← A (cid:88) i,j √ σ i σ j | ii (cid:105)(cid:104) jj | AA . Since ˜ σ = (cid:80) i,j √ σ i σ j | i (cid:105)(cid:104) j | = | φ (cid:105)(cid:104) φ | : | φ (cid:105) = (cid:80) i √ σ i | i (cid:105) isby assumption a valid quantum state, we showed that Y is an element of X . Lemma 7.
The two sets X := (cid:40) X AB = id A ⊗ E B ← A (cid:88) i,j ρ i,j | ii (cid:105)(cid:104) jj | AA (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ∈ DI , ρ = (cid:88) i,j ρ i,j | i (cid:105)(cid:104) j | quantum state (cid:41) , Y := { Y AB | Y ≥ , Tr B Y = ∆( σ A ) , diag ( (cid:104) i | A Y | j (cid:105) A ) = 0 ∀ i (cid:54) = j, σ A quantum state } are equal.Proof. A quantum operation E is detection incoherent iff∆ E = ∆ E ∆. Assume X ∈ X . Using Lem. 6, and the factthat diag ( (cid:104) i | A X | j (cid:105) A ) = diag (∆ E ( ρ i,j | i (cid:105)(cid:104) j | ))= diag (∆ E ∆ ( ρ i,j | i (cid:105)(cid:104) j | )) , we find that X ∈ Y .Now assume Y ∈ Y . From the proof of Lem. 6, we knowthat the assumptions ensure that we can write Y =id ⊗ E (cid:88) i,j √ σ i σ j | ii (cid:105)(cid:104) jj | (B3)where E B ← A is a quantum operation which is composedof ˜ E B ← ˜ A Π ˜ A ← A . Thendiag ( (cid:104) i | A Y | j (cid:105) A ) = √ σ i σ j diag ( E | i (cid:105)(cid:104) j | ) = 0 ∀ i (cid:54) = j ensures that∆ E ( | i (cid:105)(cid:104) j | ) = ∆ E ∆( | i (cid:105)(cid:104) j | ) ∀ i (cid:54) = j. This allows us to conclude that E is detection incoherent.Thanks to the previous Lemmas, we propose the fol-lowing method to evaluate the pre-processed improve-ments numerically. Theorem 8.
Consider a quantum channel C C ← B and let N = dim( C ) . Let further ( s m,n ) m,n be the matrix of di-mension N × N that contains as rows all N -dimensionalvectors (cid:126)s m whose entries are ± . The solution of the op-timization problem F λ,(cid:126)φ ( C C ← B ) = max E∈DI (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ C C ← B E B ← A (cid:16) λ − µ Λ (cid:126)φ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (B4) is then equivalent to the maximum of the solutions of the following N semidefinite programsmaximize: t m subject to: t m ≤ N − (cid:88) n =0 s m,n (cid:104) n | C C C ← B ( Z ) | n (cid:105) C Z = (cid:88) i,j (cid:16) λ − µe i ( φ i − φ j ) (cid:17) (cid:104) i | A X AB | j (cid:105) A X AB ≥ B ( X AB ) = ∆( σ A ) σ A ≥ σ A ) = 1diag ( (cid:104) i | A X AB | j (cid:105) A ) = 0 ∀ i (cid:54) = j. (B5) Proof.
We begin by rewriting the optimization problemin Eq. (B4) asmaximize: Tr (cid:12)(cid:12)(cid:12) ∆ CE (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) (cid:12)(cid:12)(cid:12) subject to: E ∈ DI ρ ≥ ρ ) = 1 . Using the index representation from App. D, the objec-tive function can now be expanded as Tr (cid:12)(cid:12)(cid:12) ∆ CE (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) (cid:12)(cid:12)(cid:12) (B6)= N − (cid:88) n =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) n | C (cid:88) i,j,k,l E i,jk,l (cid:16) λ − µe i ( φ i − φ j ) (cid:17) ρ i,j | k (cid:105)(cid:104) l | | n (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = N − (cid:88) n =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) n | B C (cid:32) (cid:88) i,j (cid:16) λ − µe i ( φ i − φ j ) (cid:17) (cid:104) i | A (cid:88) o,p ρ o,p | o (cid:105)(cid:104) p | A ⊗ (cid:88) k,l E o,pk,l | k (cid:105)(cid:104) l | B | j (cid:105) A (cid:33) | n (cid:105) B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = N − (cid:88) n =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) n | B C (cid:32) (cid:88) i,j (cid:16) λ − µe i ( φ i − φ j ) (cid:17) (cid:104) i | A id A ⊗ E B ← A (cid:32)(cid:88) o,p ρ o,p | oo (cid:105)(cid:104) pp | AA (cid:33) | j (cid:105) A (cid:33) | n (cid:105) B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Hence, we reformulate our problem into maximize: N − (cid:88) n =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) n | B C (cid:32) (cid:88) i,j (cid:16) λ − µe i ( φ i − φ j ) (cid:17) (cid:104) i | A id A ⊗ E B ← A (cid:32)(cid:88) o,p ρ o,p | oo (cid:105)(cid:104) pp | AA (cid:33) | j (cid:105) A (cid:33) | n (cid:105) B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) subject to: E ∈ DI ρ ≥ ρ ) = 1 . Finally, using Lem. 7 and (cid:88) n | f n | = max (cid:126)s m ( (cid:126)s m · (cid:126)f ) , where the vectors (cid:126)s m have been introduced in the state-ment of the Theorem, we proved that Eq. (B4) and thegreatest value of the solutions of Eq. (B5) are equiva-lent.As one can see, the evaluation method that we pro-pose in the above Theorem unfortunately requires us tosolve a number of semidefinite programs that grows ex-ponentially in the output dimension of C . However, thesolution gives direct access to an optimal pair E opt , ρ opt of input state and pre-processing. This is a consequenceof the constructive proofs of Lem. 6 and Lem. 7 on whichthe method relies. Let us assume that we solved thesemidefinite program in Eq. (B5) that leads to the maxi-mal t m and denote its optimal X with X opt , from whichone obtains σ opt ,i viaTr B ( X opt ) = (cid:88) i σ opt ,i | i (cid:105)(cid:104) i | . Combining Eq. (B3) with Eq. (B6), an optimal ρ isthus given by ρ opt = (cid:88) i,j √ σ opt ,i σ opt ,j | i (cid:105)(cid:104) j | . With ˜ A = span {| i (cid:105) : σ opt ,i (cid:54) = 0 } , we now define an oper-ation ˜ E B ← ˜ A opt via Eq. (B1), i.e.,˜ E B ← ˜ A opt ( | i (cid:105)(cid:104) j | ˜ A ) = (cid:104) i | A X opt | j (cid:105) A √ σ opt ,i σ opt ,j . Note that due to the definition of ˜ A , division by zero isexcluded, and ˜ E B ← ˜ A opt is determined uniquely. Togetherwith the operation Π ˜ A ← A introduced in Eq. (B2), whichis also well defined due to the knowledge of σ opt ,i , anoptimal E B ← A corresponding to the ρ opt given above istherefore E B ← A opt = ˜ E B ← ˜ A opt Π ˜ A ← A . (B7) Appendix C: Interpretation of faithfulness
In this Appendix, we discuss the intuition behind theresults of Thm. 3 concerning the faithfulness of the pre-processed improvements. As example, we consider a de-tecting qubit operation C that is a stochastic mixture ofthe Hadamard gate H and the identity channel, i.e., C ( ρ ) = p HρH † + p ρ. (C1)For (cid:126)φ = (cid:18) π , (cid:19) , (C2)in Fig. 8 and Fig. 9, we plotted M λ,(cid:126)φ ( C ) with the helpof Thm. 8. These plots clearly show that M λ,(cid:126)φ ( C ) isnot faithful for λ (cid:54) = µ . This implies that, for λ (cid:54) = µ , FIG. 8. With (cid:126)φ and C ( p ) as in the main text, where p de-notes the probability of applying the Hadamard gate, M λ,(cid:126)φ ( C )is plotted for different choices of λ ≥ µ . One clearly sees that M λ,(cid:126)φ ( C ) is not faithful for λ (cid:54) = .FIG. 9. Plot corresponding to Fig. 8 for λ ≤ µ . not every operation able to detect coherence allows Bobto increase his probability of guessing correctly if Aliceapplied Λ (cid:126)φ or not.In the following, we will analyze the reasons for thisfact. Firstly, we notice that Bob can always guess cor-rectly with a minimum probability of by not takinginto account any information about Alice’s actions butjust announcing his guesses randomly with equal proba-bility.Secondly, by purely taking into account his knowledgeof λ , he can increase this probability by the trivial bias B t := 12 | λ − µ | . (C3)To further increase his probability of guessing correctly,he needs to make use of information that he gains fromthe incoherent POVM.Now suppose that we want to distinguish two states σ and σ occurring with probabilities λ and µ via an incoherent measurement. In this case, the maximal biasover that we can obtain is [33, Prop. 16] B m := 12 || ∆( λσ − µσ ) || , (C4)which we will call the measurement bias.Intuitively, one might expect that B m is greater than B t for all pairs of states σ and σ that have different pop-ulations, because then one can find an incoherent POVMthat leads to different statistics for the two states. How-ever, one deduces from Fig. 8 and Fig. 9 that there existqubit states σ = CE ( ρ ) and σ = CE Λ (cid:126)φ ( ρ ) (with differ-ent populations for our costly C and ideal ρ and E ), forwhich B m = B t . In addition, we note that for λ (cid:54) = µ these two plots show a discontinuity in the gradient of M λ,(cid:126)φ ( C ) with respect to p .Since both σ and σ are qubit states, the respectivemeasurement bias is given by B m = 12 Tr | ∆( λσ − µσ ) | = 12 Tr (cid:12)(cid:12)(cid:12) ( λσ (0)00 − µσ (1)00 ) | (cid:105)(cid:104) | + ( λσ (0)11 − µσ (1)11 ) | (cid:105)(cid:104) | (cid:12)(cid:12)(cid:12) = 12 max {| Tr( λσ − µσ ) | , | Tr( σ z ( λσ − µσ )) |} = 12 max {| λ − µ | , | Tr( σ z ( λσ − µσ )) |} . (C5)From Ref. [33, Prop. 16] and its proof, we know that anoptimal guessing strategy for the distinction of σ and σ that involves only incoherent measurements consistsof the following: measure the POVM { P , P = − P } ,where P is the projector onto the positive part of∆( λσ − µσ ) and P the projector onto its negative part,and announce i = 0 , • If λσ (0)00 − µσ (1)00 and λσ (0)11 − µσ (1)11 are both non-negative, we have P = . In other words, Bobdoes not need to do any measurement and alwaysclaims that the phases have not been attached. • If neither λσ (0)00 − µσ (1)00 nor λσ (0)11 − µσ (1)11 are pos-itive, we find P = and Bob always claims thatthe phases have been attached. Thus, in these firsttwo cases, no free measurement leads to useful in-formation. • If λσ (0)00 − µσ (1)00 is positive and λσ (0)11 − µσ (1)11 neg-ative, we choose P = | (cid:105)(cid:104) | , i.e., actually performa measurement. Here, according to the outcome,Bob should declare that Alice encoded (cid:126)φ or thatshe did not. • In the remaining case where λσ (0)00 − µσ (1)00 is nega-tive and λσ (0)11 − µσ (1)11 positive, we find P = | (cid:105)(cid:104) | .In the last two cases, we can actually gain addi-tional information via a free measurement.0This also explains the discontinuities in the gradients,which occur at the points from which on performing anincoherent measurement leads to an actual advantage.The above four cases are collected in the third line ofEq. (C5) (where the POVMs are expressed by the mea-surement of the observable ± σ z ).This behavior is actually not due to the incoherentmeasurements or any other quantum property, but canalready be explained with the following purely classicalexample. Imagine that Alice has two marbles, a darkgreen one and a blue one. She chooses one of them withan a priori probability that is known to Bob and throwsit into a dark room. Now Bob looks at it, but since theroom is dark, blue and dark green are hard to distinguish,hence Bob is not sure about the color of the marble hesees. What is now his best strategy to guess correctlywhich one it is? Betting based on the knowledge of the apriori probability or based on the color he believes to see?Or combining both? Intuitively, he should make the betbased on the information that is most secure. Therefore,if he knows that Alice throws the blue marble with aprobability of 95%, but according to his eyes (which weassume to be reliable with a probability of 60% undersuch conditions) it is the green one, he should not trusthis eyes. Independent of whether he looked at the marbleor not, Bob’s ideal guessing strategy is to always claimthat the marble is blue. As the percentages change, atsome point, the ideal guessing strategy starts to dependon what he sees. Considering the specific case that the apriori probability for the marbles is 50%, this will alwaysbe the case (assuming that his eyes gather any usefulinformation), which is the reason why M λ,(cid:126)φ ( C ) is faithfulfor λ = . Appendix D: Technical results
In this Appendix, we collect some technical results thatare needed for the proofs of the results in the main textpresented in App. E. In both Appendices, we often rep-resent the action of a quantum channel C on matrix ele-ments | i (cid:105)(cid:104) j | as C ( | i (cid:105)(cid:104) j | ) = (cid:88) k,l C i,jk,l | k (cid:105)(cid:104) l | . (D1)With this notation, we have [33, Prop. 15] C i,ik,l ∝ δ k,l ∀ i, k, l, (D2)for all creation incoherent operations C , whilst for anoperation that cannot detect coherence C i,jk,k ∝ δ i,j ∀ i, j, k (D3)is satisfied. Moreover, the following Proposition holds. Proposition 9. If C is a quantum channel, the corre-sponding coefficients C i,jk,l fulfill the following properties: 1. C n,nm,m ≥ ∀ m, n ,2. C i,jk,l = C j,i ∗ l,k ∀ i, j, k, l ,3. (cid:80) m C i,jm,m = δ i,j ∀ i, j, m .Proof. The first claim follows from complete positivity.Let us write the Choi state corresponding to the quantumoperation C as J C = C ⊗ id (cid:88) i,j | ii (cid:105)(cid:104) jj | = (cid:88) i,j,k,l C i,jk,l | ki (cid:105)(cid:104) lj | . Complete positivity of C is then equivalent to (cid:104) v | J C | v (cid:105) ≥ ∀ | v (cid:105) . Choosing | v (cid:105) = | nm (cid:105) , we hence find (cid:104) mn | J C | mn (cid:105) = C n,nm,m ≥ ∀ m, n as a necessary but not sufficient condition for completepositivity.The second claim follows from the fact that a quantumoperation preserves hermiticity. The hermitian conjugateof our Choi matrix is given by J †C = (cid:88) i,j,k,l C i,j ∗ k,l | lj (cid:105)(cid:104) ki | = (cid:88) i,j,k,l C j,i ∗ l,k | ki (cid:105)(cid:104) lj | . Now, since J †C = J C must hold, we have C j,i ∗ l,k = C i,jk,l ∀ i, j, k, l .The third claim follows from trace preservation. Letus write a state as ρ = (cid:80) i,j ρ i,j | i (cid:105)(cid:104) j | and the action ofthe channel C on it as C ( ρ ) = (cid:88) i,j,k,l ρ i,j C i,jk,l | k (cid:105)(cid:104) l | . Choosing ρ such that ρ n,n = 1 for a fixed n and ρ i,j = 0for all other coefficients, we find the following necessarycondition for trace preservation1 = Tr ( C ( ρ )) = (cid:88) m,i,j ρ i,j C i,jm,m = (cid:88) m C n,nm,m ∀ n. The case for i (cid:54) = j can be proven by changing thechoice of ρ . Let us take a state such that ρ ˜ i, ˜ i = ρ ˜ j, ˜ j = and ρ ˜ i, ˜ j = e iξ = ρ ∗ ˜ j, ˜ i for a pair of indices ˜ i (cid:54) = ˜ j . Thisautomatically implies that all other coefficients are zero.Thus, using again trace preservation and the previousresult, we find (cid:88) m (cid:16) C ˜ i, ˜ im,m + C ˜ j, ˜ jm,m (cid:17) + (cid:88) m (cid:16) C ˜ i, ˜ jm,m e iξ + C ˜ j, ˜ im,m e − iξ (cid:17) = 1 ∀ ξ ⇐⇒ Re (cid:32) e iξ (cid:88) m C ˜ i, ˜ jm,m (cid:33) = 0 ∀ ξ ⇐⇒ (cid:88) m C ˜ i, ˜ jm,m = 0 . Lemma 10.
An operator α = (cid:16) id A ⊗ ∆ B (cid:17) C AB ← C DI (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) , (D4) where C AB ← C DI is a detection incoherent quantum channel,can be decomposed as α = (cid:88) b p b (cid:16) ˜ C A ← C DI ,b (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) (cid:17) ⊗ | b (cid:105)(cid:104) b | B , (D5) where ˜ C A ← C DI ,b are detection incoherent quantum channelsand p b are probabilities.Proof. Let us write down α using a Kraus representationof (cid:16) id A ⊗ ∆ B (cid:17) C AB ← C DI , i.e., α = (cid:88) n,b ( A ⊗ | b (cid:105)(cid:104) b | B ) K n (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) K † n ( A ⊗ | b (cid:105)(cid:104) b | B ) . Now, we define the operator α | b := (cid:104) b | B α | b (cid:105) B = (cid:88) n (cid:104) b | B K n (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) K † n | b (cid:105) B and evaluate its trace using the index representation,Eq. (D3), and Prop. 9,Tr (cid:0) α | b (cid:1) = (cid:88) a (cid:104) a, b | AB C AB ← C DI (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) | a, b (cid:105) AB = (cid:88) a (cid:104) a, b | AB (cid:32) (cid:88) i,j,k,l,m,n C i,jkl,mn (cid:16) λ − µe i ( φ i − φ j ) (cid:17) ρ i,j | kl (cid:105)(cid:104) mn | AB (cid:33) | a, b (cid:105) AB = ( λ − µ ) (cid:88) a (cid:88) i C i,iab,ab ρ i,i = ( λ − µ ) Tr (cid:0) P b C AB ← C DI ( ρ ) (cid:1) = ( λ − µ ) p b , where p b is the probability of collapsing the state C AB ← C DI ( ρ ) to the subspace onto which the operator P b = (cid:88) a | a, b (cid:105)(cid:104) a, b | AB projects. At this point, we define ˜ K n,b := (cid:104) b | B K n √ p b ∀ b suchthat p b (cid:54) = 0. This allows us to write α = (cid:88) b α | b ⊗ | b (cid:105)(cid:104) b | B = (cid:88) b : p b (cid:54) =0 p b (cid:88) n ˜ K n,b (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) ˜ K † n,b ⊗ | b (cid:105)(cid:104) b | B . For all b with p b (cid:54) = 0, we now interpret { ˜ K n,b } n a as set ofKraus operators associated to an operation ˜ C A ← C DI ,b , whichis thus hermiticity preserving and completely positive.Trace preservation of ˜ C A ← C DI ,b follows fromTr (cid:16) ˜ C A ← C DI ,b ( ρ ) (cid:17) = 1 p b Tr (cid:32) ( A ⊗ (cid:104) b | B ) (cid:88) n K n ρK † n ( A ⊗ | b (cid:105) B ) (cid:33) = 1 p b (cid:88) a (cid:104) a | A ⊗ (cid:104) b | B (cid:88) n K n ρK † n | a (cid:105) A ⊗ | b (cid:105) B = 1 p b Tr (cid:0) P b C AB ← C DI ( ρ ) (cid:1) = p b p b = 1 . It remains to prove that that C A ← C DI ,b is detection incoher-ent. From˜ C A ← C DI ,b ( ρ ) = (cid:104) b | B √ p b C AB ← C DI ( ρ ) | b (cid:105) B √ p b = 1 p b (cid:104) b | B (cid:88) i,j,k,l,m,n C i,jkl,mn ρ i,j | kl (cid:105)(cid:104) mn | | b (cid:105) B = (cid:88) i,j,k,m C i,jkb,mb ρ i,j p b | k (cid:105)(cid:104) m | A = (cid:88) i,j,k,l ˜ C i,jk,l ( b ) ρ i,j | k (cid:105)(cid:104) l | A follows that ˜ C i,jk,k ( b ) = C i,jkb,kb p b ∝ δ i,j . Due to Ref. [33,Prop. 15] the ˜ C A ← C DI ,b are thus detection incoherent be-cause C AB ← C DI was.In addition, we need the following Lemmas in the maintext. Lemma 11.
Consider an operation C B ← A with dim( B ) = 2 . If there exists an m such that C i,jm,m =0 ∀ i, j , then C i,jk,l = 0 ∀ k (cid:54) = l, ∀ i, j .Proof. Let us assume that m = 0. If this is not thecase we just need to relabel our Hilbert space. Applying C B ← A to a state ρ , we get C ( ρ ) = (cid:88) i,j C i,j , ρ i,j | (cid:105)(cid:104) | + (cid:88) i,j C i,j , ρ i,j | (cid:105)(cid:104) | + (cid:88) i,j (cid:88) m (cid:54) = n C i,jm,n ρ i,j | m (cid:105)(cid:104) n | = (cid:88) i,j C i,j , ρ i,j | (cid:105)(cid:104) | + (cid:88) i,j C i,j , ρ i,j | (cid:105)(cid:104) | + (cid:88) i,j C i,j , ρ i,j | (cid:105)(cid:104) | . At this point, we recall that C ( ρ ) must be positivesemidefinite and that a necessary condition for positivesemidefiniteness of an operator is that its determinant is2non-negative. Moreover, this must hold for all states ρ .Using Proposition 9.2, we getdet( C ( ρ )) = − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) i,j C i,j , ρ i,j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We now consider two particular choices of states forwhich the previous condition must hold. The first oneis ρ = | i (cid:105)(cid:104) i | . With this, we obtaindet( C ( ρ )) = − (cid:12)(cid:12)(cid:12) C i,i , (cid:12)(cid:12)(cid:12) ≥ ∀ i ⇐⇒ C i,i , = 0 ∀ i. The second choice is ρ = 12 (cid:0) | i (cid:105)(cid:104) i | + | j (cid:105)(cid:104) j | + | i (cid:105)(cid:104) j | e iξ + | j (cid:105)(cid:104) i | e − iξ (cid:1) . We getdet( C ( ρ )) = − (cid:12)(cid:12)(cid:12) C i,i , + C j,j , + C i,j , e iξ + C j,i , e − iξ (cid:12)(cid:12)(cid:12) = − (cid:12)(cid:12)(cid:12) C i,j , e iξ + C j,i , e − iξ (cid:12)(cid:12)(cid:12) ≥ ∀ i (cid:54) = j, ∀ ξ ⇐⇒ C i,j , = 0 ∀ i (cid:54) = j. Hence, C i,j , = 0 ∀ i, j . Using again Proposition 9.2 finishesthe proof. Lemma 12.
For all states | ψ (cid:105) AZ there exist a detectionincoherent operation E AZ ← A and a state | ϕ (cid:105) A such that E AZ ← A (cid:16) λ − µ Λ A(cid:126)φ (cid:17) ([ | ϕ (cid:105)(cid:104) ϕ | A )= (cid:16)(cid:16) λ − µ Λ A(cid:126)φ (cid:17) ⊗ id Z (cid:17) ( | ψ (cid:105)(cid:104) ψ | AZ ) . (D6) Proof.
Given | ψ (cid:105) AZ := (cid:80) m,n ψ m,n | m, n (cid:105) AZ , we choose | ϕ (cid:105) A such that | ϕ (cid:105) A := (cid:88) m ϕ m | m (cid:105) A with | ϕ m | = (cid:88) n | ψ m,n | ∀ m. Next, we define the operator U := (cid:88) m,n v m,n | m, n (cid:105) AZ (cid:104) m | A , where v m,n = ψ m,n ϕ m ∀ m s.t. ϕ m (cid:54) = 0 , ∀ n,v m,n = 1dim( Z ) ∀ m s.t. ϕ m = 0 , ∀ n, and find that (cid:88) n | v m,n | = (cid:88) n | ψ m,n | | ϕ m | = 1 ∀ m s.t. ϕ m (cid:54) = 0 , (cid:88) n | v m,n | = (cid:88) n Z ) = 1 ∀ m s.t. ϕ m = 0 . This ensures that U † U = (cid:88) m (cid:32)(cid:88) n | v m,n | (cid:33) | m (cid:105)(cid:104) m | A = A , hence U is an isometry and it defines a CPTP map.Let us now verify that this map is detection incoherent.Given a state ω A = (cid:80) i,j ω i,j | i (cid:105)(cid:104) j | A , we have∆ (cid:0) U ∆( ω A ) U † (cid:1) = ∆ (cid:32) (cid:32)(cid:88) m,n v m,n | m, n (cid:105) AZ (cid:104) m | A (cid:33)(cid:32)(cid:88) i ω i,i | i (cid:105)(cid:104) i | A (cid:33) (cid:32)(cid:88) o,p v o,p | o, p (cid:105) AZ (cid:104) o | A (cid:33) † (cid:33) = ∆ (cid:88) i,n,p v i,n ω i,i v ∗ i,p | in (cid:105)(cid:104) ip | AZ = (cid:88) i,n | v i,n | ω i,i | in (cid:105)(cid:104) in | AZ , while∆ (cid:0) U ω A U † (cid:1) = ∆ (cid:32) (cid:32)(cid:88) m,n v m,n | m, n (cid:105) AZ (cid:104) m | A (cid:33)(cid:88) i,j ω i,j | i (cid:105)(cid:104) j | A (cid:32)(cid:88) o,p v o,p | o, p (cid:105) AZ (cid:104) o | A (cid:33) † (cid:33) = ∆ (cid:88) i,j,n,p v i,n ω i,j v ∗ j,p | in (cid:105)(cid:104) jp | AZ = (cid:88) i,n | v i,n | ω i,i | in (cid:105)(cid:104) in | AZ = ∆ (cid:0) U ∆( ω A ) U † (cid:1) . Therefore, the map defined by U is detection incoherent.Let us call this map E AZ ← A and use it to show Eq. (D6).We obtain E AZ ← A (cid:16) λ − µ Λ A(cid:126)φ (cid:17) (( | ϕ (cid:105)(cid:104) ϕ | A )= U (cid:88) i,j (cid:16) λ − µe i ( φ i − φ j ) (cid:17) ϕ i ϕ ∗ j | i (cid:105)(cid:104) j | A U † = (cid:88) i,j,n,p (cid:16) λ − µe i ( φ i − φ j ) (cid:17) ϕ i ϕ ∗ j v i,n v ∗ j,p | in (cid:105)(cid:104) jp | AZ = (cid:88) i,j,n,p (cid:16) λ − µe i ( φ i − φ j ) (cid:17) ψ i,n ψ ∗ j,p | in (cid:105)(cid:104) jp | AZ = (cid:16)(cid:16) λ − µ Λ A(cid:126)φ (cid:17) ⊗ id Z (cid:17) ( | ψ (cid:105)(cid:104) ψ | AZ ) , where the third equality follows from the definition of v m,n . We recall that ϕ m = 0 for some m iff for that m we had ψ m,n = 0 ∀ n .3 Lemma 13.
Let D C ← BZ , C B ← A be quantum channelsand ρ AZ an incoherent state. Then there exists a prob-ability distribution { p i } i , incoherent states { ρ | i } i andquantum channels {D C ← Bi } i such that D C ← BZ (cid:16) C B ← A ⊗ id Z (cid:17) ( ρ AZ ) = (cid:88) i p i D C ← Bi C B ← A ( ρ | i ) . Moreover, if D C ← BZ ∈ MIO , one can choose D C ← Bi ∈MIO too.Proof. The probability to obtain outcome b if subsystem Z of the state ρ AZ is projectively measured in its inco-herent basis is given by p b = Tr {(cid:104) b | Z ρ AZ | b (cid:105) Z } , and, for p b (cid:54) = 0, we denote the corresponding post mea-surement states of system A by ρ | b := (cid:104) b | Z ρ AZ | b (cid:105) Z p b . Since ρ AZ ∈ I by assumption, the ρ | b are incoherent tooand we have (cid:16) C B ← A ⊗ id Z (cid:17) ( ρ AZ )= (cid:16) id B ⊗ ∆ Z (cid:17) (cid:16) C B ← A ⊗ id Z (cid:17) ( ρ AZ )= (cid:88) b ( B ⊗ | b (cid:105)(cid:104) b | Z ) (cid:16) C B ← A ⊗ id Z ( ρ AZ ) (cid:17) ( B ⊗ | b (cid:105)(cid:104) b | Z )= (cid:88) b (cid:104) b | Z (cid:16) C B ← A ⊗ id Z ( ρ AZ ) (cid:17) | b (cid:105) Z ⊗ | b (cid:105)(cid:104) b | Z = (cid:88) b C B ← A ( (cid:104) b | Z ρ AZ | b (cid:105) Z ) ⊗ | b (cid:105)(cid:104) b | Z = (cid:88) b : p b (cid:54) =0 p b C B ← A (cid:0) ρ | b (cid:1) ⊗ | b (cid:105)(cid:104) b | Z . Now let { K n } n be a set of Kraus operators correspond-ing to D C ← BZ and define L n,b := K n | b (cid:105) Z . Since D C ← BZ is a channel, we have (cid:80) n K † n K n = BZ , and hence (cid:88) n L † n,b L n,b = (cid:88) n (cid:104) b | Z K † n K n | b (cid:105) Z = (cid:104) b | Z BZ | b (cid:105) Z = B . For fixed b , the set { L n,b } n is thus a Kraus decompositionof a channel D C ← Bb , and we can write D C ← BZ (cid:16) C B ← A ⊗ id Z (cid:17) ( ρ AZ )= (cid:88) b : p b (cid:54) =0 p b D C ← BZ (cid:0) C B ← A (cid:0) ρ | b (cid:1) ⊗ | b (cid:105)(cid:104) b | Z (cid:1) = (cid:88) b : p b (cid:54) =0 p b (cid:88) n K n (cid:0) C B ← A (cid:0) ρ | b (cid:1) ⊗ | b (cid:105)(cid:104) b | Z (cid:1) K † n = (cid:88) b : p b (cid:54) =0 p b (cid:88) n L n,b (cid:0) C B ← A (cid:0) ρ | b (cid:1)(cid:1) L † n,b = (cid:88) b : p b (cid:54) =0 p b D C ← Bb C B ← A ( ρ | b ) , which completes the first part of the proof.If D C ← BZ ∈ MIO , we further obtain D C ← Bb ∆( ρ B ) = (cid:88) n K n | b (cid:105) Z ∆ ( ρ B ) (cid:104) b | Z K † n = (cid:88) n K n ∆ ( ρ B ⊗ | b (cid:105)(cid:104) b | Z ) K † n = D C ← BZ ∆ ( ρ B ⊗ | b (cid:105)(cid:104) b | Z )= ∆ D C ← BZ ∆ ( ρ B ⊗ | b (cid:105)(cid:104) b | Z )= ∆ D C ← Bb ∆( ρ B ) . Appendix E: Proofs of the results in the main text
Here we collect the proofs of the results in the maintext, which we repeat for readability.
Theorem 1.
The functionals M λ,(cid:126)φ ( C ) are convex mea-sures in the detection incoherent setting for all λ ∈ [0 , and for all (cid:126)φ ∈ R M .Proof. Let us start by proving nullity, i.e., we assume that C is detection incoherent. Then, since the compositionof free operations is a free operation, we have ∆ CE =∆ CE ∆.Defining the complementary dephasing operator as∆ c := id − ∆, we can write∆ (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) = ∆ (cid:16) λ − µ Λ (cid:126)φ (cid:17) (∆ + ∆ c )( ρ )= ∆( λ − µ )∆( ρ ) + ∆ (cid:16) λ − µ Λ (cid:126)φ (cid:17) ∆ c ( ρ )= ( λ − µ )∆( ρ ) . Here, the second line follows from the fact that Λ (cid:126)φ doesnot act on diagonal elements, while the third is due to thejoint action of ∆ c and ∆ which cancel firstly the diagonaland then the off-diagonal terms. Therefore, we find M λ,(cid:126)φ ( C ) = max ρ, E Tr (cid:12)(cid:12)(cid:12) ∆ CE (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) (cid:12)(cid:12)(cid:12) − | λ − µ | = max ρ, E Tr (cid:12)(cid:12)(cid:12) ∆ CE ∆ (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) (cid:12)(cid:12)(cid:12) − | λ − µ | = | λ − µ | max ρ, E Tr | ∆ CE ∆( ρ ) | − | λ − µ | = | λ − µ | − | λ − µ | = 0 , where the last line is due to complete positivity and tracepreservation of the operations ∆, C , and E .Non-negativity follows from M λ,(cid:126)φ ( C ) = max ρ, E Tr (cid:12)(cid:12)(cid:12) ∆ CE (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) (cid:12)(cid:12)(cid:12) − | λ − µ |≥ max ρ, E (cid:12)(cid:12)(cid:12) Tr (cid:16) ∆ CE (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) (cid:17)(cid:12)(cid:12)(cid:12) − | λ − µ | = | λ − µ | − | λ − µ | = 0 . F λ,(cid:126)φ ( C ) := max E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ CE (cid:16) λ − µ Λ (cid:126)φ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (E1)The extension to M λ,(cid:126)φ ( C ) is straightforward since thefunctionals only differ by a constant.We start proving that the functionals F λ,(cid:126)φ ( C ) satisfyEq. (6) by exploiting that a detection incoherent channel C DI cannot turn an incoherent POVM into a coherentone [59]. We therefore find F λ,(cid:126)φ ( C DI C ) = max E ,ρ Tr (cid:12)(cid:12)(cid:12) ∆ C DI CE (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) (cid:12)(cid:12)(cid:12) = max E ,ρ max P ∈P I Tr (cid:16) P C DI CE (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) (cid:17) = max E ,ρ max P ∈P I ,P (cid:48) = P C DI Tr (cid:16) P (cid:48) CE (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) (cid:17) ≤ max E ,ρ max P (cid:48) ∈P I Tr (cid:16) P (cid:48) CE (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) (cid:17) = F λ,(cid:126)φ ( C ) , where the inequality is due to an extension of the set overwhich we maximize.That the F λ,(cid:126)φ ( C ) also satisfy Eq. (7) is proven by F λ,(cid:126)φ ( CC DI ) = max E∈DI ,ρ Tr (cid:12)(cid:12)(cid:12) ∆ CC DI E (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) (cid:12)(cid:12)(cid:12) = max E∈DI , E (cid:48) = C DI E ,ρ Tr (cid:12)(cid:12)(cid:12) ∆ CE (cid:48) (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) (cid:12)(cid:12)(cid:12) ≤ max E (cid:48) ∈DI ,ρ Tr (cid:12)(cid:12)(cid:12) ∆ CE (cid:48) (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) (cid:12)(cid:12)(cid:12) = F λ,(cid:126)φ ( C ) . Next, we prove satisfaction of Eq. (8). Indeed, we willprove the slightly more general statement that the func-tionals F λ,(cid:126)φ ( C ) are constant under tensor product withthe identity. We begin with F λ,(cid:126)φ (cid:16) C A ⊗ id B (cid:17) = max E AB ← C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ (cid:16) C A ⊗ id B (cid:17) E AB ← C (cid:16) λ − µ Λ (cid:126)φ (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = max E AB ← C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) ∆ A C A ⊗ id B (cid:17) (cid:16) id A ⊗ ∆ B (cid:17) E AB ← C (cid:16) λ − µ Λ (cid:126)φ (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max p b , ˜ E A ← Cb ,ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) ∆ A C A ⊗ id B (cid:17)(cid:88) b p b (cid:16) ˜ E A ← Cb (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) (cid:17) ⊗ | b (cid:105)(cid:104) b | B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max p b , ˜ E A ← Cb ,ρ (cid:88) b p b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) ∆ A C A ⊗ id B (cid:17)(cid:16) ˜ E A ← Cb (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) (cid:17) ⊗ | b (cid:105)(cid:104) b | B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = max p b , ˜ E A ← Cb ,ρ (cid:88) b p b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ A C A (cid:16) ˜ E A ← Cb (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = max ˜ E A ← C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ A C A (cid:16) ˜ E A ← C (cid:16) λ − µ Λ (cid:126)φ (cid:17)(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = F λ,(cid:126)φ (cid:16) C A (cid:17) , where the first inequality follows from Lem. 10 and thesecond from convexity of the trace norm.On the other hand, we can also prove the inverse in-equality. We have F λ,(cid:126)φ (cid:16) C A ⊗ id B (cid:17) = max E AB ← C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ (cid:16) C A ⊗ id B (cid:17) E AB ← C (cid:16) λ − µ Λ C(cid:126)φ (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ max E A ← C ,ρ B ,ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ (cid:16) C A ⊗ id B (cid:17)(cid:16) E A ← C ⊗ ρ B (cid:17) (cid:16)(cid:16) λ − µ Λ C(cid:126)φ (cid:17) ( ρ ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = max E A ← C ,ρ,ρ B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ C A E A ← C (cid:16) λ − µ Λ C(cid:126)φ (cid:17) ( ρ ) ⊗ ∆ B ( ρ B ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = max E A ,ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ C A E A ← C (cid:16) λ − µ Λ C(cid:126)φ (cid:17) ( ρ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = F λ,(cid:126)φ (cid:16) C A (cid:17) , where the inequality is due to a restriction of the set overwhich we maximize. We conclude that M λ,(cid:126)φ ( C ⊗ id B ) = M λ,(cid:126)φ ( C )for all B , C , λ , and (cid:126)φ . Finally, convexity follows straight-forwardly from convexity of the trace norm. Theorem 2.
An auxiliary system does not increase thepre-processed improvements, i.e., for (cid:126) ˜ φ ( AZ ) such that Λ (cid:126) ˜ φ ( AZ ) := Λ A(cid:126)φ ⊗ id Z , it holds that M λ,(cid:126) ˜ φ ( AZ ) ( C ) = M λ,(cid:126)φ ( C ) ∀C ∀ λ ∀ (cid:126)φ ∀ Z. Proof.
Our proof rests upon showing an inequality inboth directions. Also here, we will prove the inequalitiesfor F λ,(cid:126) ˜ φ ( AZ ) ( C ) (defined in Eq. (E1)), with the extensionto M λ,(cid:126) ˜ φ ( AZ ) ( C ) being straightforward.In one direction, we have F λ,(cid:126) ˜ φ ( AZ ) ( C C ← B ) = F λ,(cid:126) ˜ φ ( AZ ) ( C C ← B ⊗ id D )= max E BD ← AZ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ (cid:16) C C ← B ⊗ id D (cid:17) E BD ← AZ (cid:16)(cid:16) λ − µ Λ A(cid:126)φ (cid:17) ⊗ id Z (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ max E B ← A , E D ← Z ,ρ A ,ρ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ (cid:16) C C ← B ⊗ id D (cid:17)(cid:0) E B ← A ⊗ E D ← Z (cid:1) (cid:16)(cid:16) λ − µ Λ A(cid:126)φ (cid:17) ⊗ id Z (cid:17) ( ρ A ⊗ ρ Z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = max E B ← A ,ρ A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ C C ← B E B ← A (cid:16) λ − µ Λ A(cid:126)φ (cid:17) ( ρ A ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) max E D ← Z ,ρ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆id D E D ← Z id Z ( ρ Z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = max E B ← A ,ρ A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ C C ← B E B ← A (cid:16) λ − µ Λ A(cid:126)φ (cid:17) ( ρ A ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = F λ,(cid:126)φ ( C C ← B ) , F λ,(cid:126)φ ( C C ← B ) = max E B ← A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ C C ← B E B ← A (cid:16) λ − µ Λ A(cid:126)φ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ max E B ← A = E B ← AZ E AZ ← A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ C C ← B E B ← AZ E AZ ← A (cid:16) λ − µ Λ A(cid:126)φ (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = max E B ← A = E B ← AZ E AZ ← A , | ϕ (cid:105) A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ C C ← B E B ← AZ E AZ ← A (cid:16) λ − µ Λ A(cid:126)φ (cid:17) ( | ϕ (cid:105)(cid:104) ϕ | A ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ max E B ← AZ , E AZ ← A ∧ | ϕ (cid:105) A : Eq. (D6) is satisfied (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ C C ← B E B ← AZ E AZ ← A (cid:16) λ − µ Λ A(cid:126)φ (cid:17) ( | ϕ (cid:105)(cid:104) ϕ | A ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = max E B ← AZ , | ψ (cid:105) AZ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ C C C ← B E B ← AZ (cid:16)(cid:16) λ − µ Λ A(cid:126)φ (cid:17) ⊗ id Z (cid:17) ( | ψ (cid:105)(cid:104) ψ | AZ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = F λ,(cid:126) ˜ φ ( AZ ) ( C C ← B ) , where the first inequality is due to a restriction of theset over which we maximize. In the following equality,we used that the maximum is always achieved on purestates due to convexity of the trace norm. The secondinequality follows from a further restriction of the setover which we maximize. The equality thereafter is dueto Lem. 12. Theorem 3.
For all (cid:126)φ ∈ R M with at least two differentcomponents, the functionals M λ,(cid:126)φ ( C ) are faithful if andonly if λ = .Proof. For this proof, we define the functionals F λ,(cid:126)φ ( C , E , ρ ) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ CE (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (E2)Let us start with λ = µ = . In this case, we have M ,(cid:126)φ ( C ) = F ,(cid:126)φ ( C ).Now assume that C can detect coherence. In otherwords, we are assuming that ∃ ˜ m, ˜ k, ˜ l such that ˜ k (cid:54) = ˜ l and C ˜ k, ˜ l ˜ m, ˜ m := (cid:104) ˜ m | C ( | ˜ k (cid:105)(cid:104) ˜ l | ) | ˜ m (cid:105) (cid:54) = 0. For simplicity, we take˜ m = 0 since we can always relabel the indices. We willshow that in this case, there exists a choice of E and ρ such that F ,(cid:126)φ ( C , E , ρ ) >
0, for F ,(cid:126)φ ( C , E , ρ ) as definedin Eq. (E2). This implies that for non-free C , we find M ,(cid:126)φ ( C ) >
0, which proves faithfulness.First, by assumption, there exists a couple ˜ i (cid:54) = ˜ j suchthat φ := φ ˜ i − φ ˜ j (cid:54) = 0. Writing again ρ = (cid:80) i,j ρ i,j | i (cid:105)(cid:104) j | , we have (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) = (cid:88) i,j | i (cid:105)(cid:104) j | (cid:16) λ − µe i ( φ i − φ j ) (cid:17) ρ i,j . (E3)Now we choose our initial state ˜ ρ such that ρ ˜ i, ˜ i = ρ ˜ j, ˜ j = and ρ ˜ i, ˜ j = e iξ = ρ ∗ ˜ j, ˜ i . Thus (cid:18) λ − µ Λ (cid:126)φ (cid:19) (˜ ρ ) = 12 (cid:0) | ˜ i (cid:105)(cid:104) ˜ i | + | ˜ j (cid:105)(cid:104) ˜ j | (cid:1) ( λ − µ )+ 12 (cid:0) | ˜ i (cid:105)(cid:104) ˜ j | e iξ (cid:0) λ − µe iφ (cid:1) + | ˜ j (cid:105)(cid:104) ˜ i | e − iξ (cid:0) λ − µe − iφ (cid:1)(cid:1) . The free operation E we choose is E SWAP defined via theunitary U SWAP = | ˜ i (cid:105)(cid:104) ˜ k | + | ˜ k (cid:105)(cid:104) ˜ i | + | ˜ j (cid:105)(cid:104) ˜ l | + | ˜ l (cid:105)(cid:104) ˜ j | + − | ˜ i (cid:105)(cid:104) ˜ i | − | ˜ k (cid:105)(cid:104) ˜ k | − | ˜ j (cid:105)(cid:104) ˜ j | − | ˜ l (cid:105)(cid:104) ˜ l | . According to our assumptions, C detects relative phasesbetween ˜ k , ˜ l and Λ (cid:126)φ encodes a relative phase between ˜ i , ˜ j .The operation E SWAP swaps the qubits that these pairsof indices define and leaves the remainder of the spaceunchanged, which ensures that the subspace in which C detects is aligned with the one in which Λ (cid:126)φ encodes (seealso App. A why this might be necessary). This will nowallow us to prove faithfulness. From E SWAP (cid:16) λ − µ Λ (cid:126)φ (cid:17) (˜ ρ ) = U SWAP (cid:16) λ − µ Λ (cid:126)φ (cid:17) (˜ ρ ) U † SWAP = 12 (cid:18) (cid:16) | ˜ k (cid:105)(cid:104) ˜ k | + | ˜ l (cid:105)(cid:104) ˜ l | (cid:17) ( λ − µ )+ | ˜ k (cid:105)(cid:104) ˜ l | e iξ (cid:0) λ − µe iφ (cid:1) + | ˜ l (cid:105)(cid:104) ˜ k | e − iξ (cid:0) λ − µe − iφ (cid:1) (cid:19) , and inserting explicit λ = , follows∆ CE SWAP (cid:16) λ − µ Λ (cid:126)φ (cid:17) (˜ ρ ) = 14 (cid:88) m (cid:18) C ˜ k, ˜ lm,m | m (cid:105)(cid:104) m | e iξ (cid:0) − e iφ (cid:1) + C ˜ l, ˜ km,m | m (cid:105)(cid:104) m | e − iξ (cid:0) − e − iφ (cid:1) (cid:19) = 12 (cid:88) m | m (cid:105)(cid:104) m | Re (cid:16) C ˜ k, ˜ lm,m e iξ (cid:0) − e iφ (cid:1)(cid:17) , where we used the index representation introduced in6App. D. Now, we have F ,(cid:126)φ ( C , E SWAP , ˜ ρ ) = 12 M − (cid:88) m =0 (cid:12)(cid:12)(cid:12) Re (cid:16) C ˜ k, ˜ lm,m e iξ (cid:0) − e iφ (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) = 12 (cid:12)(cid:12)(cid:12) Re (cid:16) C ˜ k, ˜ l , e iξ (cid:0) − e iφ (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) + 12 M − (cid:88) m =1 (cid:12)(cid:12)(cid:12) Re (cid:16) C ˜ k, ˜ lm,m e iξ (cid:0) − e iφ (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12) Re (cid:16) C ˜ k, ˜ l , e iξ (cid:0) − e iφ (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) + 12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Re (cid:32) M − (cid:88) m =1 C ˜ k, ˜ lm,m e iξ (cid:0) − e iφ (cid:1)(cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Re (cid:16) C ˜ k, ˜ l , e iξ (cid:0) − e iφ (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) , where the inequality stems from the triangle inequalityand the last line from Proposition 9.3 in App. D.Finally, recalling that max ξ Re( Ae iξ ) = | A | , we con-clude that M ,(cid:126)φ ( C ) ≥ max ξ F ,(cid:126)φ ( C , E SWAP , ˜ ρ ) ≥ (cid:12)(cid:12)(cid:12) C ˜ k, ˜ l , (cid:0) − e iφ (cid:1)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) C ˜ k, ˜ l , (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) − e iφ (cid:12)(cid:12) > C ˜ k, ˜ l , (cid:54) = 0 and φ (cid:54) = 0. Togetherwith Thm. 1, this finishes the case λ = µ = .Next, we show that whenever λ (cid:54) = µ , there exists achannel ˜ C / ∈ DI such that M λ,(cid:126)φ ( ˜ C ) = 0 for all (cid:126)φ . With E ∈ DI and starting from Eq. (E3), we have E (cid:16) λ − µ Λ (cid:126)φ (cid:17) ( ρ ) = (cid:88) i,k | k (cid:105)(cid:104) k | E i,ik,k ρ i,i ( λ − µ )+ (cid:88) i,j,k (cid:54) = l | k (cid:105)(cid:104) l | E i,jk,l ρ i,j (cid:16) λ − µe i ( φ i − φ j ) (cid:17) , (E4)where we made use of Eq. (D3) of App. D. Using thisrepresentation, we get F λ,(cid:126)φ ( C , E , ρ ) = (cid:88) m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) i,k C k,km,m E i,ik,k ρ i,i ( λ − µ )+ (cid:88) i,j,k (cid:54) = l C k,lm,m E i,jk,l ρ i,j (cid:16) λ − µe i ( φ i − φ j ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:88) m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) i,k C k,km,m E i,ik,k ρ i,i ( λ − µ ) (E5)+2 (cid:88) k>l,i,j Re (cid:16) C k,lm,m E i,jk,l ρ i,j (cid:16) λ − µe i ( φ i − φ j ) (cid:17)(cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . For a more convenient notation, we use the quantities A m := 2 (cid:88) k>l,i,j Re (cid:16) C k,lm,m E i,jk,l ρ i,j (cid:16) λ − µe i ( φ i − φ j ) (cid:17)(cid:17) from here on. In order for M λ,(cid:126)φ ( C ) to be faithful, weneed that F λ,(cid:126)φ ( C ) > | λ − µ | ∀C / ∈ DI . We further noticethat, if the terms inside the absolute value in Eq. (E5)have the same sign for all m , then F λ,(cid:126)φ ( C ) = | λ − µ | dueto trace preservation. Hence, a necessary condition for M λ,(cid:126)φ ( C ) > m such that (cid:88) i,k C k,k ˜ m, ˜ m E i,ik,k ρ i,i ( λ − µ ) + A m < . From here on, we consider a specific detecting operationthat will violate this condition, namely˜ C ( ρ ) = p QρQ † + p ρ, (E6)where p + p = 1, p i ≥
0, and Q a costly unitary (e.g.,a quantum Fourier transform). As long as p >
0, ˜ C isthus non-free. With this choice, we find that˜ C k,km,m = p | Q m,k | + p δ m,k , ˜ C k,lm,m = p Q m,k Q ∗ m,l ∀ k (cid:54) = l, where Q m,k are the matrix elements of the unitary Q .For faithfulness to hold, we thus need that there existsan ˜ m such that (cid:88) i,k ( p | Q ˜ m,k | + p δ ˜ m,k ) E i,ik,k ρ i,i ( λ − µ ) + A ˜ m < , or, using p = 1 − p , that p ( λ − µ ) (cid:88) i,k | Q ˜ m,k | E i,ik,k ρ i,i − (cid:88) i E i,i ˜ m, ˜ m ρ i,i + A ˜ m < − ( λ − µ ) (cid:88) i E i,i ˜ m, ˜ m ρ i,i . (E7)At this stage, we make some further assumptions. Firstly,we only consider the case λ > µ (the other case is anal-ogous). Secondly, we choose the detecting channel suchthat its input dimension is two. This forces the outputdimension of the pre-processing to be two as well.To finish the proof, we now examine two different cases,beginning with the one in which ∃ m (cid:48) s.t. ∀ i E i,im (cid:48) ,m (cid:48) = 0.Then, using Lem. 11, we obtain E i,jk,l = 0 ∀ k (cid:54) = l, ∀ i, j .This means that Eq. (E7), assuming w.l.o.g. ˜ m = 0,becomes p ( λ − µ ) | Q , | < m (cid:48) = 0 ,p ( λ − µ ) | Q , | < ( λ − µ )( p −
1) if m (cid:48) = 1 , where we used the fact that the trace of a density oper-ator is one. One easily verifies that the inequalities areindependent of ρ and E and cannot be satisfied in eithercase.7The second case is the one in which ∀ m ∃ i s.t. E i,im,m (cid:54) =0. Recalling Proposition 9.1 of App. D, we thus deducethat the right hand side of Eq. (E7) is negative. We nextchoose p > p ≤ min m ( λ − µ ) (cid:80) i E i,im,m ρ i,i (cid:12)(cid:12)(cid:12) ( λ − µ ) (cid:16)(cid:80) i,k | Q m,k | E i,ik,k ρ i,i − (cid:80) i E i,im,m ρ i,i (cid:17) + A m (cid:12)(cid:12)(cid:12) . Note that this is always possible, because all quantities inthe above fraction are finite and the numerator cannot bezero by assumption. Moreover, when the denominator iszero, we choose an arbitrary p with 0 < p ≤
1. For thischoice of p , the inequality (Eq. (E7)) cannot be satisfied.We conclude that, if λ (cid:54) = µ , there exist non-free oper-ations for which M λ,(cid:126)φ ( C ) = 0. Theorem 4.
The functionals N λ,(cid:126)φ ( C ) are convex mea-sures in the creation incoherent setting for all λ ∈ [0 , and for all (cid:126)φ ∈ R M .Proof. Large parts of this proof are very similar to theproof of Thm. 1. For completeness, we give the full proofnevertheless, starting with nullity. Let us assume that
C ∈ MIO , i.e., C ∆ = ∆ C ∆. If D is also creation inco-herent, then DC ∆ = ∆ DC ∆. Noticing that Λ (cid:126)φ ∆ = ∆,we thus get N λ,(cid:126)φ ( C ) = max D∈MIO (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) λ − µ Λ (cid:126)φ (cid:17) DC ∆ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − | λ − µ | = max D∈MIO (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) λ − µ Λ (cid:126)φ (cid:17) ∆ DC ∆ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − | λ − µ | = max D∈MIO || ( λ − µ ) ∆ DC ∆ || − | λ − µ | = | λ − µ | max D∈MIO || ∆ DC ∆ || − | λ − µ | = | λ − µ | − | λ − µ | = 0 . Non-negativity holds due to N λ,(cid:126)φ ( C ) = max D∈MIO ,ρ ∈I Tr (cid:12)(cid:12)(cid:12)(cid:16) λ − µ Λ (cid:126)φ (cid:17) DC ( ρ ) (cid:12)(cid:12)(cid:12) − | λ − µ |≥ max D∈MIO ,ρ ∈I (cid:12)(cid:12)(cid:12) Tr (cid:16)(cid:16) λ − µ Λ (cid:126)φ (cid:17) DC ( ρ ) (cid:17)(cid:12)(cid:12)(cid:12) − | λ − µ | = | λ − µ | − | λ − µ | = 0 . We now proceed with the three proofs of monotonicity,again for the functionals G λ,(cid:126)φ ( C ) := max D∈MIO (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) λ − µ Λ (cid:126)φ (cid:17) DC ∆ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (E8)since the extensions to N λ,(cid:126)φ ( C ) are straightforward.We begin showing that all G λ,(cid:126)φ ( C ) satisfy Eq. (6). Letus assume C ∈ MIO . Then G λ,(cid:126)φ ( C C ) = max D∈MIO ,ρ ∈I Tr (cid:12)(cid:12)(cid:12)(cid:16) λ − µ Λ (cid:126)φ (cid:17) DC C ( ρ ) (cid:12)(cid:12)(cid:12) = max D (cid:48) = DC ∈MIO ,ρ ∈I Tr (cid:12)(cid:12)(cid:12)(cid:16) λ − µ Λ (cid:126)φ (cid:17) D (cid:48) C ( ρ ) (cid:12)(cid:12)(cid:12) ≤ max D (cid:48) ∈MIO ,ρ ∈I Tr (cid:12)(cid:12)(cid:12)(cid:16) λ − µ Λ (cid:126)φ (cid:17) D (cid:48) C ( ρ ) (cid:12)(cid:12)(cid:12) = G λ,(cid:126)φ ( C ) . Eq. (7) is always satisfied too because, with C ∈ MIO again, we have G λ,(cid:126)φ ( CC ) = max D∈MIO ,ρ ∈I Tr (cid:12)(cid:12)(cid:12)(cid:16) λ − µ Λ (cid:126)φ (cid:17) DCC ( ρ ) (cid:12)(cid:12)(cid:12) = max D∈MIO ,ρ (cid:48) = C ( ρ ) ∈I Tr (cid:12)(cid:12)(cid:12)(cid:16) λ − µ Λ (cid:126)φ (cid:17) DC ( ρ (cid:48) ) (cid:12)(cid:12)(cid:12) ≤ max D∈MIO ,ρ (cid:48) ∈I Tr (cid:12)(cid:12)(cid:12)(cid:16) λ − µ Λ (cid:126)φ (cid:17) DC ( ρ (cid:48) ) (cid:12)(cid:12)(cid:12) = G λ,(cid:126)φ ( C ) . To conclude, we need to prove validity of Eq. (8). Again,we will show that the G λ,(cid:126)φ ( C ) are constant under tensorproduct by establishing inequalities in both directions.We begin with G λ,(cid:126)φ (cid:16) C B ← A ⊗ id Z (cid:17) = max D C ← BZ ∈MIO ,ρ AZ ∈I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) λ − µ Λ (cid:126)φ (cid:17) D C ← BZ (cid:16) C B ← A ⊗ id Z (cid:17) ( ρ AZ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = max D C ← BZ ∈MIO ,ρ AZ ∈I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) λ − µ Λ (cid:126)φ (cid:17)(cid:88) i p i ( ρ AZ ) D C ← Bi ( D C ← BZ ) C B ← A (cid:0) ρ | i ( ρ AZ ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max p i , D C ← Bi ∈MIO ,ρ | i ∈I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) λ − µ Λ (cid:126)φ (cid:17)(cid:88) i p i D C ← Bi C B ← A ( ρ | i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max p i , D C ← Bi ∈MIO ,ρ | i ∈I (cid:88) i p i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) λ − µ Λ (cid:126)φ (cid:17) D C ← Bi C B ← A ( ρ | i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max D C ← B ∈MIO ,ρ ∈I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) λ − µ Λ (cid:126)φ (cid:17) D C ← B C B ← A ( ρ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = G λ,(cid:126)φ ( C B ← A ) , where the second equality is to be understood in the senseof Lem. 13. The first inequality is due to an extension ofthe set over which we maximize and the second inequalityfollows from convexity of the trace norm.Finally, we prove the inverse inequality with G λ,(cid:126)φ ( C B ← A ⊗ id Z ) ≥ G Λ (cid:126)φ (cid:16) Tr Z (cid:16) C B ← A ⊗ id Z (cid:17)(cid:17) = G λ,(cid:126)φ ( C B ← A ) , where we used monotonicity under right composition,and conclude that the post-processed improvements areconstant under tensor product with the identity channel.Also here, convexity follows straightforwardly fromconvexity of the trace norm. The post-processed im-provements are thus convex measures in the detectionincoherent setting. Theorem 5.
The functionals N ,(cid:126)φ ( C ) are faithful for all (cid:126)φ ∈ R M with at least two different components. Proof.
The proof is similar to the corresponding part ofthe proof of Thm. 3. By assumption, Λ (cid:126)φ cannot add onlya global phase, thus ∃ ˜ m, ˜ n s.t. ˜ m (cid:54) = ˜ n and φ ˜ m (cid:54) = φ ˜ n . We further assume that the operation C : A → B is notfree, i.e., ∃ ˜ i, ˜ k, ˜ l s.t. ˜ k (cid:54) = ˜ l and C ˜ i, ˜ i ˜ k, ˜ l (cid:54) = 0 . As in the proof of Thm. 3, we now consider an explicitchoice of an incoherent state, ˜ ρ = (cid:12)(cid:12) ˜ i (cid:11)(cid:10) ˜ i (cid:12)(cid:12) , and a free op-eration ˜ D that is composed of two creation incoherentchannels. The first one is a SWAP operation defined viathe unitary U SWAP = | ˜ m (cid:105)(cid:104) ˜ k | + | ˜ k (cid:105)(cid:104) ˜ m | + | ˜ n (cid:105)(cid:104) ˜ l | + | ˜ l (cid:105)(cid:104) ˜ n | + − | ˜ m (cid:105)(cid:104) ˜ m | − | ˜ k (cid:105)(cid:104) ˜ k | − | ˜ n (cid:105)(cid:104) ˜ n | − | ˜ l (cid:105)(cid:104) ˜ l | that aligns again the relevant subspaces (see the proofof Thm. 3) and the second channel is defined via Krausoperators K := | ˜ m (cid:105)(cid:104) ˜ m | + | ˜ n (cid:105)(cid:104) ˜ n | , L j := | j (cid:105)(cid:104) j | ∀ j (cid:54) = ˜ m, ˜ n. We therefore have C (˜ ρ ) = (cid:88) k,l C ˜ i, ˜ ik,l | k (cid:105)(cid:104) l | , and˜ DC (˜ ρ )= C ˜ i, ˜ i ˜ k, ˜ l | ˜ m (cid:105)(cid:104) ˜ n | + C ˜ i, ˜ i ˜ l, ˜ k | ˜ n (cid:105)(cid:104) ˜ m | + C ˜ i, ˜ i ˜ k, ˜ k | ˜ m (cid:105)(cid:104) ˜ m | + C ˜ i, ˜ i ˜ l, ˜ l | ˜ n (cid:105)(cid:104) ˜ n | + C ˜ i, ˜ i ˜ m, ˜ m | ˜ k (cid:105)(cid:104) ˜ k | + C ˜ i, ˜ i ˜ n, ˜ n | ˜ l (cid:105)(cid:104) ˜ l | + (cid:88) j / ∈{ ˜ k, ˜ l, ˜ m, ˜ n } C ˜ i, ˜ ij,j | j (cid:105)(cid:104) j | . This time, we define A := (cid:16) id − Λ (cid:126)φ (cid:17) ˜ DC (˜ ρ )= (cid:16) − e i ( φ ˜ m − φ ˜ n ) (cid:17) C ˜ i, ˜ ik,l | ˜ m (cid:105)(cid:104) ˜ n | + (cid:16) − e − i ( φ ˜ m − φ ˜ n ) (cid:17) C ˜ i, ˜ il,k | ˜ n (cid:105)(cid:104) ˜ m | , and have thus A † A = (cid:12)(cid:12)(cid:12)(cid:16) − e i ( φ ˜ m − φ ˜ n ) (cid:17) C ˜ i, ˜ ik,l (cid:12)(cid:12)(cid:12) | ˜ m (cid:105)(cid:104) ˜ m | + (cid:12)(cid:12)(cid:12)(cid:16) − e i ( φ ˜ m − φ ˜ n ) (cid:17) C ˜ i, ˜ ik,l (cid:12)(cid:12)(cid:12) | ˜ n (cid:105)(cid:104) ˜ n | and √ A † A = (cid:12)(cid:12)(cid:12)(cid:16) − e i ( φ ˜ m − φ ˜ n ) (cid:17) C ˜ i, ˜ ik,l (cid:12)(cid:12)(cid:12) ( | ˜ m (cid:105)(cid:104) ˜ m | + | ˜ n (cid:105)(cid:104) ˜ n | ) . 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