Measuring coherence of quantum measurements
Valeria Cimini, Ilaria Gianani, Marco Sbroscia, Jan Sperling, Marco Barbieri
MMeasuring coherence of quantum measurements
Valeria Cimini, Ilaria Gianani,
1, 2
Marco Sbroscia, Jan Sperling, ∗ and Marco Barbieri
1, 4 Dipartimento di Scienze, Universit`a degli studi Roma Tre, Via della vasca Navale 84, 00146 Rome, Italy Dipartimento di Fisica, Sapienza Universit`a di Roma, Piazzale Aldo Moro 5, 00185 Rome, Italy Integrated Quantum Optics Group, Applied Physics, University of Paderborn, 33098 Paderborn, Germany Consiglio Nazionale delle Ricerche, Largo E. Fermi 6, 50125 Florence, Italy (Dated: October 14, 2019)The superposition of quantum states lies at the heart of physics and has been recently found to serve as aversatile resource for quantum information protocols, defining the notion of quantum coherence. In this con-tribution, we report on the implementation of its complementary concept, coherence from quantum measure-ments. By devising an accessible criterion which holds true in any classical statistical theory, we demonstratethat noncommutative quantum measurements violate this constraint, rendering it possible to perform an op-erational assessment of the measurement-based quantum coherence. In particular, we verify that polarizationmeasurements of a single photonic qubit, an essential carrier of one unit of quantum information, are alreadyincompatible with classical, i.e., incoherent, models of a measurement apparatus. Thus, we realize a methodthat enables us to quantitatively certify which quantum measurements follow fundamentally different statisticallaws than expected from classical theories and, at the same time, quantify their usefulness within the modernframework of resources for quantum information technology.
I. INTRODUCTION
Quantum interference phenomena are a key property thatenable us discern classical physics from the quantum realm[1–5]. Different forms of quantum coherence constitute thebasis for a variety of notions of nonclassicality, such as en-tanglement which is a result of nonlocal superpositions [6–10]. The application of quantum coherence as a resource forquantum information protocols recently gained a lot of atten-tion (see Refs. [11–13] for introductions) because it connectsfundamental question about the physical nature with practicalaspects of upcoming quantum technologies.In order to show how classical expectations are supersededby quantum physics, a number of measurable criteria havebeen proposed. Most prominently, Bell’s inequality [14] en-ables us to show that local hidden-variable models do not suf-ficiently describe general correlations between quantum sys-tems. More generally, the concept of contextuality provides abroadly applicable approach which demonstrates the superior-ity of quantum-mechanical joint probabilities over their classi-cal counterparts [15, 16]. The underlying constraints for bothexamples provide criteria which were derived in a classicalpicture to demonstrate how quantum physics overcomes clas-sical limitations; see Ref. [17] for a recent experiment.Conversely, the resource-theoretic framework of quantumcoherence is already formulated in the quantum formalismand quantifies the operational usefulness of superpositionswhen compared to quantum-statistical ensembles [11–13]. Infact, quantum superpositions themselves can directly serve asa measure of quantum coherence [18–20]. Moreover, the gen-eral concept of quantum coherence encompasses previous no-tions of quantumness, e.g., quantum-optical nonclassicality asformulated by Glauber and others [21–25], and thereby ren-ders it possible to distinguish classical interference phenom-ena from coherence effects genuine to quantum physics. ∗ [email protected] Except for some recent approaches with remarkable impli-cations [26–31], the state-based approach to quantum coher-ence does not address the quantumness of measurement itself[32]. However, there can be no doubt that the understandingof the nature of measurements is vital to the fundamentals ofphysics and its practical applications, e.g., allowing us to im-plement measurement-based quantum computation [33, 34].Other scenarios in which the coherence of measurements be-comes essential relate to the manipulation and preparationof quantum states [35–38], conditional quantum correlations[39, 40], and questions concerning the collapse of the wavefunction [26, 41–44]. The other way around, Heisenberg’sseminal uncertainty relation [45] poses a fundamental preci-sion limitation to quantum measurements of multiple observ-ables [46–48], which is not the case in classical models. Thus,an experimentally accessible distinction between classical andquantum statistics, based on the outcomes of measurements, isvital for many applications. While some measurements havebeen performed, for example, to confirm the noncommutativ-ity of certain observables [49, 50], a general connection be-tween the quantumness of measurements and the state-basednotion of quantum coherence, together with its experimentalcertification, is still missing.In this contribution, we close this gap between the theoryof quantum coherence of states and experiments with incom-patible quantum measurements. To derive our experimentallyaccessible and generally applicable criteria, we first perform aderivation in a purely classical framework; second, we relateour findings to quantum coherence of measurements. Then,we apply our technique to data obtained in our experimentof polarization measurements of photons, detecting one qubitof information. Our results not only verify with high statis-tical significance if and when a classical interpretation of ameasurement ultimately fails in quantum systems, but it alsoprovides a quantifier of the measurement-based quantum co-herence. Thus, we provide and implement a practical tool tostudy the fundamentals and application-oriented properties ofquantum measurements. a r X i v : . [ qu a n t - ph ] O c t II. CLASSICAL LAW OF TOTAL PROBABILITIES
Like the approach by Bell and others, let us formulate ourclassical constraints solely based on universally valid featuresof classical statistics. For this reason, we consider a probabil-ity distribution P , where P ( x ) and P ( y ) are the probabilitiesto measure the outcomes x and y for two random variables.Further, the probability to measure y after a measurement of x is given by the conditional probability P ( y | x ) = P ( y , x ) / P ( x ) ,where P ( y , x ) is the joint probability for the given outcomes.Consequently, the probability to detect y regardless of theprior outcome x is given by P (cid:48) ( y ) = ∑ x P ( y | x ) P ( x ) . Accordingto the law of total probability [51, 52], we have P ( y ) cl. = P (cid:48) ( y ) (1)for any classical system. It is worth emphasizing that the lawof total probability applies to any classical model even if themeasurement is not an ideal one.Using the classical identity (1), we can now formulate avariance-based constraint for classical statistics, V P ( y ) [ y ] cl. = V P (cid:48) ( y ) [ y ] , (2)where V denotes the variance. This classical relation is knownas the law of total variance [51, 52] and follows from thedecomposition V P (cid:48) ( y ) [ y ] = E P ( x ) [ V P ( y | x ) [ y ]] + V P ( x ) [ E P ( y | x ) [ y ]] ,which is based on the construction of P (cid:48) via conditional prob-abilities and where E denotes the mean value. A violation ofthe classically universal law in Eq. (2) certifies the incompat-ibility of the measurement with classical statistics. It is worthemphasizing that beyond the second-order criterion (2), moresophisticated generalizations are possible using Eq. (1). III. RELATION TO QUANTUM COHERENCE
Let us now establish the relation of the above criterion tothe notion of quantum coherence. For this reason, we identifyan observable, represented through the operator ˆ x , to serveas our incoherent gauge when compared to a second, generalobservable ˆ y . The decomposition of those observables readsˆ x = ∑ x x ˆ Ξ x and ˆ y = ∑ y y ˆ Π y , using the positive operator-valuedmeasures { ˆ Ξ x } x and { ˆ Π y } y .Measuring the outcome x is achieved with the proba-bility P ( x ) = tr ( ˆ ρ ˆ Ξ x ) = (cid:104) ˆ Ξ x (cid:105) ˆ ρ and leaves us with a post-measurement state ˆ ρ x = ˆ Ξ / x ˆ ρ ˆ Ξ / x / P ( x ) . In analogy to theclassical case, we now ignore the first outcome, resulting inˆ ρ (cid:48) = ∑ x P ( x ) ˆ ρ x = ∑ x ˆ Ξ / x ˆ ρ ˆ Ξ / x . (3)For our purpose, it is now convenient to define that a state isincoherent if the map in Eq. (3) leaves the state unchanged,i.e., ˆ ρ (cid:55)→ ˆ ρ (cid:48) = ˆ ρ . Conversely, quantum coherence is given byˆ ρ (cid:48) (cid:54) = ˆ ρ . In this sense, ˆ ρ (cid:55)→ ˆ ρ (cid:48) is a so-called strictly incoherentoperation [2, 53]. In general, assessing coherence demands achoice of a preferred basis on grounds of physical considera-tions. Here, it is motivated through a detection of ˆ x because, in itself, it does have a completely classical model in terms ofthe measured statistics P ( x ) .For comparing the two cases, the measurement of ˆ y withoutand with a prior measurement of ˆ x yields V P ( y ) [ y ] = (cid:104) ( ∆ ˆ y ) (cid:105) ˆ ρ and V P (cid:48) ( y ) [ y ] = (cid:104) ( ∆ ˆ y ) (cid:105) ˆ ρ (cid:48) , (4)respectively, corresponding to the variances for the previouslydiscussed classical case. Here, however, the classical law oftotal variances does not apply, and we can find (cid:104) ( ∆ ˆ y ) (cid:105) ˆ ρ (cid:54) = (cid:104) ( ∆ ˆ y ) (cid:105) ˆ ρ (cid:48) in the presence of quantum coherence, ˆ ρ (cid:54) = ˆ ρ (cid:48) .It is worth emphasizing that our approach does indeed mea-sure the incompatibility of the performed measurements be-cause [ ˆ Ξ x , ˆ Π y ] = ∀ x , y ) implies P ( y ) = P (cid:48) ( y ) , regardless ofthe coherence of the initial state ˆ ρ [54]. Therefore, whenthe classical constraint (2) is violated, we can directly inferthat the quantum measurement ˆ y exhibits quantum coherencewith respect to the detection of ˆ x . In an ideal scenario, wherethe measurements are represented through orthonormal bases,i.e., ˆ Ξ x = | x (cid:105)(cid:104) x | and ˆ Π y = | y (cid:105)(cid:104) y | , this means that the mea-surement of ˆ y is not described through incoherent mixtures( ˆ Π y (cid:54) = ∑ x q y , x | x (cid:105)(cid:104) x | ); rather, it requires quantum superpositions( | y (cid:105) = ∑ x c y , x | x (cid:105) ), i.e., quantum coherence in the measurementoperators. More specifically, our intermediate definition ofthe coherence of the states, based on the measurement of ˆ x [cf. Eq. (3)], actually serves as a proxy to infer the quantumcoherence of the second measurement ˆ y when compared to ˆ x .Consequently, we have formulated an observable criterion thatassesses the quantum coherence of measurements and whichis based off of the classical law of total probabilities. IV. IMPLEMENTATION
We explore the previously devised concepts for qubits.While our approach applies to arbitrary system, qubits arefundamental quantum objects as they represent the basic unitof quantum information science [33, 55, 56]. Our qubitsare encoded in the polarization of single photons. The pre-ferred basis is given by the horizontal ( H ) and vertical ( V )polarization, also defining the reference measurement ˆ x = −| H (cid:105)(cid:104) H | + | V (cid:105)(cid:104) V | = (cid:2) − (cid:3) . The states we prepare take thegeneral form ˆ ρ = (cid:20) − p (cid:112) p ( − p ) γ (cid:112) p ( − p ) γ p (cid:21) , (5)where p ∈ [ , ] indicates the population unbalance betweenthe two levels, and the parameter γ ∈ [ , ] determines the co-herence in the state when compared to its incoherent counter-part, ˆ ρ (cid:48) = ( − p ) | H (cid:105)(cid:104) H | + p | V (cid:105)(cid:104) V | , cf. Eq. (3). Note that γ can be additionally equipped with a complex phase factor, e i φ ,to account for the most general case of a qubit; this, however,does not lead to any conceptional advantage and is, therefore,fixed to one ( e i φ =
1) in our treatment. The qubit is subjectedto two consecutive measurements. The first one measures thePauli- z operator (here, denoted as ˆ x ); the second one measuresan arbitrary observableˆ y = cos θ (cid:20) − (cid:21) + sin θ (cid:20) (cid:21) , (6)parametrized through the angle θ .In our experiment, we prepare linear polarization statesfrom H -polarized photons by means of a half-wave plate(HWP) at an angle α —hence, p = sin ( α ) . The statisticsfor mixed states is obtained by inputting states correspond-ing to the setting + α and − α with the respective weights w + , w − ≥
0, chosen in such a way that w + + w − = w + − w − = γ hold true.In order to implement the ˆ x measurement without destroy-ing the signal photon, we couple the photon with an ancillarymeter by means of a controlled sign gate [57–59]. This is con-stituted by a beam splitter with polarization-dependent trans-mittivities, T H = T V = /
3. Two-photon nonclassical in-terference occurs selectively on the beam splitter only for thevertical components of the signal and the meter photon polar-izations. These consequently acquire a π phase shift with re-spect to the other three terms when post-selecting those eventsin which the two photons emerge on distinct outputs of thebeam splitter—this then demands to register only coincidenceevents between the arms. In our setup, we encounter slight im-perfections, such as T H = . (cid:54) = T V = . (cid:54) = / ◦ , are used to balance polarization-dependent lossinduced by the first one [60]. Remaining discrepancies be-tween our data and the theoretical modeling can be attributedto reduced visibility of the two-photon interference, as well asthe Sagnac interferometer. This, however, does not affect theworking of our criterion (2) as it applies to arbitrary measure-ments, including their imperfections.The action of the described gate can be switched on andoff by controlling the polarization of the meter [61–63]. Con-sider a pure signal state state ( γ =
1) arriving at the gate. If a H -polarized meter is injected, no coupling can occur; hence,the joint state remains separable, (cid:0) √ − p | H (cid:105) s + √ p | V (cid:105) s (cid:1) ⊗| H (cid:105) m . No information can be inferred from the meter aboutthe signal. If the meter is, however, injected in a diago-nal polarization, | + (cid:105) m = ( | H (cid:105) m + | V (cid:105) m ) / √
2, the output two-photon polarization state is entangled, √ − p | H (cid:105) s ⊗ | + (cid:105) m + √ p | V (cid:105) s ⊗ |−(cid:105) m , due to the phase shift imparted on the gate.By measuring the meter in the diagonal basis, one can extractthe full information about the ˆ x measurement of the originalsignal state. The observable ˆ y is then measured conventionallyby a HWP at β = θ / (cid:104) ( ∆ ˆ y ) (cid:105) ˆ ρ = − [( p − ) cos θ + (cid:112) p ( − p ) γ sin θ ] ; second, we mea-sure the variance resulting from a prior measurement of ˆ x , ex-pected to follow (cid:104) ( ∆ ˆ y ) (cid:105) ˆ ρ (cid:48) = − ( − p ) cos θ . In orderto account for experimental artifacts, both measurements areperformed with the photons passing through the gate, with the FIG. 1. Variance difference, representing the deviation from ourclassical constraint, ∆ V = p and θ ; the points show the experimental data.(b) Cut of the plot (a) for p = . p = . polarization of the meter set accordingly, and registering co-incidences counts. In both cases, the polarization of the meteris not analyzed since we are ignoring the outcome x , as ex-pressed in Eq. (3). V. RESULTS
To assess the amount of measurement-induced quantum co-herence when compared to the classical constraint (2), it isconvenient to consider the following difference: ∆ V = V P (cid:48) ( y ) [ y ] − V P ( y ) [ y ] = (cid:104) ( ∆ ˆ y ) (cid:105) ˆ ρ (cid:48) − (cid:104) ( ∆ ˆ y ) (cid:105) ˆ ρ . (7)Because of Eq. (2), a significant deviation from ∆ V = ∆ V as a func-tion of θ and p for a measurement of ˆ y [Eq. (6)] for pure states[ γ = θ ≈
0, i.e., [ ˆ x , ˆ y ] = ∆ V =
0) regardlessof the input state as predicted in the theory part. Moreover, wecan rule out that our system can be mimicked by any classi-cal model of a measurement as ∆ V significantly deviates fromzero in almost all other cases. For specific values of p , thedeviation from the classical bound is shown in Figs. 1(b) and1(c). In particular, Fig. 1(c) certifies that the maximal viola-tion is obtained for θ ≈ π /
2, which corresponds to a detection
FIG. 2. Variance difference between the two measurement configu-rations [Eq. (7)]. (a) The surface is the expected theoretical behavioras a function of γ and θ ; the points depict our data. (b) Cut of theplot (a) along θ = ◦ . (c) Cut of the plot (a) along θ = ◦ .FIG. 3. Maximal violation of ∆ V = θ ≈ π / p and γ in plots (a) and (b), respectively. Plot (a) is expected to besymmetric with respect to p = / of ˆ y that is a Pauli- x measurement, and thus maximally incom-patible with the reference measurement ˆ x , the Pauli- z operator.In addition, we explore mixed states to probe the quantumcoherence between the measurements in Fig. 2(a). For thispurpose, we fix the value of p = sin ( α ) at α = ◦ and studythe difference of the variances as a function of γ [Eq. (5)].We can observe that, in general, the highest purity ( γ → ∆ V (cid:54) =
0, which also represents the scenario with the highestcoherence of the probe state ˆ ρ . Again, Figs. 2(b) and 2(c)show the cuts with the optimal deviations ∆ V from the classi-cal bound zero.Finally, we can also measure the maximal deviation of thestate ˆ ρ prior to the measurement ˆ x and ˆ ρ (cid:48) after the detectiontook place [64, 65], cf. Eq. (3). The result is shown inFig. 3 as functions of p and γ , defining the prepared statein Eq. (5). The shown results enable us to quantify themeasurement-induced decoherence because one can straight-forwardly prove [66] that the square of trace distance be-tween ˆ ρ and ˆ ρ (cid:48) is identical to ∆ V for θ ≈ π /
2. For instance,we can conclude from Figs. 3(a) and 3(b) that the maxi-mal decoherence occurs for p ≈ / γ ≈
1. This corre- sponds to a maximally coherent input state, ˆ ρ = | ψ (cid:105)(cid:104) ψ | with | ψ (cid:105) = ( | H (cid:105) + | V (cid:105) ) / √
2, that is converted into a maximally in-coherent one, ˆ ρ (cid:48) = ( | H (cid:105)(cid:104) H | + | V (cid:105)(cid:104) V | ) /
2, through the detec-tion of ˆ x . VI. DISCUSSION
In summary, we formulated and implemented a method thatenables us to certify quantum coherence between two mea-surements. We applied the law of total probabilities (andvariances) to formulate conditions that apply to all classicalmeasurements. The translation to the quantum domain en-abled us to violate these classical requirements, and therebywe revealed a connection to the notion of quantum coherencebetween measurements. We confirmed our theory by prob-ing the quantumness of different and essential qubit measure-ments, encoded in the polarization of photons. This allowedus to experimentally verify the fundamental incompatibilityof quantum measurements with classical statistical models ona quantitative basis. Furthermore, we were able to assessthe measurement-induced decoherence which occurs when ameasurement is performed on a quantum system.Our studies reveal fundamental and application-orientedproperties genuine to the quantum description of measure-ments. First, we confirmed—with an easily accessible, al-ternative approach and high statistical significance—that thequantum statistics of measurements has fundamentally differ-ent properties than expected from any classical perspective.Second, we were able to connect the resource-theoretic notionof quantum coherence of quantum states to the coherence be-tween two measurement scenarios. Specifically, one measure-ment defines a classical reference, the incompatibility of thisreference with the employed state and the second measure-ment then lead to quantum effects beyond classical physics. Inthis scenario, the coherence of the state serves as a medium toprove that the description of the second measurement requiresquantum superpositions since for commuting observables, anyquantum coherence of the state become meaningless. Thisfurther demonstrates that, in quantum physics, it makes a pro-found difference if one measures a second observable in thecontext of preceding one or not—even if one is ignorant tothe outcome of the first detection event. Note that the role ofthe first and second measurements is fixed by reasons of ex-perimental practicality and can be exchanged in our treatmentwithout affecting any of our general observations.Our approach also enables us to quantify the loss of co-herence as a result of the alteration of a state after a quantum-measurement process took place, relating to the collapse of thewave function. In particular, we show that intervening with ameasurement has a disruptive action on the quantum informa-tion carried by the state’s coherence. Indeed, a prior measure-ment cancels the presence of coherence in the state, affecting asubsequent measurement, which is also the basis of our quan-tumness criteria. Furthermore, our second-order criterion canbe straightforwardly extended to higher order correlations andother nonlinear statistical quantifiers, such as the entropy, out-lining possible generalizations. Moreover, our method can bealso generalized to compare more than two measurements ina pairwise manner or when measured successively. We alsowant to stress that our approach applies to discrete and con-tinuous variables alike, and its purely classical derivation doesnot presuppose any knowledge about quantum physics, suchas notions of eigenstates, measurement operators, collapse ofthe wave function, etc. This is in contrast to other attempts toquantify the coherence of measurements that require a quan-tum description.It is also worth emphasizing that measurement-based quan-tum protocols rely on realizing measurements which are in-compatible. Here, we were able to assess the quantum co-herence between such measurements to quantify this resourceof incompatibility, which is analogous to the requirementson quantum protocols which exploit the coherence of thestate. Based on our method interpreted as a means to quantifythe measurement-induced decoherence, we have additionally demonstrated how to infer the coherence in the trace distanceof a qubit state via the performed measurements, providing thenecessary information for the success of certain quantum tasks[67, 68]. Furthermore, the variance-based form of our crite-rion enables us to predict the precision to estimate a quantityin settings which consists of noncommuting measurements,which is useful, e.g., when comparing classical and quantummetrology [69]. Thus, we also provide a useful tool to quan-tify the coherence of measurements for practical purposes.
ACKNOWLEDGMENTS
M. S. acknowledges support from the ADAMO projectof Distretto Tecnologico Beni e Attivit`a Culturali, RegioneLazio. The Integrated Quantum Optics group acknowledgesfinancial support from the Gottfried Wilhelm Leibniz-Preis(Grant No. SI1115/3-1). [1] T. Baumgratz, M. Cramer, and M. B. Plenio,
Quantifying Co-herence , Phys. Rev. Lett. , 140401 (2014).[2] A. Winter and D. Yang,
Operational Resource Theory of Co-herence , Phys. Rev. Lett. , 120404 (2016).[3] T. Theurer, N. Killoran, D. Egloff, and M. B. Plenio,
ResourceTheory of Superposition , Phys. Rev. Lett. , 230401 (2017).[4] T. Biswas, M. G. D´ıaz, and A. Winter,
Interferometric visibil-ity and coherence , Proc. Roy. Soc. London A , 20170170(2017).[5] Y.-T. Wang, J.-S. Tang, Z.-Y. Wei, S. Yu, Z.-J. Ke, X.-Y. Xu, C.-F. Li, and G.-C. Guo,
Directly Measuring the Degree of Quan-tum Coherence using Interference Fringes , Phys. Rev. Lett. ,020403 (2017).[6] W. Vogel and J. Sperling,
Unified quantification of nonclassi-cality and entanglement , Phys. Rev. A , 052302 (2014).[7] A. Streltsov, U. Singh, H. S. Dhar, M. N. Bera, and G. Adesso, Measuring Quantum Coherence with Entanglement , Phys. Rev.Lett. , 020403 (2015).[8] N. Killoran, F. E. S. Steinhoff, and M. B. Plenio,
Convert-ing Nonclassicality into Entanglement , Phys. Rev. Lett. ,080402 (2016).[9] E. Chitambar and M.-H. Hsieh,
Relating the Resource Theoriesof Entanglement and Quantum Coherence , Phys. Rev. Lett. ,020402 (2016).[10] L.-F. Qiao et al. , Entanglement activation from quantum coher-ence and superposition , Phys. Rev. A , 052351 (2018).[11] G. Adesso, T. R. Bromley, and M. Cianciaruso, Measures andapplications of quantum correlations , J. Phys. A: Math. Theor. , 473001 (2016).[12] A. Streltsov, G. Adesso, and M. B. Plenio, Quantum coherenceas a resource , Rev. Mod. Phys. , 041003 (2017).[13] E. Chitambar and G. Gour, Quantum Resource Theories , Rev.Mod. Phys. , 025001 (2019).[14] J. S. Bell, On the Einstein Podolsky Rosen paradox , Physics ,195 (1964).[15] M. Howard, J. Wallman, V. Veitch, and J. Emerson, Contex-tuality supplies the ’magic’ for quantum computation , Nature(London) , 351 (2014).[16] R. Raussendorf,
Contextuality in measurement-based quantumcomputation , Phys. Rev. A , 022322 (2013). [17] A. Zhang, H. Xu, J. Xie, H. Zhang, B. J. Smith, M. S. Kim, andL. Zhang, Experimental Test of Contextuality in Quantum andClassical Systems , Phys. Rev. Lett. , 080401 (2019).[18] J. ˚Aberg,
Quantifying Superposition , arXiv:quant-ph/0612146.[19] J. ˚Aberg,
Catalytic Coherence , Phys. Rev. Lett. , 150402(2014).[20] J. Sperling and W. Vogel,
Convex ordering and quantificationof quantumness , Phys. Scr. , 074024 (2015).[21] R. J. Glauber, Coherent and incoherent states of the radiationfield , Phys. Rev. , 2766 (1963).[22] U. M. Titulaer and R. J. Glauber,
Correlation functions for co-herent fields , Phys. Rev. , B676 (1965).[23] L. Mandel,
Non-classical states of the electromagnetic field ,Phys. Scr. T , 34 (1986).[24] L. Mandel and E. Wolf,
Optical Coherence and Quantum Op-tics , (Cambridge University Press, Cambridge, 1995).[25] W. Vogel and D.-G. Welsch,
Quantum Optics (Wiley-VCH,Weinheim, 2006).[26] Yao Yao, G. H. Dong, Xing Xiao, Mo Li, and C. P. Sun,
Inter-preting quantum coherence through a quantum measurementprocess , Phys. Rev. A , 052322 (2017).[27] F. Bischof, H. Kampermann, and D. Bruß, Resource the-ory of coherence based on positive-operator-valued measures ,arXiv:1812.00018.[28] P. Skrzypczyk and N. Linden,
Robustness of Measurement, Dis-crimination Games, and Accessible Information , Phys. Rev.Lett. , 140403 (2019).[29] P. Skrzypczyk, I. ˇSupi´c, and D. Cavalcanti,
All Sets of Incom-patible Measurements give an Advantage in Quantum StateDiscrimination , Phys. Rev. Lett. , 130403 (2019).[30] R. Takagi and B. Regula,
General resource theories in quan-tum mechanics and beyond: operational characterization viadiscrimination tasks , arXiv:1901.08127.[31] T. Guff, N. A. McMahon, Y. R. Sanders, and A. Gilchrist,
A Re-source Theory of Quantum Measurements , arXiv:1902.08490.[32] D. Walls and G. J. Milburn,
Quantum Coherence and Measure-ment Theory , In: D. Walls and G. J. Milburn,
Quantum Optics (Springer, Berlin, Heidelberg, 2008).[33] M. Van den Nest, A. Miyake, W. D¨ur, and H. J. Briegel,
Univer-sal Resources for Measurement-Based Quantum Computation , Phys. Rev. Lett. , 150504 (2006).[34] H. J. Briegel, D. E. Browne, W. D¨ur, R. Raussendorf, and M.Van den Nest, Measurement-based quantum computation , Nat.Phys. , 19 (2009).[35] X. Hu and H. Fan, Extracting quantum coherence via steering ,Sci. Rep. , 34380 (2016).[36] E. Chitambar, A. Streltsov, S. Rana, M. N. Bera, G. Adesso, andM. Lewenstein, Assisted Distillation of Quantum Coherence ,Phys. Rev. Lett. , 070402 (2016).[37] T. Ma, M.-J. Zhao, S.-M. Fei, and G.-L. Long,
Remote creationof quantum coherence , Phys. Rev. A , 042312 (2016).[38] D. Girolami, How Difficult is it to Prepare a Quantum State? ,Phys. Rev. Lett. , 010505 (2019).[39] J. Sperling, T. J. Bartley, G. Donati, M. Barbieri, X.-M. Jin,A. Datta, W. Vogel, and I. A. Walmsley,
Quantum Correlationsfrom the Conditional Statistics of Incomplete Data , Phys. Rev.Lett. , 083601 (2016).[40] E. Agudelo, J. Sperling, L. S. Costanzo, M. Bellini, A. Zavatta,and W. Vogel,
Conditional Hybrid Nonclassicality , Phys. Rev.Lett. , 120403 (2017).[41] M. Fuwa, S. Takeda, M. Zwierz, H. M. Wiseman, A. Furusawa,
Experimental proof of nonlocal wavefunction collapse for a sin-gle particle using homodyne measurements , Nat. Commun. ,6665 (2015).[42] G. C. Knee, K. Kakuyanagi, M.-C. Yeh, Y. Matsuzaki, H. Toida,H. Yamaguchi, S. Saito, A. J. Leggett, and W. J. Munro, A strictexperimental test of macroscopic realism in a superconductingflux qubit , Nat. Commun. , 13253 (2016).[43] X.-Y. Xu et al. , Measurements of Nonlocal Variables andDemonstration of the Failure of the Product Rule for a Pre-and Postselected Pair of Photons , Phys. Rev. Lett. , 100405(2019).[44] J. Sperling,
Continuity of measurement outcomes ,arXiv:1805.12404.[45] W. Heisenberg, ¨Uber den anschaulichen Inhalt der quanten-theoretischen Kinematik und Mechanik , Z. Physik , 172(1927).[46] U. Singh, A. K. Pati, and M. N. Bera, Uncertainty Relations forQuantum Coherence , Mathematics , 47 (2017).[47] X. Yuan, G. Bai, T. Peng, and X. Ma, Quantum uncertaintyrelation using coherence , Phys. Rev. A , 032313 (2017).[48] H. Dolatkhah, S. Haseli, S. Salimi, and A. S. Khorashad, Tight-ening the entropic uncertainty relations for multiple measure-ments and applying it to quantum coherence , Quantum Inf. Pro-cess. , 13 (2019).[49] V. Parigi, A. Zavatta, M. Kim, and M. Bellini, Probing Quan-tum Commutation Rules by Addition and Subtraction of SinglePhotons to/from a Light Field , Science , 1890 (2007).[50] A. Zavatta, V. Parigi, M. S. Kim, H. Jeong, and M. Bellini,
Experimental Demonstration of the Bosonic Commutation Re-lation via Superpositions of Quantum Operations on ThermalLight Fields , Phys. Rev. Lett. , 140406 (2009).[51] D. R. Brillinger,
The calculation of cumulants via conditioning ,Ann. Inst. Stat. Math. , 215 (1969).[52] M. J. Schervish, Theory of Statistics (Springer, New York, NY, 1995).[53] B. Yadin, J. Ma, D. Girolami, M. Gu, and V. Vedral,
Quan-tum Processes Which Do Not Use Coherence , Phys. Rev. X ,041028 (2016).[54] The relations [ ˆ Ξ x , ˆ Π y ] = P (cid:48) ( y ) = (cid:104) ˆ Π y (cid:105) ˆ ρ (cid:48) = ∑ x tr ( Ξ / x ˆ ρ Ξ / x ˆ Π y ) = tr ( ˆ ρ ˆ Π y ∑ x ˆ Ξ x ) = P ( y ) , using the identi-ties tr ( ˆ a ˆ b ˆ c ) = tr ( ˆ c ˆ a ˆ b ) ( ∀ ˆ a , ˆ b , ˆ c ) and ˆ1 = ∑ x ˆ Ξ x .[55] E. Knill and R. Laflamme, Power of One Bit of Quantum Infor-mation , Phys. Rev. Lett. , 5672 (1998).[56] M. A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information (Cambridge University Press, Cam-bridge, England, 2000).[57] N. K. Langford et al. , Demonstration of a simple entanglingoptical gate and its use in Bell-state analysis , Phys. Rev. Lett. , 210504 (2005).[58] N. Kiesel, C. Schmid, U. Weber, R. Ursin, and H. Weinfurter, Linear optics controlled- phase gate made simple , Phys. Rev.Lett. , 210505 (2005).[59] R. Okamoto, H. F. Hofmann, S. Takeuchi, and K. Sasaki, Demonstration of an optical quantum Controlled-NOT gatewithout path interference , Phys. Rev. Lett. , 210506 (2005).[60] E. Roccia, I. Gianani, L. Mancino, M. Sbroscia, F. Somma, M.G. Genoni, and M. Barbieri, Entangling Measurements for Mul-tiparameter Estimation with Two Qubits , Quantum Sci. Tech. ,01LT01 (2018).[61] G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T.C. Ralph, Measuring a Photonic Qubit without Destroying It ,Phys. Rev. Lett. , 190402 (2004).[62] L. Mancino, M. Sbroscia, E. Roccia, I. Gianani, F. Somma, P.Mataloni, M. Paternostro, and M. Barbieri, The Entropic Costof Quantum Generalized Measurements , npj Quantum Inf. , 20(2018).[63] L. Mancino, M. Sbroscia, E. Roccia, I. Gianani, V. Cimini, M.Paternostro, and M. Barbieri, Information-reality complemen-tarity in photonic weak measurements , Phys. Rev. A , 062108(2018).[64] M. Sawerwain and J. Wi´sniewska, Quantum Coherence Mea-sures for Quantum Switch , arXiv:1803.03321.[65] A. Venugopalan, S. Mishra, and T. Qureshi,
Monitoring Deco-herence via Measurement of Quantum Coherence , Physica A , 308 (2019).[66] The trace distance between ˆ ρ [Eq. (5)] and ˆ ρ (cid:48) [Eq. (5) for γ = (cid:107) ˆ ρ − ˆ ρ (cid:48) (cid:107) = γ (cid:112) p ( − p ) . For θ = π / ∆ V = γ p ( − p ) . A more general relation, includingcomplex γ , can be found in Ref. [44].[67] T. Theurer, D. Egloff, L. Zhang, and M. B. Plenio, QuantifyingOperations with an Application to Coherence , Phys. Rev. Lett. , 190405 (2019).[68] M. Oszmaniec and T. Biswas,
Operational relevance of re-source theories of quantum measurements , Quantum , 133(2019).[69] S. V. Moreira and M. Terra Cunha, Quantifying quantum inva-siveness , Phys. Rev. A99