Realism-information complementarity in photonic weak measurements
Luca Mancino, Marco Sbroscia, Emanuele Roccia, Ilaria Gianani, Valeria Cimini, Mauro Paternostro, Marco Barbieri
RRealism-information complementarity in photonic weak measurements
Luca Mancino, Marco Sbroscia, Emanuele Roccia, Ilaria Gianani, Valeria Cimini, Mauro Paternostro, and Marco Barbieri
1, 3 Dipartimento di Scienze, Universit`a degli Studi Roma Tre, Via della Vasca Navale 84, 00146, Rome, Italy Centre for Theoretical Atomic, Molecular and Optical Physics,School of Mathematics and Physics, Queen’s University Belfast, Belfast BT7 1NN, United Kingdom Istituto Nazionale di Ottica - CNR, Largo Enrico Fermi 6, 50125, Florence, Italy
The emergence of realistic properties is a key problem in understanding the quantum-to-classical transition.In this respect, measurements represent a way to interface quantum systems with the macroscopic world: thesecan be driven in the weak regime, where a reduced back-action can be imparted by choosing meter states ableto extract different amounts of information. Here we explore the implications of such weak measurement forthe variation of realistic properties of two-level quantum systems pre- and post-measurement, and extend ourinvestigations to the case of open systems implementing the measurements.
I. INTRODUCTION
The classical interpretation of the result of a measurementis merely the disclosure of a property of the system at the mo-ment of its observation. It is now clear that this view failsto capture the more intricate process of measuring a quantumobject. The latter breaks the normal evolution of the quan-tum state of the object, and results in the observable assuminginstantaneously the measured value. An extensive body of lit-erature has been dedicated to discussing this matter, with in-terpretations ranging from the standard operative Copenhagenview (“shut up and calculate” [1]), to Bohmian mechanics [2],to the Bayesian concept of the state collapse as an informationupdate [3], to more exotic suggestions such as the many-worldtheory [4] and collapse models [5, 6]. Regardless of the pref-erence to the possible solution of the measurement problem,we are confronted with the need to understand how the classi-cal world, where realistic values for observable might not beinherent, but are certainly tenable, emerges from the quantumun-realistic world.In this debate, a prominent role is reserved to the notionof elements of the reality, that Einstein, Podolsky and Rosenintroduced as intrinsic properties of the system that can bepredicted with certainty without any disturbance [7]. This no-tion complemented with that of locality is unable to explainpeculiar quantum phenomena such as entanglement [8, 9].These elements are generally associated to the wavefunction,the only description of the reality quantum mechanics is ableto provide. The current debate is centred on whether the wave-function itself has an ontic nature, i.e. it has a realistic conno-tation, or it is merely epistemic paraphernalia to describe anunderlying realistic nature [10–18].Recently, Bilobran and Angelo introduced a notion of real-ism based on both quantum states and observables, and con-nected it to an experimental procedure [19]: an element ofthe reality is introduced for the observable O whenever thequantum states, here considered as a density matrix ρ , is notaltered by a measurement of O ; this adheres to the standardnotion of classicality as that of a state that commutes with anymeasurement operator. A measurement of the realistic contentof ρ is then defined based on the entropy of the pre- and post-measurement states. If weak monitoring replaces the projec- tive measurement [20], it has been shown that under ideal con-ditions the change in the entropic measure of reality content of ρ , ∆ R , is in a duality relation with the amount of informationextracted, ∆ I [21]: ∆ R + ∆ I = 0 , (1)close to those introduced in Ref. [22] for coherence andwhich-path. In this article, we explore this relation in an ex-periment based on a photonic weak measurement device [23–31]. We show to what extent this equality can guide observa-tion in actual experiments. In addition, we extend our inves-tigations to the case of an open-system implementing a quan-tum measurement [31], and draw considerations on how theinitial entropy connected to the measuring device connects tothe emergence of realistic characters, according to the defini-tion of Bilobran and Angelo. II. A MEASURE OF REALITY.
Our intent is to investigate how a realistic description be-comes possible as we tune the invasivity of the measurementfrom negligible (weak measurement) to the standard projec-tive regime (strong measurement) [20]. Therefore, the figureof merit we must use should be capable of addressing mix-tures and should be related to measurable quantities. We con-sider the case where an observable O is measured on a genericquantum state ρ by means of a suitable device. The defini-tion in Ref.[19] considers the degree of irreality of the ob-servable O , described in quantum mechanics by the operator ˆ O = (cid:80) k o k | k (cid:105)(cid:104) k | , associated to the state ρ as: I ( O | ρ ) = S (Φ O ( ρ )) − S ( ρ ) , (2)where S is the Von Neumann entropy and the map Φ O ( ρ ) = (cid:80) k p k | k (cid:105)(cid:104) k | , with p k = (cid:104) k | ρ | k (cid:105) , describes the action of themeasuring device. The degree of irreality of O vanishes if thiscan be measured without affecting the state, and it is maxi-mum when the measurement of O is disruptive to the point ofbringing a pure state into a complete mixture; the latter cor-respond to a case in which ρ is an equal superposition of all a r X i v : . [ qu a n t - ph ] A p r possible eigenstates | k (cid:105) . This definition then reveals an epis-temic approach, as it is only concerned with our ignorance ofthe realistic value of the observable O .We need to extend these positions to the case of a weakmeasurement: this is a generalisation of the standard projec-tive measurement, for which the output state is not unambigu-ously identified, although some information is obtained. Theimplementation of a weak measurement is typically carriedout by coupling the system with a pointer object, which isthen measured [20]. Due to the coupling, the value of the ob-servable O modifies the distribution of a related observable Q on the pointer. When the effect of such modification al-lows to discriminate different states of the pointer - ı.e. theinduced shift of the mean value of Q is significantly largerthan the width δQ - the measurement functions in the stan-dard conditions. In opposite limit in which the size of theshift is comparable to δQ , we operate in the weak regime. Forany measurement strength, an element of the reality can bedefined whenever there exist a procedure to predict with cer-tainty what shift will occur: the element can then attributed tothe shift itself [32].The generalisation of the map Φ O to the weak regime isperformed as follows [19, 21]. Upon collecting the outcome k , the state emerging from the weak measurement is writ-ten as: C (cid:15)k ( ρ ) = (1 − (cid:15) ) ρ + (cid:15) | k (cid:105)(cid:104) k | . The limit (cid:15) → cor-responds to the projective case extracting maximal informa-tion, and (cid:15) → corresponds to performing to measurement thatclearly delivers no information. If the outcomes are not sorted,the average state is M (cid:15)O ( ρ ) = (cid:80) k p k C (cid:15)k ( ρ ) [21]. The map M (cid:15)O has the remarkable property of commuting with Φ O : M (cid:15)O Φ O = Φ O M (cid:15)O for all strengths. This implies that themap M (cid:15)O ( ρ ) can not be invoked as introducing any elementof the reality, whenever Φ O did not.We can use these definitions to calculate the variation of thedegree of reality of O following a weak measurement as [19]: ∆ R = − ∆ I = I ( O | ρ ) − I ( O |M (cid:15)O ( ρ ))= S ( M (cid:15)O ( ρ )) − S ( ρ ) , (3)where we have used the definition of irreality (2) and the com-mutation properties of M (cid:15)O to obtain the last equality. Byinvoking the concavity of the Von Neumann entropy, the vari-ation Eq. (3) can be bound as [21]: ∆ R ≥ (cid:15) I ( O | ρ ) , (4)demonstrating that the degree of reality of O is always nondecreasing upon monitoring. III. REALITY-INFORMATION DUALITY
The variation of the degree of reality can be directly re-lated to a change in the information content of the initial state ρ [21]. In order to define a proper quantifier, we analyse themeasurement strategy in detail. The weak monitoring is per-formed by introducing an ancillary system ρ A = | A (cid:105)(cid:104) A | , andcoupling it to the system by means of the unitary dynamics ˆ U : M (cid:15)k ( ρ ) = Tr A (cid:16) ˆ U ρ ⊗ | A (cid:105)(cid:104) A | ˆ U † (cid:17) . (5) FIG. 1. Experimental behaviour of ∆ R as a function of the measure-ment strength (cid:15) . The measured values correspond to the points. Theblue solid line corresponds to the prediction, and the solid orangeline corresponds to the bound 4. All relevant changes need being evaluated between theinitial separable state ρ SA = ρ ⊗ ρ A , and the final state ρ (cid:48) SA = ˆ U ρ ⊗ | A (cid:105)(cid:104) A | ˆ U † . The overall information available inthe bipartite state can be decomposed as the sum of three con-tributions: I tot = I S + I A + I S : A , where I S : A is the mu-tual information of the bipartite state [33], and the local in-formation content is I S = ln d − S ( ρ ) for the system, and I A = ln d A − S ( ρ A ) = ln d A for the ancilla, with d ( d A ) thedimension of the Hilbert space of the system (ancilla). Sincethe evolution is unitary, the total information content of thejoint state of the system and the ancilla can not change. There-fore, if we evaluate the new amount of information availableafter the measurement I (cid:48) tot = I (cid:48) S + I (cid:48) A + I (cid:48) S : A , we expect nodifference in the total values I tot = I (cid:48) tot , but only a redistribu-tion among the three terms. Here, the final state of the ancillais given by ρ (cid:48) A = Tr S (cid:0) U ρ ⊗ | A (cid:105)(cid:104) A | U † (cid:1) . Since the differ-ence in information of the system ∆ I S equals the variation ofits degree of reality up to a sign, we find the relation [21]: ∆ I = ∆ I S : A + ∆ I A = − ∆ I , (6)leading to its interpretation as a complementarity relation: ∆ I = − ∆ I S = − ∆ R , (7)that rigorously holds only when the coupling is unitary, hencereversible. In this limit, the changes in the degree of realityassociated to O are the only source of the variations in themutual information, and in the marginal information contentof the ancilla. IV. PHOTONICS EXPERIMENT
We employ the measurement device in [23, 24, 31] to in-vestigate the experimental behaviour of Eqs. (4) and (7) forsingle qubits. Both system S and ancilla A are encoded in FIG. 2. Experimental behaviour of ∆ I as a function of the measure-ment strength (cid:15) . The measured values correspond to the points. Thered solid line corresponds to the prediction 7. the polarisation of single photons. These interact in an inter-ferometric setup that implements a controlled-phase interac-tion ˆ U = | (cid:105)(cid:104) | ⊗ ˆ I + | (cid:105)(cid:104) | ⊗ ˆ Z , where ˆ I is the 2 × ˆ Z is the z Pauli operator. This can be usedas a weak measurement device of the observable O = Z ,corresponding to a measurement of the populations of thehorizontal H (1) and vertical V (0) components of the sys-tem [34]. The state of the system is kept fixed in the pure state | + (cid:105) = ( | (cid:105) + | (cid:105) ) / √ , while the ancilla is initially taken as | ψ ( θ ) (cid:105) = cos θ | (cid:105) A +sin θ | (cid:105) A , and, after the measurement, itis measured in the basis {(cid:104) + | A , (cid:104)−| A } . For θ = 0 no changeis imparted to the ancilla being it an eigenstate of ˆ Z , henceit will eventually deliver no information on the system; for θ = π/ , the ancilla is unaffected, if the system is in | (cid:105) , androtated by 180 ◦ around the z -axis of the Bloch sphere, if thesystem is in | (cid:105) . Discriminating between the two possibilitieson the ancilla provides full information on the system. In be-tween these two extremes, the state of the ancilla defines themeasurement strength as (cid:15) = 1 − cos(2 θ ) [34]. In our experi-ment, we started with a fiducial bipartite state prepared closelyto a pure state - we will then assume that the initial entropiesare zero with an error comparable to the experimental uncer-tainties. We then performed full tomography of the bipartitefinal state, and obtain the relevant quantities in the inequalities(4) and (7), following the original suggestion [19].The results shown in Fig.1 illustrate the measured change inthe degree of reality of Z following a weak measurement withtuneable strength. The experimental points have been esti-mated from the experimentally reconstructed bipartite densitymatrices ρ (cid:48) SA after the measurement, tracing out the ancilla.The data follows the predicted behaviour and clearly satis-fies the bound Eq. (4). It is seen how the linear lower limit (cid:15) I ( O | ρ ) , for the pure initial state ρ = | + (cid:105)(cid:104) + | remains farfrom the experimental data and from the predictions, exceptfor extremal values of (cid:15) , where it is most useful.We have also evaluated the difference in the information ∆ I from the experimental ρ (cid:48) SA : ∆ I = S ( M (cid:15)O ( ρ )) − S ( ρ (cid:48) AS ) ,due to the fact we start with pure states. The corresponding FIG. 3. Experimental behaviour of ∆ R as a function of the entropyof the initial meter state S m , for θ = 16 ◦ . The measured valuescorrespond to the points. The purple solid line corresponds to theprediction. results are reported in Fig.2. The experiment shows that thecomplementarity relation Eq. (7) is more sensitive to externalfactors, since ∆ I S saturates at a lower value than expected.This comes from the fact that the coupling between the sys-tem and the ancilla photons is not unitary, and the dissipationincreases with the measurement strength. Part of the infor-mation available is then lost to undetected degrees of freedomof the photon pair acting effectively as an environment: this,however, still contributes to the emergence of realistic proper-ties, much in the spirit of quantum Darwinism [35–37].The emergence of realism in such open systems can be in-vestigate more systematically by using a mixed meter, i.e. astate presenting uncontrolled correlations with the environ-ment. This can be mimicked by mixing the counting statisticsrelative to the state | ψ ( θ ) (cid:105) with that for the orthogonal state | ψ ( θ + π/ (cid:105) [31] with weights p and − p respectively: inthe latter case, the measurement strength is the same , how-ever, due to the action of ˆ U , an extra ˆ Z rotation is impartedto the signal state after the measurement. This results in in-creased entropy with respect to that resulting uniquely fromthe measurement back action [31, 38, 39].In Fig. 3, the data for the variation ∆ R when the systemis measured by means of a meter state with initial entropy S m = H ( p ) , where H ( p ) is the Shannon entropy [33], in theweak measurement regime, θ = 16 ◦ . As the mixing of themeter increases, the state of the signal starts presenting a morepronounced change in its degree of realism. This is due to thefact that information about the value of Z is present in the me-ter as well as in the environment to which this was originallycorrelated: this, clearly, can not be fully retrieved by observ-ing the meter only, but still dictates how realistic propertiesappear in the system. V. CONCLUSIONS
The matter of establishing when realistic properties emergein quantum systems can be quantified by using the definitionof the degree of irreality I ( O | ρ ) in Eq. (2), based on Von Neu-mann entropy. This is equivalent to giving a prominent role tothe notion of information: indeed, the degree of irreality sodefined is given by the amount of information one needs todescribe ρ in full, if the observable O is known. On the otherhand, our experiment shows how a metric based on definitionare largely insensitive to the imperfections of the measure-ment device; however, the impact of the measurement itselfcan be retrieved by looking at changes in the total information I tot contained in the joint state of system and ancilla. We havealso been able to comment on the implications of mixednessin the ancilla: the coupling to an environment makes informa-tion on the system available, and this is sufficient for realisticproperties to emerge, even if no one can gather it by lookingat the ancilla only. ACKNOWLEDGEMENTS
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