M Trassinelli

Conditional probability and interferences in generalized measurements with or without definite causal order

In the context of generalized measurement theory, the Gleason-Busch theorem assures the unique form of the associated probability function. Recently, in Flatt et al. Phys. Rev. A 96, 062125 (2017), the case of subsequent measurements has been treated, with the derivation of the Lüders rule and its generalization (Kraus update rule). Here we investigate the special case of subsequent measurements where an intermediate measurement is a composition of two measurements (a or b) and the case where the causal order is not defined (a and b or b and a). In both cases interference effects can arise. We show that the associated probability cannot be written univocally, and the distributive property on its arguments cannot be taken for granted. The two probability expressions correspond to the Born rule and the classical probability; they are related to the intrinsic possibility of obtaining definite results for the intermediate measurement. For indefinite causal order, a causal inequality is also deduced. The frontier between the two cases is investigated in the framework of generalized measurements with a toy model, a Mach-Zehnder interferometer with a movable beam splitter.

I. INTRODUCTION

In Quantum Mechanics, probabilities are obtained by the squared modulus of complex amplitudes, which give rise to interference phenomena. In the common example of Young's setup composed of a source, two slits and a screen or movable detector as represented in Fig. 1, the probability to detect an emitted particle in a position x on the backstop wall is given by

where interference terms are not present. The probability function for the quantum case is strictly connected to the Hilbert space structure, where systems are described with respect to a defined basis and where the complex numbers mentioned above correspond to coordinates. With some minimal requirements on the probability function P r, in 1957 Gleason [START_REF] Gleason | Measures on the closed subspaces of a hilbert space[END_REF] demonstrated that P r is univocally defined in a Hilbert space by the trace rule P r(i) = tr(ρP i ),

(3) * martino.trassinelli@insp.jussieu.fr where ρ = |ϕ ϕ| is the density matrix of the prepared system and P i = |i i| is the projector on the state of interest. In the case of an initial pure state |ϕ = |s , Eq. ( 3) corresponds to the Born rule with P r(i) = | i|s | 2 . Gleason's theorem has some limitations; it is valid only for Hilbert spaces with a dimension larger than two and for projective von Neumann measurements [START_REF] Neumann | Mathematical Foundations of Quantum Mechanics[END_REF].

In the framework of the general measurement formalism of positive-operator-valued measures (POVM, also called probability operator measures), in 2003, Busch [START_REF] Busch | Quantum states and generalized observables: A simple proof of Gleason's theorem[END_REF] extended Gleason's theorem for any dimension and for imperfect measurements described by positive operators, effects E i , instead of projectors. Recently (2017), in the same context of POVM, Flatt, Barnett and Croke applied the Gleason-Busch theorem to subsequent measurements [START_REF] Flatt | Gleason-busch theorem for sequential measurements[END_REF]. Considering the operators E i and F j associated with the measurements i and j, with i before j, Flatt and coworkers proved that P r(i, j) takes the general form

P r(i, j) = tr F j k K ik ρK † ik , (4)
where the operators K ik are related to the effects by

E i = k K † ik K ik .
From the above equation and the corresponding one for the conditional probability P r(i|j), the Kraus update rule [START_REF] Kraus | General state changes in quantum theory[END_REF][START_REF] Kraus | States, Effects, and Operations Fundamental Notions of Quantum Theory[END_REF]

ρ → ρ ′ i = k K ik ρK † ik tr k K † ik K ik (5)
for the state update of the system ρ ′ i after the measurement i is recovered. It is worth noting that the Kraus update rule, and its particular case of the Lüders rule [START_REF] Lüders | Über die zustandsänderung durch den meßprozeß[END_REF] valid for ideal measurement and determined by the von Neumann projection postulate, are derived from first principles. There is no need of other postulates than the description of states via the Hilbert space and a few basic requirements for the probability function.

In this article, we apply the formalism of subsequent generalized measurements to the case with two possible and mutually exclusive intermediate measurements (a two-way interferometer) with indefinite measurement order. For both cases, interferences can occur. With the introduction of a new notation of the probability function arguments with respect to Flatt et al., we will show that two possible expressions of the final probability P r can be derived from Eq. ( 3). These two expressions correspond to Eqs. ( 1) and (2) in the example of Young's slits, and are related to the possibility of distinguishing or not the intermediate measurement. The difference between the two forms is the order of the arguments of the probability expressions, where the distributive property can not be taken for granted. The violation of the distributive property in Quantum Mechanics is not new and it has been pointed out since the early years of its formulation [START_REF] Birkhoff | The logic of quantum mechanics[END_REF][START_REF] Piron | Foundations of quantum physics[END_REF] and extensively discussed in Quantum Logic. Its connection to the extension of classical probability to quantum probability is well discussed in the literature in the case of perfect projective measurement [START_REF] Beltrametti | The Logic of Quantum Mechanics[END_REF][START_REF] Cassinelli | Conditional probabilities in quantum mechanics. i.-conditioning with respect to a single event[END_REF][START_REF] Cassinelli | Conditional probabilities in quantum mechanics. ii. additive conditional probabilities[END_REF][START_REF] Hughes | The Structure and Interpretation of Quantum Mechanics[END_REF][START_REF] Trassinelli | Relational quantum mechanics and probability[END_REF]. For imperfect general measurements, when positive operators are considered instead of projectors, some work has been performed by Busch and collaborators [START_REF] Busch | Complementarity and uncertainty in mach-zehnder interferometry and beyond[END_REF][START_REF] Busch | Quantum Measurement[END_REF]. Here we present a general discussion about the probability function for the distinguishable and indistinguishable path cases (the particle-like and wave-like behaviors) in the case of imperfect (unsharp) measurements.

For Young's slits, the frontier between the different cases and the domain of validity of Eqs. ( 1) and ( 2), has been extensively discussed in the past. Experimentally, it has been explored in the last decades through investigations of interference effects with molecules with larger and larger masses. Diffraction of large inorganic and organic molecules with masses beyond 25000 atomic mass units has been obtained [START_REF] Nairz | Diffraction of complex molecules by structures made of light[END_REF][START_REF] Fein | Quantum superposition of molecules beyond 25 kDa[END_REF][START_REF] Brand | Bragg diffraction of large organic molecules[END_REF]. Here, we discuss this frontier in the context of generalized measurements considering a Mach-Zehnder interferometer with movable beam splitter. This toy model, introduced in the past by Haroche et al. [START_REF] Bertet | A complementarity experiment with an interferometer at the quantum-classical boundary[END_REF][START_REF] Haroche | Exploring the Quantum: Atoms, Cavities, and Photons[END_REF], has the interesting feature of allowing to pass from one case to the other continuously, simply considering a variation of the mass of the movable beam splitter.

In the case of measurements with indefinite order, extensively discussed in the literature in the last years [START_REF] Oreshkov | Quantum correlations with no causal order[END_REF][START_REF] Brukner | Quantum causality[END_REF][START_REF] Branciard | The simplest causal inequalities and their violation[END_REF][START_REF] Castro-Ruiz | Dynamics of quantum causal structures[END_REF][START_REF] Zych | Bell's theorem for temporal order[END_REF][START_REF] Wechs | On the definition and characterisation of multipartite causal (non)separability[END_REF][START_REF] Henderson | Quantum temporal superposition: The case of quantum field theory[END_REF], we will show that they can be treated with the same approach as for the two-path interferometer. Here too, the presence of interference or not is related in this case to two possible, but not equivalent, expressions of the probability function.

II. PROBABILITY FOR SUBSEQUENT MEASUREMENTS A. Introduction of new notations

Taking inspiration from the Quantum Logic approach [11-15, 30, 31] and the propositional definition of probability [START_REF] Cox | Algebra of Probable Inference[END_REF][START_REF] Fine | Theories of probability: an examination of foundations[END_REF][START_REF] Jaynes | Probability Theory: The Logic of Science[END_REF], we introduce a new notation with the logical operators "∧" and "∨" to unambiguously discuss the joint probability of series of subsequent measurements. The conjunction operator "∧" is equivalent to "AND" in normal language and to the comma in the previously introduced notation P r(i, j). The disjunction operator "∨" is equivalent to "OR" also indicated with the "+" operator (in Refs. [START_REF] Busch | Quantum states and generalized observables: A simple proof of Gleason's theorem[END_REF][START_REF] Flatt | Gleason-busch theorem for sequential measurements[END_REF] as example). Particular attention has to be payed for measurements i, j that are incompatible. In this case, the logical operator "∧" is not well defined [START_REF] Cassinelli | Conditional probabilities in quantum mechanics. i.-conditioning with respect to a single event[END_REF][START_REF] Cassinelli | Conditional probabilities in quantum mechanics. ii. additive conditional probabilities[END_REF][START_REF] Hughes | The Structure and Interpretation of Quantum Mechanics[END_REF][START_REF] Ballentine | Probability theory in quantum mechanics[END_REF], except if the order of subsequent measurements is defined. As already pointed out in the consistent histories interpretation of Quantum Mechanics [START_REF] Griffiths | Consistent histories and the interpretation of quantum mechanics[END_REF][START_REF] Griffiths | Consistent Quantum Theory[END_REF][START_REF] Omnès | Consistent interpretations of quantum mechanics[END_REF], differently from standard logic, the operator "∧" is not symmetric with respect to i, j with i ∧ j = j ∧ i. With this notation, the joint probability defined above for a measurement j obtained after a measurement i can be written as

℘(j ∧ i|s) ≡ P r(i, j), (6)
where we explicitly indicate the system preparation s, which is in fact connected to the possible measurement outcomes. We also invert the order of i, j to clearly indicate the sequential order of the measurement or preparation from right to left (preparation s, first measurement i and second measurement j).

B. Rewriting probabilities

Before treating in detail the Young's slits problem with the new introduced notation, we shall rewrite the properties and assumptions of the probability function used by Flatt et al. [START_REF] Flatt | Gleason-busch theorem for sequential measurements[END_REF] that lead to Eq. ( 5). We consider a set of positive-semidefinite operators (effects) E i of the same POVM with i E i = I. The requirement properties of the probability function ν(E i ) = P r(i) for the

S E i E (ij) F j FIG.
2. Scheme of two subsequent measurements. Adapted from Ref. [START_REF] Flatt | Gleason-busch theorem for sequential measurements[END_REF].

Gleason-Busch theorem are

(P1) 0 ≤ ν(E i ) ≤ 1. (P2) ν(I) = 1. (P3) ν(E i + E j . . .) = ν(E i ) + ν(E j ) + . . .

The function ν(E i

) is in fact a map from the full set of effects E(H) acting on the Hilbert space H:

E → ν(E) with ν(E) ∈ [0, 1].
With our notation, the previous propositions become

(P1*) 0 ≤ ℘(i|s) ≤ 1. (P2*) ℘(I|s) = 1. (P3*) ℘(i ∨ j ∨ . . . |s) = ℘(i|s) + ℘(j|s) + . . .
where i, j are the measurements that correspond to the effects E i , E j and I = i i measurement correspond to the identity operator I.

When two subsequent measurements are considered together, Flatt et al. introduced the new function

µ i ν (F j ) = ν(E (ij) ) = P r(i, j) (7)
for the action of the effect F j after the action of E i and E (ij) indicating the cumulative effect (see Fig. 2). In addition, the following assumptions are considered by Flatt et al.

(A1) 0 ≤ µ i ν (F j ) ≤ ν(E i ) < 1. (A2) µ i ν (I) = ν(E i ). (A3) µ i ν (F j + F k + . . .) = µ i ν (F j ) + µ i ν (F k ) + . . . .
With our notation, we consider on the same level the measurements j and i and the operator "∧" indicates the measurement order. The assumptions (A1-2) can simply be rewritten as

(A1*) 0 ≤ ℘(j ∧ i|s) ≤ ℘(i|s) ≤ 1. (A2*) ℘(I ∧ i|s) = ℘(i|s).
(A2*) is now a tautology. For (A3), the rewriting is ambiguous. µ i ν (F j + F k + . . .) can be written in fact in two different forms:

℘((j ∧ i) ∨ (k ∧ i)|s) (8)
or

℘((j ∨ k) ∧ i|s). (9)
Another situation that cannot be easily treated with the formalism of Flatt et al. is the case of subsequent measurements with indefinite causal order. Equation [START_REF] Kraus | States, Effects, and Operations Fundamental Notions of Quantum Theory[END_REF] implies in fact a defined causal order because of the nesting of the two probability functions. The treatment of indefinite causal order phenomena and the associated possible interferences is well defined for elementary processes in the context of quantum field theory (e.g. in the relativistic Compton scattering [START_REF] Itzykson | Quantum Field Theory[END_REF]). When a subsequent interaction with measuring detectors is considered, the situation is more complicated and it has been the center of interest of several works in the last years [START_REF] Oreshkov | Quantum correlations with no causal order[END_REF][START_REF] Brukner | Quantum causality[END_REF][START_REF] Branciard | The simplest causal inequalities and their violation[END_REF][START_REF] Henderson | Quantum temporal superposition: The case of quantum field theory[END_REF]. Considering two measurements represented by two effects E i and E j with an indefinite order and a final measurement represented by F d , similarly to Eqs. ( 8) and ( 9), the associated probability can be written in two ways

℘(k ∧ [(i ∧ j) ∨ (j ∧ i)]|s) (10)
or

℘((k ∧ i ∧ j) ∨ (k ∧ j ∧ i)|s). (11)
In the following sections, it will be shown that the formulas ( 8)-( 9) and ( 10)- [START_REF] Beltrametti | The Logic of Quantum Mechanics[END_REF] will lead to different probability expressions. These different expressions, obtained from a unique definition of the probability function, will be the key point of the present work.

C. General considerations for POVM operators

To investigate the difference between Eqs. ( 8) and ( 9), we come back the specific example of Young's slits where we consider the possibility to measure or flag the passage through each slit. Before that, a short introduction to generalized measurements is mandatory. In the framework of POVM, the single measurements are described by the positive-valued operators E ℓ = K † ℓ K ℓ , where K ℓ operators are determined by the unitary interaction between the system we want to study and the detector, both considered as quantum systems. The general expression for K ℓ is given by [START_REF] Barnett | Quantum Information[END_REF][START_REF] Laloë | Do We Really Understand Quantum Mechanics?[END_REF]

K ℓ = i,j α ij ℓ det |Φ det i |j i| , (12)
where α ij depend on the action of the unitary matrix U int describing the interaction between the system and the detector. The initial state is described by

|Ψ 0 i = |i |Φ 0 , E a E (ad) E b E (bd) S F d S E a E b F d E (abd) E (ab)
FIG. 3. Schemes of subsequential measurements corresponding to the Young's slits experiment for the case where the path of the particle can be detected (top) or not (bottom).

where |i and |Φ 0 describe the initial state of the system and the detector, respectively. After their mutual interaction, the system and detector states are described by

|Ψ 0 i → |Ψ i = |ϕ i |Φ det i = j α ij |j |Φ det i . |Φ det i
describes the detector state after the interaction with the system in an initial state |i . Finally, |ℓ det represents the detector state corresponding to the macroscopic outcome of the measurement device. In the case of a nondestructive measurement, the above formula is simplified to

K ℓ = i ℓ det |Φ det i |i i| . (13)

III. INTERFERENCES IN A TWO-PATH INTERFEROMETER A. Distinguishable paths

In the case of Young's slits, we consider that the detection of the path taken by the particle is possible and is non-destructive. The formula corresponding to Eq. ( 8) becomes ℘((d ∧ a) ∨ (d ∧ b)|s) and depends on the operators K a , K b and K d . K a , K b are related to the detection of the path a or b, and the corresponding effects are

E a = K † a K a , E b = K † b K b . K d is related to the detec- tion d on the screen with F d = K † d K d
The combination of E a and E b with F d can be assimilated to the effects E (ad) and E (bd) as in Eq. ( 7), and for which the property (P3)/(P3*) can be applied. In this case we have

℘((d ∧ a) ∨ (d ∧ b)|s) = ℘(d ∧ a|s) + ℘(d ∧ b|s) = = tr(F d K a ρK † a ) + tr(F d K b ρK † b ). ( 14
)
The above equation corresponds to the classic probability sum rule, i.e. the particle-like probability in Eq. ( 2). The fact that we can decompose the measurement in two separate operators E (ad) and E (bd) (Fig. 3, top) implicitly means that the different paths can be distinguished and we have just a duplicated version of the basic subsequent measurement represented in Fig. 2. This case can be easily treated with the formalism introduced by Flatt et al. with the introduction of the probability functions µ a ν (F d ) and µ b ν (F d ). In the case of ideal projective measurements, we have

E a = K a = P a = |a a| and E b = K b = P b = |b b|
where we used the properties of projectors P † i = P i and P i P i = P i . The above equation then becomes [START_REF] Trassinelli | Relational quantum mechanics and probability[END_REF]

℘((d ∧ a) ∨ (d ∧ b)|s) = = | d|U (ad) |a a|U (sa) |s | 2 + | d|U (bd) |b b|U (sb) |s | 2 , (15)
where the unitary operators U correspond to the evolution of the different parts of the apparatus. The expression of ℘((d ∧ a) ∨ (d ∧ b)|s) can also be directly obtained by the trace reduction of the density matrix ρ with respect to detector base |a det and |b det . In this case we have

℘((d ∧ a) ∨ (d ∧ b)|s) = tr(F d ρ r ), (16)
with ρ r = tr a det ,b det (|Ψ i Ψ i |) and where

|Ψ i = j α ij |j |Φ i .
From the linearity of the trace operator, it is easy to verify that the previous expression is equivalent to Eq. ( 14). This indicates that the use of the trace over the undetected |a det , |b det states implicitly implies an interaction between the system and the which-path detectors, even if they are not directly involved in the measurement.

B. Indistinguishable paths and discussion

In the case we can not distinguish which path is taken by the particle, the a ∨ b cannot be decomposed and we have to deal with the expression

℘(d ∧ (a ∨ b)|s) = tr(F d K a∨b ρK † a∨b ). ( 17
)
The operator K a∨b can be built in three different ways:

1. from path-detectors with the same final state after the interaction with the system (|a det = |b det ), 2. from a complementary measurement c (e.g. a series of detectors on the slit walls),

3. via a detector state |q det belonging to the span generated by the vectors |a det and |b det .

As we will see, a genuine a ∨ b measurement is related to the first two cases only. The third approach is in fact related to the quantum eraser case and it is discussed separately in the next section.

In the case where the measurement of the passage of a particle in one path or the other induces the same detector state |ab det ≡ |a det = |b det , the operator corresponding to a ∨ b can be written as

K a∨b = i=a,b ℓ det |ab det |i i| = i=a,b K i , (18)
where K a and K b are associated to the effects E a and E b represented in the bottom of Fig. 3 and |ℓ det is a generic state corresponding to the a ∨ b measure (|ℓ det = |ab det for an ideal measurement). Note that this is not the case for Kraus operators K ik as in Eq. ( 5) where different detector states Φ det k correspond to the same system state |i . Here in opposite, different system states |a and |b correspond to the same detector state |ab det , and the trace operator in Eqs. ( 5) and ( 17) cannot be separated into different terms.

If we consider a complementary measurement c to both a and b measurements, we have that c ∧ a = 0, c ∧ b = 0 and c = I -a ∨ b. E a∨b corresponds to the absence of signal in the measurement E c , then, using the property of the set of effects of the POVM for which i=a,b,c E i = I, we have E a∨b = I -E c = E a + E b . K a∨b can be written as [START_REF] Busch | Quantum Measurement[END_REF][START_REF] Barnett | Quantum Information[END_REF][START_REF] Auletta | Quantum Mechanics[END_REF]]

K a∨b = U a∨b E a + E b , (19)
where U a∨b is a unitary matrix that depends on the details of the interaction between |a det , b det and the propagating particle-wave.

In the case of ideal projective measurements for both situations discussed above, K a∨b can be explicitly written. In this case we have that E a∨b = E a + E b = P a + P b and Eq. ( 17) becomes [START_REF] Trassinelli | Relational quantum mechanics and probability[END_REF] (see also Refs. [START_REF] Beltrametti | The Logic of Quantum Mechanics[END_REF][START_REF] Cassinelli | Conditional probabilities in quantum mechanics. ii. additive conditional probabilities[END_REF][START_REF] Busch | Quantum Measurement[END_REF])

℘(d ∧ (a ∨ b)|s) = = | d|U (ad) |a a|U (sa) |s + d|U (bd) |b b|U (sb) |s | 2 , ( 20
)
which is equivalent to the quantum form of the probability in Eq. ( 1), i.e. equivalent to the Born rule.

In the general case represented in Eqs. ( 14) and (17) (and in the particular case in Eqs. [START_REF] Trassinelli | Relational quantum mechanics and probability[END_REF] and ( 20)),

℘(d ∧ (a ∨ b)|s) = ℘((d ∧ a) ∨ (d ∧ b)|s) (21)
and the distributive property on the arguments of ℘ is violated. Equations ( 14) and ( 15) reproduce the sum rule valid for the classical probability P r C (Eq. ( 2)). However, Eqs. ( 17) and ( 20) present additional interference terms and are compatible with the Born rule. If the distributive property is considered valid, the two expressions should be equivalent. But the validity of distributivity cannot in fact be taken for granted. As anticipated in the introduction, the violation of the distributive law in quantum phenomena is well known since the early years of Quantum Mechanics [START_REF] Birkhoff | The logic of quantum mechanics[END_REF][START_REF] Putnam | Is logic empirical?[END_REF]. In particular in Quantum Logic [START_REF] Piron | Foundations of quantum physics[END_REF][START_REF] Beltrametti | The Logic of Quantum Mechanics[END_REF][START_REF] Hughes | The Structure and Interpretation of Quantum Mechanics[END_REF][START_REF] Luisa | The mathematical environment of quantum information[END_REF][START_REF] Piron | Survey of general quantum physics[END_REF][START_REF] Auletta | Foundations and Interpretation of Quantum Mechanics: In the Light of a Critical-historical Analysis of the Problems and of a Synthesis of the Results[END_REF][START_REF] Dalla Chiara | Reasoning in Quantum Theory: Sharp and Unsharp Quantum Logics[END_REF] this is related to the properties of orthomodular lattices, associated to sets of yes/no experiments, where the distributivity on their elements is not always valid. For the indistinguishable case, the measurement a ∨ b corresponds to an atomic operator E a∨b ≡ E (ab) that cannot be decomposed in terms of E a , E b . The cumulative effect E (abd) depends then on all three measurements a, b and d and can be represented by the scheme in the bottom of Fig. 3.

C. The quantum eraser revisited

In the Quantum Logic context, a measurement representing a ∨ b can be built from a vector |q det = α |a det + β |b det [START_REF] Hughes | The Structure and Interpretation of Quantum Mechanics[END_REF], with α, β = 0, which belongs to the span generated by the vectors |a det and |b det . Using Eq. ( 13) with ℓ = q, we can then write

K a∨b ≡ K q = α * K a + β * K b , (22)
where |α| 2 + |β| 2 = 1 for a normalized probability. Once inserted in Eq. ( 17), the above expression gives rise to mixed a det |b det terms and then to interference phenomena. This is in fact the case of the quantum eraser [START_REF] Busch | Complementarity and uncertainty in mach-zehnder interferometry and beyond[END_REF][START_REF] Scully | Quantum optical tests of complementarity[END_REF][START_REF] Herzog | Complementarity and the quantum eraser[END_REF][START_REF] Kim | Delayed "choice" quantum eraser[END_REF][START_REF] Weisz | An electronic quantum eraser[END_REF][START_REF] Ma | Delayed-choice gedanken experiments and their realizations[END_REF], where instead of the direct path detection via |a det , |b det , a combination of them is considered and interference terms appear. This is a situation not equivalent to the case with a complementary measurement c = I -a ∨ b. Even if we recover the presence of interferences with the use of |q det instead of |a det or |b det , we are dealing with a single measurement q that corresponds to the probability ℘(d ∧ q|s), and not ℘(d ∧ (a ∨ b)|s). Similarly to a, b measurements, we could consider the alternative measurement given by the vector |r det = -e iφ β |a det + e iφ α |b det orthogonal to |q det . When both possible measurements q and r are considered, we can write down the probabilities ℘((d∧q)∨(d∧r))|s) and ℘(d∧(q ∨r)|s). |r det , |q det and |a det , |b det are two different bases describing the detection and they are related by a unitary transformation. Because of the property of the unitary transformation, it can be demonstrated (see App. A for the detailed calculations) that the combination of the two measurements r and q and the which-path a and b are completely equivalent and

℘((d ∧ q) ∨ (d ∧ r))|s) = ℘((d ∧ a) ∨ (d ∧ b))|s). ( 23
)
The interference terms present in the separate terms ℘(d ∧ q|s) and ℘(d ∧ r|s), completely compensate in ℘((d ∧ q) ∨ (d ∧ r))|s) = ℘(d ∧ q|s) + ℘(d ∧ r|s) like in the well known results on the quantum eraser.

For the case of ℘(d ∧ (q ∨ r))|s) probability, the situation is more complicated because it depends on the values of α, β, φ but also on the choice of α ′ , β ′ for building K q∨r = α ′ K q + β ′ K r . With this last consideration, we can conclude that in fact the construction of K a∨b via Eq. ( 22) is not equivalent to a genuine which-path ignorance, but it is a special case where a different detector state basis is considered.

D. A toy model with a Mach-Zehnder interferometer

The fundamental difference between distinguishable and indistinguishable cases, i.e., the use of Eq. ( 16) or Eq. ( 19) for the probability function, is the coupling between the considered system and the possible which-path detector(s) and/or the environment, but also the information that can be extracted from the detector(s) outputs. Such a coupling has been extensively studied in the context of decoherence theory [START_REF] Hubert | Decoherence, einselection, and the quantum origins of the classical[END_REF][START_REF] Schlosshauer | Decoherence, the measure-ment problem, and interpretations of quantum mechanics[END_REF]. In this section we consider a very simple case to investigate the limits of Eqs. ( 16) and ( 19) in terms of effects thanks to a toy system where we can continuously tune the detectability of the taken path.

We consider a Mach-Zehnder interferometer with a movable beam splitter (represented in Fig. 4), an example discussed in the literature and realized experimentally with atoms in resonant cavities [START_REF] Bertet | A complementarity experiment with an interferometer at the quantum-classical boundary[END_REF][START_REF] Haroche | Exploring the Quantum: Atoms, Cavities, and Photons[END_REF]. Here we treat the problem in terms of effects in a POVM framework. A discussion of the Mach-Zehnder interferometer in terms of unsharp detection has been already discussed by Busch and Shilladay [START_REF] Busch | Complementarity and uncertainty in mach-zehnder interferometry and beyond[END_REF]. In this past work, the unsharpness of the detection was studied in terms of measurement mixing between the two paths, like in the quantum eraser case discussed in Sec. III C. The cases of distinguishability or indistinguishability of the paths were also treated, but not the frontier between them, which on the contrary is the main subject of the following paragraphs. A more general treatment of the continuous passage between distinguishable and indistinguishable cases has been done in the past by B.-G. Englert [START_REF] Englert | Fringe visibility and which-way information: An inequality[END_REF]. The visibility of interferences and of the distinguishability quantity (in terms of the amount of which-way information can be extracted) is discussed, but not the relation to the probability function in the context of POVM.

The system considered here is composed by a singlephoton source emitting monochromatic photons |1 in interacting with: a movable beam splitter BS1, two mirrors M, M ′ , a phase shifter P S that induces a phase φ, a second (fixed) beam splitter BS2 and two detectors d 1 and d 2 , following the scheme represented in Fig. 4. The movable beam splitter, with a mass m, can move with respect to a pivot and is connected to a fixed part by a spring that corresponds to a resonant angular frequency ω. The beam splitter-spring system is described by a harmonic oscillator with energy spectrum E n = ω (n + 1 2 ). When the photon is reflected from the first beam splitter, a momentum kick ∆P = √ 2p, with p = 2π /λ the impulse of the photon, is transferred to BS1 with a translation from its ground state |0 BS to the coherent state |α BS with α = ip/ √ mω [START_REF] Bertet | A complementarity experiment with an interferometer at the quantum-classical boundary[END_REF][START_REF] Haroche | Exploring the Quantum: Atoms, Cavities, and Photons[END_REF]. In analogy to the Young's slits, we can consider the interferometer arm with the reflection from the movable beam splitter as the path a, and path b otherwise (see Fig. 4).

!"# !"$ %& % ' # ' $ (
In the case of a fixed beam splitter, the state of the beam splitter itself does not change after the passage of the photon and the state corresponding to the photon is

|1 in → |φ = - 1 2 |1 out - i 2 |2 out - e iφ 2 |1 out + ie iφ 2 |2 out (24)
The probability of detecting something on the detector d 1 depends on the operator K d1 = |0 out 1 out | and the corresponding effect

F d1 = K † d1 K d1 = |1 out 1 out |.
Because of the impossibility of determining the path taken by the photon, the corresponding probability is ℘(d 1 ∧ (a ∨ b)|s). The complementary detection c representing K a∨b could be constitued by a series of detectors around the beam splitter BS1, like the wall detection in the case of the Young's slits, to insure the interaction (reflection or transmission) of the incoming photon |1 in with BS1. The probability is then given by

℘(d 1 ∧ (a ∨ b)|s) = tr(F d1 ρ ′ ) = 1 2 [1 + cos(φ)] , (25)
with ρ ′ = |φ φ| and |φ given by Eq. ( 24). We recover the standard formula of the Mach-Zehnder interferometer [START_REF] Busch | Complementarity and uncertainty in mach-zehnder interferometry and beyond[END_REF][START_REF] Auletta | Quantum Mechanics[END_REF].

We consider now that the beam splitter BS1 can move and that its state after the recoil is described by the coherent state |α BS . Considering the initial state |1 in |0 BS describing the photon-beam splitter system, after the interaction between the incoming photon with the first beamsplitter BS1 (and the mirrors M and M ′ and the second beam splitter BS2), the photon/mirror state |φ is described by

|1 in |0 BS →= |φ |Φ BS = = - 1 2 |1 out |α BS - i 2 |2 out |α BS + - e iφ 2 |1 out |0 BS + ie iφ 2 |2 out |0 BS , ( 26
)
where |φ is state of the photon at the exit of the interferometer and |Φ BS is the state of the movable beam splitter after the passage of the photon. The operator K b = 0 BS |Φ BS |φ 1 in | can be associated to the branch b where there is no momentum transfer to BS1, which remains in the |0 BS state. For the branch a, we cannot directly use α BS |Φ BS |φ 1 in | as K a operator. Due to the non-orthogonality of |α BS and |0 BS , this leads to the possibility of having E a + E b > 1, violating the basic POVM properties. Considering that we can identify a coherent state only if its corresponding signal is above the quantum shot noise of the system, instead of |α BS we can consider its Gram-Schmidt orthogonalization |α ′ BS with respect to

|0 BS |α ′ BS = |α BS -0 BS |α BS |0 BS 1 -| 0 BS |α BS | 2 . ( 27
)
The corresponding which-path operators are then

K a = α ′ BS |φ 1 in | = 1 2 1 -| 0 BS |α BS | 2 |1 out 1 in | + - i 2 1 -| 0 BS |α BS | 2 α ′ BS |α BS |2 out 1 in | (28)
and

K b = 0 BS |Ψ 1 in | = = - 1 2 0 BS |α BS |1 out 1 in | -0 BS |α BS i 2 |2 out 1 in | + - e iφ 2 |1 out 1 in | + ie iφ 2 |2 out 1 in | . ( 29
)
The corresponding probabilities of the single paths become Finally, we have then

℘(d 1 ∧ a|s) = tr(F d1 K a ρK † a ) = 1 4 1 -| 0 BS |α BS | 2 (30)
℘(d 1 ∧ b|s) = tr(F d1 K b ρK † b ) = 1 4 1 + e iφ 0 BS |α BS 2 = (31)
℘((d 1 ∧ a) ∨ (d 1 ∧ b)|s) = 1 2 1 + e -|α| 2 2 cos(φ) , (32)
where we used α BS |0 BS = e -|α| 2 2 . As we can see in Fig. 5, for each probability relative to a specific path, an interference term is always present and is proportional to the overlap between the |0 BS and |α BS states. Only the probability corresponding to the path b is sensitive to the phase of the P S.

In the limit 0 BS |α BS → 0 (corresponding to α → ∞ and m → 0, see Fig. 5 bottom), we have a pure particlelike behavior with ℘(d

1 ∧ a|s) = ℘(d 1 ∧ b|s) = 1/4 and ℘((d 1 ∧ a) ∨ (d 1 ∧ b)|s) = 1/2.
In the limit 0 BS |α BS → 1 (corresponding to α → 0 and m → ∞, see Fig. 5 bottom), we have instead ℘(d 1 ∧ a|s) = 0 and ℘(d 1 ∧b|s) = 1. From the detection of |0 BS , no information on the taken path can be extracted. This is similar to the case 3 discussed in Sec. III B, where both path detectors provide the same output. In this limit case ℘(d Except to the limit case with 0 BS |α BS → 0 (m → 0), the two equations ( 25) and [START_REF] Cox | Algebra of Probable Inference[END_REF] lead to different forms of the probability function. Once more, the expressions

IV. INTERFERENCES IN SUBSEQUENT MEASUREMENTS WITH INDEFINITE CAUSAL ORDER

A. General considerations for indefinite causal order measurements

Another two-way system where interference effects occur is the case of subsequent measurements with an indefinite causal order. To treat such class of subsequent measurements, the new formalism of the process matrices has been developed in the last years [START_REF] Oreshkov | Quantum correlations with no causal order[END_REF][START_REF] Brukner | Quantum causality[END_REF][START_REF] Henderson | Quantum temporal superposition: The case of quantum field theory[END_REF], and the presence of interference effects in measurements with indefinite order has been proven experimentally [START_REF] Rubino | Experimental verification of an indefinite causal order[END_REF]. In particular it has been demonstrated that, for two subsequent measurements a and b, the probability relation

valid for a probabilistic composition between two measurements with defined causal order (where the measurement order is indicated by the operator " " and with 0 ≤ λ ≤ 1) is violated for indefinite causal phenomena.

With the probability definition and notation introduced in the previous sections, both cases of two-way interferences and two causal orders are simply discussed on the same footing and without introducing new notations in the probability function like P r a b .

In the case a single causal sequence of emissions from a source s, with a measurement a followed by a measurement b and final detection d, the corresponding probability is

℘(d ∧ b ∧ a|s) = tr(F d K b K a ρK † a K † b ), (34)
where K a and K b are the Kraus operators associated to the effects E a and E b corresponding to the intermediate measurements, respectively, and F d is the effect corresponding to the final detection. Note that due to the implicit prior of the order "a after b" in the notation P r a b , in our notation P r a b (a, b) corresponds in fact to ℘(d|b

∧ a ∧ s) = ℘(d ∧ b ∧ a|s)/℘(b ∧ a|s).
Similarly to the which-path case, if we consider the same subsequents measurements but where the order be-

B. Distinguishable causal order

This case corresponds to the situation where we can implicitly determine the measurement order, similarly to the case where each path in the Young's slits can be determined, and the associated probability writing is

℘((d ∧ b ∧ a) ∨ (d ∧ a ∧ b))|s.
The different sequences of measurements can be associated to the effects E (abd) and E (bad) and each corresponding probability can be constructed from a nested version of Eq. [START_REF] Kraus | States, Effects, and Operations Fundamental Notions of Quantum Theory[END_REF]. Moreover, in this case the property (P3)/(P3*) can be applied:

℘((d∧b∧a)∨(d∧a∧b)|s) = ℘(d∧a∧b|s)+℘(d∧b∧a|s) = = tr(F d K b K a ρK † a K † b ) + tr(F d K a K b ρK † b K † a ) = = ℘(a ∧ b|s)℘(d|a ∧ b ∧ s) + ℘(b ∧ a|s)℘(d|b ∧ a ∧ s). (35)
The last equality of the above equation allows to appreciate the equivalence to Eq. ( 33), which is manifestly not violated where ℘(a ∧ b|s) = λ.

C. Indistinguishable causal order and discussion

If the order of the measurement is intrinsically indef-

inite, ℘(d ∧ [(b ∧ a) ∨ (a ∧ b)]|s)] has to be considered with ℘(d ∧ [(b ∧ a) ∨ (a ∧ b)])|s) = tr(F d K a∧b ρK † a∧b ), (36)
where the Kraus operator K a∧b corresponds to the indefinite ordered measurement (b ∧ a) ∨ (a ∧ b). Compared to Eq. ( 35), additional interference terms can be present and we have

℘(d∧[(b∧a)∨(a∧b)])|s) = ℘((d∧b∧a)∨(d∧a∧b)|s). (37)
In order to appreciate the differences between the two expressions, we can consider ideal measurements for a and b with the assumption that they can be described by non-commutative and non-orthogonal projectors P a and P b acting in the large Hilbert space composed by the inputs and outputs of the two measurements [START_REF] Flatt | Gleason-busch theorem for sequential measurements[END_REF][START_REF] Oreshkov | Quantum correlations with no causal order[END_REF]. With these assumptions, Eq. ( 35) can be written as The above equation violates Eq. ( 33), as obtained in the work of Brukner and coworkers with the process matrices formalism [START_REF] Oreshkov | Quantum correlations with no causal order[END_REF][START_REF] Brukner | Quantum causality[END_REF][START_REF] Branciard | The simplest causal inequalities and their violation[END_REF].

℘((d ∧ b ∧ a) ∨ (d ∧ a ∧ b)|s) = = tr

V. CONCLUSION

In conclusion, we present a formulation of the probability function in the context of generalized measurements for subsequent detections with several possible paths, and for subsequent detections with indefinite order. From the assumption of the Hilbert space structure for the description of systems, Gleason-Busch theorem assures that the trace of the density operator univocally defines the form of the probability function. Flatt and coworkers demonstrate that from this result, when subsequent measurements are considered, the Kraus updating rule is reconstructed. Here we apply the same methodology to a two-path case with a renewed notation but also to the case of intermediate measurements without define causal order.

For the two-path case, two different expressions of the probability are found, ℘((d ∧ a) ∨ (d ∧ b)|s) and ℘(d ∧ (a ∨ b)|s), which are related to the possibility of distinguishing or not the trajectory in the measurement system. In fact, the distributive property of the probability function arguments cannot be taken for granted. From the first expression, the classical law or probability P r C (a ∨ b) = P r C (a) + P r C (b) is recovered. The use of the reduced trace over the undetected states of the path-detectors leads to this same expression. With regards to the ℘(d ∧ (a ∨ b)|s), the associated operator K a∨b to the a ∨ b measurement, can be interpreted ambiguously. K a∨b can be built from measurements corresponding to the same final detector state independently from the path, or from a complementary measurement of a and b (c = N OT (a ∨ b)). Both approaches lead to a final expression corresponding to the standard Born rule for the case of perfect projective measurements. If K a∨b is constructed by a mixing of path-detector states, we recover the situation of the quantum eraser. We are in fact considering the probability ℘(d ∧ q|s) associated with the state |q det = α |a det + β |b det , which depends on the choice of constants α and β, i.e. a special case of which-path ignorance.

The frontier between the intrinsic possibility to distinguish a path or not is related to the coupling of the studied system with the path-detectors and/or the environ-ment. This topic is widely studied in the literature, and in particular in the context of decoherence theory. Here we consider the very simple case of a Mach-Zehnder interferometer with a movable beam splitter, which is also well known in the literature but treated here in the context of generalized measurements. We demonstrate here that by varying the mass of the beam splitter, we can continuously pass from the distinguishable path case, where ℘((d ∧ a) ∨ (d ∧ b)|s) is valid, to the indistinguishable path case, where ℘(d ∧ (a ∨ b)|s) should be used instead. This toy model reveals once more the complementarity of nature, but also underlines once more the advantages of generalized measurement theory with respect to ideal projective measurements, where unsharp detections revealing particle-like and wave-like behavior at the same time can be treated unambiguously.

With the same probability notation introduced here, another two-way problem is treated, namely the sequence of detections with an indefinite causal order. Similarly to the two-path case, the difference between the two expressions is related to the possibility to distinguish or not the causal order of the measurements. The first expression corresponds to a statistical sum of probabilities relative to the two possible measurement sequences with definite order. In this case the "causal equality" (Eq. ( 33)) is valid. In the second expression, additional interference terms appear with the violation of the causal equality, similarly to the results obtained with the process matrices formalism.

  P r(ab) = |ψ a + ψ b | 2 = = P r(a) + P r(b) + 2 P r(a)P r(b) cos (arg(ψ a ψ * b )) , (1) where ψ a , ψ b are the complex probability amplitudes associated with each slit and P r(a) = |ψ a | 2 , P r(b) = |ψ b | 2 are the probabilities associated to the opening of the single slits. The above expression is substantially different from the classical probability sum rule P r C (ab) = P r(a) + P r(b),

FIG. 1 .

 1 FIG. 1. Scheme of Young's slits experiment. Adapted from Ref. [1].

FIG. 4 .

 4 FIG. 4. Scheme of the Mach-Zehnder interferometer with a movable beam splitter BS1, a phase shifter with φ with two possible incoming beams |1in and |2in and outputs |1out and |2out measured by the detectors d1 and d2. Photons parallel to the incoming photon (horizontal propagation in the figure) are indicated by the states |1 and with |2 otherwise (vertical propagation).

= 1 4 1 +FIG. 5 .

 15 FIG. 5. Dependency of the different probability expressions on φ and α with the fixed value α = 1.5 (top) and with the fixed value φ = 0 (bottom).

1 ∧

 1 b|s) (and then ℘((d 1 ∧ a) ∨ (d 1 ∧ b)|s)) is de facto equivalent to ℘(d 1 ∧ (a ∨ b)|s) treated in the previous section. The behavior of the different formulas as function of φ and α is shown in Fig. 5.

d 1 ∧

 1 (a ∨ b) and (d 1 ∧ a) ∨ (d 1 ∧ b) cannot be considered equivalent with the violation of the distributivity property.

  [23-25] P r(a, b) = λP r a b (a, b) + (1 -λ)P r b a (a, b),

  tween a and b is unknown, the probabilities ℘((d ∧ b ∧ a) ∨ (d ∧ a ∧ b))|s and ℘(d ∧ [(b ∧ a) ∨ (a ∧ b)]|s)] lead to two different final expressions.

  (P d P b P a ρP a P b ) + tr(P d P a P b ρP b P a ). (38) On the other hand, Eq. (36) becomes ℘(d ∧ [(b ∧ a) ∨ (a ∧ b)]|s) = = tr(P d P b P a ρP a P b ) + tr(P d P a P b ρP b P a )+ + tr(P d P b P a ρP b P a ) + tr(P d P b P a ρP b P a ) + . . . (39) where the cross terms tr(P d P b P a ρP b P a ) and tr(P d P b P a ρP b P a ) are present (plus additional terms related to the non-commutativity between P a and P b ).

  For the case of two indefinite ordered measurements a and b, two different formulation of the probability function are allowed, ℘((d ∧ b ∧ a) ∨ (d ∧ a ∧ b)|s) and ℘(d ∧ [(b ∧ a) ∨ (a ∧ b)])|s).

ACKNOWLEDGMENTS

I would like to thank very much M. Romanelli and M. Walschaers for their constructive critics to the previous versions of the manuscript, but also C. Fabre, S. Reynaud, V. Parigi and N. Paul for the stimulating discussions and support. I would like also to thank A. Caticha and N. Carrara for the encouragement and suggestions after a first talk on a primordial version of the presented work.

Appendix A: Quantum eraser probabilities

We consider two path detector bases |p 1 , |p 2 , e.g. corresponding to |a det , |b det states of the Young's slit case, and a final detection d. We consider a perfect which-path measurement where |p j are associated to the system state |j . The associated operators K j = i p j |p i |i i| is then equivalent to the projectors P j = |j j|. We consider two different orthogonal states |q 1 , |q 2 related by the unitary transformation |q j = V ji |p i . For each measurement q j , the associated operator

For each single measurement q j , we have

(A2)

When we consider the probability relative to the measurement (d ∧ q 1 ) ∨ (d ∧ q 2 ), we have

where we used the unitary matrix property

The second term of the expression is in fact equal to zero because of other property of unitarity of V matrices

) Independently of the choice of the orthogonal and complete base, probability is always the same and equal to a particle-like behavior, even if the single d ∧ q j measurement can provoke interference effects.