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State transformations after quantum fuzzy measurements

Using a standard fuzzification procedure and the dynamical map in Heisenberg picture, a new expression for the state transformation after a fuzzy filter measurement, subject to covariance conditions, was obtained and some calculations were done to distinguish its properties from the those of the usual solution.


Introduction
If a quantum system interaction with the surroundings is subject to some restrictions like "quasi-isolation", i.e. it can interact up to a non-relativistic conservative quasiclassical field of forces, its evolution is given by the Schrödinger equation, which yields the unitary dynamical group (UG) { U t } t∈R : Otherwise, if it cannot be considered quasi-isolated, its evolution is well-described by master-type equations (which are the most general evolution equations which preserve the trace of the density operator ρ [1]). They actually yield some completely positive quantum dynamical semigroups (SG) [2,3] { V t } t≥0 . However, if one can accept freely an ontological interpretation of the conceptual apparatus of Quantum Physics in the first case, one has to be careful when asserting ontological-type sentences for the open quantum systems. This is due to pure quantum correlations which appear between the system and the surroundings, which cannot be destructed by any type of interaction. Of course, one can work perfectly, from the epistemological point of vue [4], with the up-mentioned SG's and master equations, but one has to be aware on the possible implications of explicitly or hidden ontological presuppositions.
Due to Born probabilistic interpretation, one has to accept -in the same epistemological perspective -the duality between states and observables, which is clearly illustrated in the two pictures of quantum evolution (Heisenberg and Schrödinger) and by the concept of instrument in Operational Quantum Physics (OQP) [2,4,5]. In OPQ the state transformer after a quantum measurement is defined as an instrument, and this aims for the mathematical formalization of the so-called Projection Postulate (a to much debated topic in Quantum Physics, often carrying an ontological ballast). The evolution due to SG yield, in the Schrödinger picture, the mixing of initially pure states, while in Heisenberg picture a sharp observable { E(B)} B∈B goes to a fuzzy observable { F (B)} B∈B [6]. If one consider the fuzzyfication map as a dynamical map in Heisenberg picture, one can compute the corresponding dual dynamical map in Schrödinger picture, an interpret it as a state transformer -which is different to the state transformer in OQP. This is a pure epistemological variant of OQP. Its study is useful in a twofold perspective: its adequacy to the real experimental physics could yield some changes in the present conceptual apparatus, while its inadequacy could emphasize the ontological importance of the usual OQP.

General topics
One of the main results of OQP is that one can define the state transformation after a filter-measurement (state-preparation procedure) using a quantum instrument [2], that is a map: E( The evolution of the system is given, in Heisenberg image, by the map: while the state transformation (Schrödinger image) is given by the dual instrument: where B ∈ B(Ω) is the Borel set of data, which the filter-measurement is looking for. Beyond positivity and monotonicity, E * (•, ρ) is asked to fulfil the following condition [2]: The sense of (1) is given by the convention used in usual OQP which accepts non-tracepreserving state transformers for B = Ω in order to limit the mathematical apparatus only to linear transformations [9]. The apparatus of OQP is trivial for sharp quantum observables and is, in general, due to von Neumann [10]. However, measurement is, by its very nature, a process of interaction which cannot, in general, fulfil the conditions for the unitary evolution of the investigated system, so one has to consider also, the case of fuzzy observables [9]. The operators: are usually named effects or fuzzy operators. The probability of obtaining the measurement result in the Borel set B is given in the terms of the instrument by: In [7] and [8] it is given a representation theorem for the most general form of a quantum instrument ∀B ∈ F , ∀Â ∈ B(H): where the operators {Ŵ in (ω)} i,n;ω have to observe the relation: (4) and (5) are to be understood in the weak sense. They give the most general form of the effects: However, the set of these mathematical objects have not mandatory correspond oneto-one to physical-significative objects, which have to observe some restrictions (for example the case of regular effects, which are neither less then 1 21 , nor greater than it). The definition of the state transformer as an instrument aims to the mathematical formalization of the so-called Projection Postulate (via relation (3)), which is a too much debated topic in Quantum Mechanics literature. It is known [13], that the conceptual status of state projection (reduction), is still polemical in measurement theory [14], whereas the concept of (nonselective) state change is rather straightforward, and can be easy handled in the Quantum Physics of Open Systems [2,17].
In [13] it is shown that E(Z, ρ) (nonselective state transformation) has a unique decomposition in (the selective) state transformations: which it is claimed to support the (selective) state reduction without assuming the Projection Postulate. However, the ideas presented above are consistent, also, with a Quantum Mechanics without Projection Postulate (without selection). If is the dynamical map in Heisenberg picture (the sharp effects go to fuzzy ones), one can write the probability using the Schrődinger image of the dynamical map (see eq. II.2.1 in [3]) Of course, this is applicable only for those generalized effects which can be obtained by fuzzyfication procedures from sharp ones. (But, for the moment, there is no physical grounded theory for the most general form of the physical effects, as it is done in [7] for the mathematical objects (6), while fuzzyfication of sharp obsevables is usual).

Purely discrete case
In [2,12] it is taken the case (4) when N 0 = 1 and the measure ν is the Heaviside one : where: On the next step, the case of a sharp observable with a purely discrete spectrum { E i } i∈Z will be considered; it is consistent with (9). The standard fuzzyfication procedure for a sharp observable is given by the relation [11,2]: where the positive constants {ω αm } α,m∈Z have to observe the relation: (simple stochasticity) and may depend on some fuzzyfication parameters (like σ which, for the moment, will be dropped).From (3), (10) and (9) one gets: Here, usually, one takes a non-complex solution: so (9) becomes: The result (15) is often taken as the main objective of OQP (see page 138 of [4]). Of course, in the sharp situation, when ω mk = δ mk , one gets the usual von Neumann result: Meanwhile, using (7) the probabilities of measuring the fuzzy observable (10) { F α } α∈Z for a system in the state ρ are given by: and the dynamical map in Schrődinger image can be to write using (17) in the terms of a completely positive map, similar to that of the dual instrument (9): In (18) one can identify: which are clearly different from (13). One is looking for a positive operator F d which yields: If one takes the real-positive solution k| F d |m = √ ω km , one can write: which can be written as (9) for The ideal case of von Neumann can be obtained, again, for F d = 1.

Purely continuous case
For the purely continuous spectrum, say the position operator case: x · |x x|dx one takes in (4) Ω = R and N(ω) = N 0 = 1. For the fuzzy effect one takes: where R f (x)dx = 1 is a condition similar to (12) and f (x) > 0 ∀x ∈ R. The next step is looking for the positive operator F c for which: and looking for a state transformer like (7) one finds: The OPQ state transformer is (equations 4.6.3 and 4.6.4 in [2]): where: while (26) can be written in the form: where: Example As an example it will be discussed only the continuous analogue of (24) which is given by: where k is the wave-number operator, conjugate to x.
In [2] it is shown that E * (B, ρ), the corresponding instrument to (27), is a solution of the covariance condition: Starting from (25) one can calculate the matrix elements in the wave-numbers space, and find that: where ( √ f )˜is the Fourier Transform of f (x). One follows that [ F c , U a ] = 0, so one can easy prove that the completely positive map (28) is also covariant w.r.t. space translations.

Discrete case
Calculating the moments of the observable E for the two states (27)  and, in general: which is a well-known formula of statistical physics. Some straightforward calculations can be done for the linear entropy [18] (see also papers: [19,20]): For (27) one has: while for (21): For a fuzzy initial state like: (31) and (32) are equal even in the general case: which for (23) is: )] Using the approximations from the example (2.2.1), for a sharp initial position (α < 0.2), the result is independent on the fuzzyfication parameter σ, S = 0 (the state remains almost pure), while for α > 1.6 (fuzzy initial position), the result is 1 − α √ π for sharp measurements (σ < 0.2) and 1 − for unsharp measurements (if still α ≪ σ this becomes the maximum value for (30), S = 1 2 )

Continuous case
One will obtain similar formulas for the position operator x = R x · E (x) (dx). For the canonically conjugate operator: k = R k · E (k) (dk) one has: The difference between (34) and (35) is given by the term i R ( ∂ρ ∂y ′ ) (y,y ′ =y) dy, which is a memory term for the quantum coherence of the initial state (prior to the fuzzy measurement of position), which, however, does not depend on the fuzzyfication parameters, as one may expect.

Conclusions
Starting with a fuzzyfication process as a dynamical map in Heisenberg picture, an alternate expression for the state transformer after a (fuzzy-)filter measurement was proposed. It is a completely positive map, similar to the dual quantum instrument of OQP, which is, also, subject to covariance conditions. Calculations distinguished some different predictions to that of the OPQ solution.