Notes on Naimark’s dilation theorem

After a short review of the main properties of unsharp observables we provide examples of commutative unsharp observables obtained as the projection of sharp observables. In particular, we consider a family of PVMs defined on the tensor product Hilbert space H⊗K which include the one introduced by Ozawa in his modification of the von Neumann position measurement model and show that their projections onto H are commutative POVMs. Then, we make some observations on the concept of Naimark dilation and its connections with the integral representation of commutative POVMs. Finally, we introduce a partial order relation in the set of Naimark dilations, show that there is a minimal element and that it is unique. Minimality is a consequence of the partial ordered structure and coincides with the usual definition of minimal Naimark dilation. In passing, we use Naimark’s theorem in order to prove some well known properties of positive operator valued measures.


Introduction
Positive operator valued measures are fruitfully used in quantum mechanics in order to generalize the concept of quantum observable which is otherwise represented by a self-adjoint operator. Such a generalization is rooted in the statistical character of quantum physics and opened new perspectives [1][2][3][4][5][6][7][8][9]. Just to recall some examples, it allowed a phase space representation of quantum mechanics where the distribution functions describing the states of the system are positive-definite [4,7,8,10,11], it made possible a rigorous description of the Heisenberg error-disturbance relations [9,12,13], it made clear that commutativity is not equivalent to compatibility [2,7], and it made it possible to define a photon localization observable [4][5][6][14][15][16][17][18][19]. As a relevant physical example that illustrates the problem of joint measurability in the framework of POVMs, one can consider the case of the position and momentum operators, Q, P , in the Hilbert space H = L 2 (R). Although they are incompatible (they do not commute) they can be smeared to two commutative POVMs, F Q , F P which are non-commuting but compatible, i.e., they are the marginals of a joint POVM [2,7]. In turn, F Q and F P provide examples of commutative POVMs obtained as the randomization of a spectral measure. That is a general  [20][21][22][23][24]. To be more precise, they can be represented as the integral of a Markov kernel with respect to a spectral measure [22][23][24]. There are other integral representations of POVMs [25,26]. In the commutative case, all of them are equivalent [23,27]. A different kind of connection between POVMs and projection valued measures (PVMs) is established by the Naimark dilation theorem which ensures that every POVM is the projection of a PVM in an extended Hilbert space [28][29][30][31]. From the physical viewpoint, the Naimark theorem ensures the existence of a measurement for any observable [7].
In the present note we make some observations on Naimark's dilation theorem. For example, it is interesting to ask under what conditions a POVM obtained as the projection of a PVM is commutative. As an example, we analyze a family of PVMs which includes the one introduced by Ozawa [32] in his modification of the von Neumann position measurement model, and show that their projection is a commutative POVM. Then, we review some of the main results on the connection between the Naimark dilation theorem and integral representation of commutative POVMs, discuss the relevance of the Naimark theorem to the compatibility problem and derive some well known properties of POVMs by means of the Naimark theorem. Finally we analyze the concept of minimality of the Naimark dilation in the framework of partially ordered sets (posets) where minimality is a consequence of the poset structure and prove the existence of a unique minimal Naimark dilation.
In what follows, we denote by B(X) the Borel σ-algebra of a topological space X and by L s (H) the space of all bounded self-adjoint linear operators acting in a Hilbert space H with scalar product ·, · . The subspace of positive operators is denoted by L + s (H).
where {∆ n } is a countable family of disjoint sets in B(X) and the series converges in the weak operator topology. It is said to be normalized if where 1 is the identity operator.
where 0 is the null operator. Let E be a PVM. By equation (2), We can then restate Definition 1.4 as follows. In quantum mechanics, non-orthogonal normalized POVMs are also called generalised or unsharp observables while PVMs are called standard or sharp observables. Definition 1.6. The von Neumann algebra (or W * -algebra) A W (F ) generated by the POVM F is the von Neumann algebra generated by the set {F (∆)} ∆∈B(X) . It is the smallest * -algebra of bounded operators closed in the weak operator topology and generated by {F (∆)} ∆∈B(X) .
In what follows, we shall always refer to normalized POVMs and we shall use the term "measurable" for the Borel measurable functions. For any unit vector ψ ∈ H, the map is a measure. In the following, we shall use the symbol d F x ψ, ψ to mean integration with respect to F (·)ψ, ψ . For any real, bounded and measurable function f and for any POVM F , there is a unique [33] bounded self-adjoint operator B ∈ L s (H) such that By the spectral Theorem [34], there is a one-to-one correspondence between spectral measures, E, and self-adjoint operators, B, the correspondence being given by Notice that the spectrum of E B coincides with the spectrum of the corresponding self-adjoint operator B. Moreover, in this case a functional calculus can be developed. Indeed, if f : R → R is a measurable real-valued function, we can define the self-adjoint operator [34] If f is bounded, then f (B) is bounded [34]. In particular, and A = t dE t is the generator of the von Neumann algebra generated by E. We point out that if F is not projection valued, equations (5) and (4) do not hold [35] and, in order to recover the generator of A W (F ), we need all the moments of F . In particular, in the case of a real commutative POVM F with bounded spectrum and such that F (∆) is discrete for any ∆, we have [36,37] where A is a generator of the von Neumann algebra A W (F ). In the following we assume X to be Hausdorff, locally compact and second countable.

Naimark's dilation theorem
A procedure to derive a POVM is to start from a PVM E in a Hilbert space H and to project to a subspace H 0 ⊂ H, i.e., F (∆) = P E(∆)P where P H = H 0 . In this context it is worth analyzing the properties of the POVM F as for example its commutativity. Here we provide a family of PVMs whose projection onto a subspace gives a commutative POVM. In particular, we consider a PVM of the kindQ whereQ(0) is a self-adjoint operator in the Hilbert space K, Q(0) is a self-adjoint operator in the Hilbert space H,Q(τ ) is a self-adjoint operator in the Hilbert space H ⊗ K and c and b are functions depending on τ . We suppose H and K are infinite-dimensional. An observable of this kind has been derived by Ozawa [32] as the solution of the Heisenberg equation in his model for the measurement of position. In the Ozawa model,Q(τ ) denotes the meter position observable at time τ and Q(0) the position observable of the system at time t = 0. The outcome of the measurement is obtained by measuring the meter observableQ at time ∆t 1 . Analogous equations can be derived for the momentum. Now we show that the projection ofQ(τ ) onto the subspace H ⊂ H ⊗ K is a commutative POVM. First we recall that the tensor product H ⊗ K can be represented as the direct sum 2 Note also that H is isomorphic to H ⊗ φ where φ is a unit vector in K. Now, let {ψ i } i and {φ j } j be bases for H and K respectively. Then Let P be the projection operator from H⊗K onto H defined by P (ξ 1 ⊗φ 1 , . . . , Let E Q(0) and EQ (0) denote respectively the spectral measures of Q(0) andQ(0) in equation (6). Then, the spectral measure ofQ(τ ) is E Q(0) ⊗ EQ (0) . Indeed, we have What about the projection of E Q(0) ⊗ EQ (0) onto H? For any ψ i ∈ H, we have 1 τ is chosen such that 0 ≤ τ ≤ ∆t with K∆t = 1 where K is the coupling constant which is supposed to be very large, K 1. 2 We recall that the direct sum of a countable family of Hilbert spaces (Hi, ·, · i) i∈N is the space where α 1 comes from the expansion EQ (0) (∆ 2 )φ 1 = l α l φ l . Moreover, ∀ψ i ∈ H, where α 1 comes from the expansion EQ (0) (∆ 2 )φ 1 = l α l φ l . We can now calculate )P should be the randomization of a sharp position observable and it would be interesting to analyze the properties of the Markov kernel that realizes the randomization. Moreover, the sharp version should be equivalent to the projection of the pointer observable cQ(0) ⊗ I + dI ⊗Q(0) (see Theorem 2.7 below). Those are problems deserving further work. Anyway, we have a relevant physical example (perhaps a family of physical examples) where the projection of a pointer observable is a commutative POVM.
An important result concerning the connections between PVMs and POVMs is due to Naimark. He proved that not only the projection of a PVM provides a POVM but that every non-orthogonal POVM is the projection of a PVM. Theorem 2.1 (Naimark [3,8,[28][29][30][31]). Let F be a POVM. Then, there exist an extended Hilbert space H + and a PVM E + on H + such that where P is the operator of projection onto H. The dilation E + : H + → H + can be chosen such that the extended Hilbert space H + coincides with the closure of span{E + (∆)ψ : ψ ∈ H, ∆ ∈ B(X)}. In this case the dilation is said to be minimal and is unique up to unitary transformations.
Therefore, up to isomorphisms, there is a unique minimal Naimark dilation of F . In subsection 2.3 we reformulate the concept of minimality of the Naimark dilation in the language of partially ordered sets (posets) where the existence of a minimal dilation can be proved by Zorn's lemma. In this framework, the definition of minimal dilation (see Theorem 2.1) becomes a consequence of the poset structure.
In the case of a commutative POVM F , Naimark's theorem can be used to define a self-adjoint operator A + in H + whose projection onto H gives the sharp reconstruction of F [36][37][38][39][40][41]. Let us first recall the integral representation of commutative POVMs. In the following Λ denotes a measurable space.
is continuous and bounded whenever f is continuous and bounded.

5) A is called the sharp version of F and is unique up to bijections.
An equivalence relation ∼ a such that A ∼ a F whenever A is discrete can be introduced. That can be interpreted as the equivalence of A and F with respect to their informational content [40].  In the case f is unbounded, the domain of definition of the operators must be taken into account [9,35]. Theorem 2.7 ([36,37]). Let F : B(R) → L + s (H) be a commutative POVM such that the operators in the range of F are discrete. 3 Let A be the sharp version of F and A + = λ dE + λ the Naimark operator corresponding to the Naimark dilation E + . Then, there are two bounded, one-to-one functions f and h such that Theorem 2.7 establishes that h(A) is the projection of f (A + ) with h and f one-to-one. According to Definition 2.5 we can say that a correspondence between A and A + is established. We denote such a correspondence by A ↔ P rA + .

Conditions for the joint measurability
In the present section we recall the definition and some of the main theorems on the joint measurability of two POVMs. Then, we show how Naimark's dilation theorem can be used in order to give necessary and sufficient conditions for joint measurability.
As the following theorem shows, in the case of two POVMs, commutativity implies compatibility but the converse is not true, i.e, commutativity is not a necessary condition for the compatibility. That is one of the main advantage in using POVMs in order to represent quantum observables.
Note that if F 1 and F 2 are PVMs, theorems 2.10 and 2.9 imply that F 1 and F 2 are compatible if and only if they commute. Indeed, if P E + 1 P and P E + 2 P are projections then [P, E + i ] = 0, i = 1, 2. Hence [F 1 , F 2 ] = 0. Moreover [35], two PVMs are compatible if and only if they are the marginals of a PVM. Theorem 2.10 is illustrated in the following diagram.
where the arrow o o c / / denotes compatibility, o o P / / denotes the relationship between a POVM and its dilation as expressed by Naimark's theorem.

Minimal dilation
In the present subsection we analyze the problem of the existence of a minimal Naimark dilation in the framework of partially ordered sets (posets). We introduce a preorder in the set of all Naimark dilations, then we consider the partial order it induces and prove the existence of a minimal element as well as its uniqueness. In this approach the definition of the minimal Naimark dilation becomes a consequence of the partial order structure. First we reformulate the definition of Naimark's dilation in a language which is more appropriate to the partial order sets framework.
Definition 2.13. Let F : B(X) → L + s (H) be a POVM on the Hilbert space H. A Naimark dilation is a couple (E + , H + ), where H + is a Hilbert space containing H and E + is a PVM, such that F (∆)ψ = P E + (∆)ψ, ∀ψ ∈ H and ∀∆ ∈ B(X), where P is the operator of projection onto H.
Let N d (F ) be the set of all the Naimark dilations of a POVM F . We can introduce a preorder in N d (F ).