Álgebras de Procesos

In Section2.1.1we showed that the quantum state space S_q associated with a quantum system is the compact convex set of all positive and normalized linear operators acting on a Hilbert space. Moreover, pure quantum states constitute the extreme points of this convex set. Hence, pure quantum states can be thought of as the building blocks of the state space. For our current purpose, it is sufficient to consider the Hilbert space of the system to be finite (say d) dimensional. So, the state space S_q will also be finite (d²) dimensional².

Now, suppose that ˆ% is an arbitrary point of the quantum state space. We ask “how can we construct ˆ% from a fixed complete set of pure states?” The answer constitutes of two parts. First, if the state is pure, i.e., ˆ% = |ψihψ|, then it is an extreme point of S_q and by Corollary 1.1.12 it possesses no nontrivial convex decomposition in terms of other quantum states. However,

2Note that, by Definition 1.1.47, the dimensionality of a convex set is the maximum number of affinely independent points belonging to that set.

given a complete set of vectors {|φii}^d_i=1 (not necessarily orthonormal) for H, it can be written as a coherent superposition of the basis elements, that is, |ψi = Pd

i=1c_i|φ_ii. Second, if ˆ% is a mixed state, then by Carath´eodory theorem 1.1.5 there exists a convex decomposition of the state into d² pure states³. Note that such a decomposition is not necessarily unique, and a quantum state may possess many convex decompositions. The important point, however, is that not every quantum state can be decomposed using the elements of a fixed set of pure states; see our discussion following Carath´eodory theorem 1.1.5. Hence, after obtaining a convex decomposition, every pure element of the decomposition can be written as a coherent superposition of the complete set of interest, just as in the first case. As a result, in general, there is a two-step construction procedure in terms of a fixed complete set of pure states for any quantum state:

i. If necessary, coherently superpose the basis states to obtain another set of pure states.

ii. Probabilistically mix the new set of pure states to obtain ˆ%.

From the above simple analysis we learn that if a quantum state ˆ% cannot be expressed as a convex combination of a set of pure states, then a coherent superposition of those pure states is essential to construct ˆ%. We then say that ˆ% contains coherence with respect to the given set of pure states.

It should now be conceivable that quantum coherence, as the manifestation of the quantum superposition principle, can be captured as a resource theory. For this purpose, we need to first define the set of free operations. This, however, depends heavily on the physical system of interest and fundamental or operational restrictions. For instance, in photonics the preparations might be restricted to the vertical and horizontal, or the diagonal and antidiagonal components of the radiation field. In contrast, in an atomic realization of qubits, the preferred preparations could be the energy eigenstates to which the systems decohere. By the above considerations and parallel to Postulate2, we know that the set of free operations for a resource theory of coherence is one that constrains an experimenter to the preparation of a specific class of quantum states and their convex combinations. The extreme points of this convex set (which is also compact), in general, does not need to satisfy any purity, orthogonalization, or completeness conditions.

This is, for example, the case for the nonclassicality theory of continuous variable bosonic systems in which the set of bosonic coherent states {|αihα| : α ∈ C}, as extreme points, are nonorthogonal and overcomplete [12].

In this chapter, we assume that the set of generating points is a finite complete orthonormal basis which is usually termed the computational basis and denoted by E = {|iihi|}. It thus follows that, in this case, the set of free states S_free = S_inc = convE is a polytope, the elements of which are commonly called incoherent states; see Fig.3.1. By making use of Theorem 1.1.9 we thus arrive at the following.

3Note that, the actual number of points required for such a decomposition is d²+ 1. However, one point serves as the origin which is assumed to be the zero operator. Hence, we are left with only d² pure states.

Figure 3.1: Geometrical illustration of the polytope of incoherent states. The black ellipse represents the quantum state space. Assuming that the generating points E of free states are pure and finite, they rely on the boundary of the state space and their closed convex hull forms a polytope, represented by the shaded area. Any point outside this polytope (the red dot) represents a quantum state that contains coherence with respect to E , i.e., its constructions inevitably requires a coherent superposition of the elements of E .

Corollary 3.2.1. The polytope of incoherent states can be characterized using finitely many separating hyperplanes, i.e., coherence witnesses.

Accordingly, the elements of the set of free operations O_free are called incoherent operations.

Importantly, incoherent states can arise from different sets of free operations. Several classes of such operations for the resource theory of coherence have been studied so far, e.g., general, strict [13], and genuine incoherent operations [14]. A review of these operations and their operational meaning can be found in Refs. [15, 16]. Here, we mainly consider two classes of incoherent operations. The first class of interest, is called the general incoherent operations as discussed in Ref. [13]. These are the most general incoherent operations possible and possess Kraus decomposition Λ( · ) = P

iFˆ_i( · ) ˆF_i^† such that P

iFˆ_i^†Fˆ_i = ˆ1 and ˆF_iS_incFˆ_i^† ⊂ S_inc for all i. The latter condition ensures that even by subselection of the operation output one cannot generate coherence from incoherent states. Every Kraus operator then must be of the form

F =ˆ X

i

c_i|iihψ_i|, (3.21)

in which c_i ∈ C and |ψii ∈ span{|ji ∈ E_i} so that E_is are disjoint subsets of E [13]. The latter means that the vectors |ψ_ii must live on disjoint orthonormal subspaces spanned by incoherent basis elements.

The second class of operations we use are called strict incoherent operations. They are simply incoherent operations of the above given form with the extra restriction that for every Kraus operator ˆF_i it holds true that ˆF_i^† is also incoherent so that the adjoint map of Λ given by Λ^‡( · ) = P

iFˆ_i^†( · ) ˆF_i is incoherent too [13]. Using Eq. (3.21) this implies that the Kraus operators of strict incoherent operations have the form

F =ˆ X

i

c_i|iihj(i)|, (3.22)

with both |ii, |ji ∈ E and j(i) is one-to-one function. As shown by Yadin et al [17], these operations correspond to those also not consuming quantum coherence within the given com-putational basis. A necessary and sufficient condition for strictness of incoherent operations is given below.

Lemma 3.2.1. [17,18] An incoherent operation Λ is strict incoherent if and only if it possesses a set of Kraus operators { ˆF_i} such that for all quantum states ˆ% holds

∀i : ∆[ ˆFi% ˆˆF_i^†] = ˆFi∆[ ˆ%] ˆF_i^†. (3.23) Here, ∆ is the fully depolarizing map and the binary operation ◦ between operations is simply the composition of superoperators, e.g., ∆ ◦ Λ[ ˆ%] = ∆[Λ[ ˆ%]]. Equation (3.2.1) is sometimes notatioally compressed int a commutation relation as [∆, ˆFi] = 0.

In what follows, however, we are interested in a subset of strict incoherent operations that we name universal strict incoherent (USI). Elements of O_USI are those generating the symmetric group (i.e., the group of permutations) on E . In mathematical terms,

symE = hhO_USIii, (3.24)

where hh · ii is the standard group generation operation via group composition, that is, the group is formed by repeatedly composing the elements of the generating set. It is also straightforward to show that symE is isomorphic to sym{1, . . . , d}, where d is the dimensionality of E . Now, the universality of O_USImust be understood over the set of incoherent states, that is, USI operations are necessary and sufficient to generate any incoherent state ˆσ ∈ Sinc via their composition and mixing, and subselection of outcomes. Note also that, starting from a pure state the subselection can be disregarded. Equivalently, any incoherent operation can be obtained from a composition of the elements in O_USI. That is,

∀Γ ∈ O_inc ∃{Λ_i} ⊆ O_USI such that Γ[ · ] = Λ₁◦ Λ₂◦ Λ₃◦ · · · [ · ]. (3.25) As an explicit example, suppose that the space is 3-dimensional with the computational basis E₃ = {|iihi|}³_i=1 and consider the set of permutation operations

O_USI;3 = {|jihi| · |iihj| + |iihj| · |jihi| + X

k=i+2

|kihk| · |kihk|}, (3.26)

for i = 1, 2, 3 and j = i + 1 mode 3. Explicitly,

Λ₁ = |2ih1| · |1ih2| + |1ih2| · |2ih1| + |3ih3| · |3ih3|, Λ₂ = |3ih2| · |2ih3| + |2ih3| · |3ih2| + |1ih1| · |1ih1|, Λ₃ = |3ih1| · |1ih3| + |1ih3| · |3ih1| + |2ih2| · |2ih2|.

(3.27)

Elements of O_USI;3 are thus permutation superoperators. It can be easily verified that O_USI;3 is

sufficient to generate any permutation of E3 via composition of its elements. In particular, for any i we have Λ_i◦ Λ_i = ∆ which is the identity over the set of incoherent states S_inc. Being able to generate any pure incoherent state starting from another arbitrary incoherent pure state using O_USI;3, it is immediately clear that any incoherent state in convE₃ can be reached by probabilistic implementation of such USI maps and their compositions as required.