Measuring Quantum Coherence

The next step in characterizing the resource theory of coherence is to provide appropriate measures that meet the axioms given in Section 1.2.2 of Chapter 1 with respect to the set of free operations defined in the previous section:

1. For any faithful measure of coherence C, a quantum state ˆσ is incoherent if and only C(ˆσ) = 0.

2. For any incoherent operation Λ ∈ O_inc acting on quantum states, the measureC has to be strongly monotonic in the following sense. Suppose that ˆ% is transformed to an ensemble of states { ˆ%_i} with the corresponding probabilities {p_i} upon the action of Λ. Then, it must be true that,

meaning that coherence should not increase on average via incoherent operations.

3. A measure C of coherence must be convex, i.e., suppose that { ˆ%i} is an ensemble of quantum states with the corresponding probabilities {p_i}, then it must hold true that

C(X

i

p_i%ˆ_i) 6X

i

p_iC(ˆ%i), (3.29)

meaning that the average coherence should not increase by mixing.

Several measures of coherence have been introduced in the literature, mostly in parallel with the measures of entanglement, e.g., coherence of formation [19] and robustness of coherence [20,21].

At this point, it is worth mentioning that there exists a preorder similar to the majorization of entanglement [22,23] for coherent states.

Theorem 3.2.1. [24] A pure state |ψihψ| can be transformed into the pure state |φihφ| by means of incoherent operations if ∆[|φihφ|] majorizes ∆[|φihφ|], i.e., ∆[|φihφ|] B∆[|ψihψ|], where

∆[ · ]=Phi| · |ii|iihi| is the fully depolarizing map.

The majorization relations B for two matrices A and B with spectrums specA = (a1, . . . , a_d)

As such, it can be easily verified that, for a given computational basis E = {|iihi|}, the pure state |Φi = ^√1

d

Pd−1

i=0|ii is the one that can be transformed into any other pure state using an incoherent operation. Even more generally, |ΦihΦ| can be transformed into any state ˆ% ∈ C^d[25].

This fact, justifies the use of the term “maximally coherent state” for |Φi, the coherence of which can be used as the unit of coherence. Defining a quantity called the entropy of coherence asCe(|ΦihΦ|) =S(∆[|ΦihΦ|]), where S is the von Neumann entropy [13], uniquely characterizes the coherence of pure states by assigning the maximum entropy to the depolarized maximally coherent state and minimum entropy to pure incoherent states. Below we will consider two measures for characterization of coherence within mixed quantum states.

Coherence of Formation

Similar to the entanglement of formation as give in Definition2.3.2, the coherence of formation characterizes the amount of coherence required to create a given state ˆ% from ensembles of pure states, particularly from maximally coherent states.

Definition 3.2.1. [19] Given a quantum state ˆ%, the coherence of formation of ˆ% is defined as Cf( ˆ%) = min

{p_i,|ψiihψ_i|}

X

i

p_iCe(|ψ_iihψ_i|), (3.31)

in which the minimization is performed over all possible decompositions of ˆ% into ensembles {p_i, |ψ_iihψ_i|}.

Using the strong monotonicity of Cf under incoherent operations, we conclude that to create ˆ% with Cf( ˆ%) one must initially have access to at least the same amount of coherence stored in a collection of pure coherent states. The majorization criterion described earlier for coherent state transformation also implies the converse, that is, one who have access toCf amount of pure state coherence in maximally coherent states can generate ˆ% with the same amount of coherence by means of incoherent operations. Hence, Cf operationally characterizes the asymptotic rate at which maximally coherent states have to be consumed to generate a given quantum coherent state [13]. Interestingly, it can be easily shown that the coherence of formation is an additive

quantity. For ˆ% =P the von Neumann entropy as given in Lemma1.4.7.

As pointed out by Winter and Yang [13], in contrast with the entanglement of formation, coherence of formation is not difficult to calculate. This might originate from the fact that the set of incoherent states is a polytope which can be characterized using a finite number of linear inequalities (i.e., witnesses) as mentioned in Corollary 3.2.1⁴. The latter implies that the minimization in Eq. (3.31) can be easily carried out using standard convex optimization techniques.

Distillable Coherence and Relative Entropy of Coherence

The converse procedure of coherence formation is coherence distillation, that is, the process of extracting pure coherent quantum states, in particular maximally coherent states, from mixed states only using incoherent operations. Winter and Yang have shown in Ref. [13] that the asymptotic rate for the transformation of a state ˆ% to maximally coherent qubit states is by the relative entropy of coherence (REC).

Recall from Section 1.4.6 that the relative von Neumann entropy is a distance function be-tween density operators which is based on von Neumann entropy. Moreover, using Lemma1.4.10, we know that the relative von Neumann entropy is faithful and can be used to measure distance of a point from a reference set, as given by Eq. (1.130). If we choose the reference set to be the set of incoherent states S_inc, we obtain REC given by

Cr( ˆ%) = δSinc( ˆ%) = inf

ˆ σ∈Sinc

D(ˆ%||ˆσ). (3.33)

Note that Cr is a basis dependent quantity. However, we omit this dependence from the function’s argument for brevity. Interestingly, a simple closed form can be obtained for this

4See also the discussion following Theorem1.1.9.

quantity as follows. Starting with the definition of D(ˆ%||ˆσ), Eq. (1.128), we have D(ˆ%||ˆσ) = Tr ˆ%(log ˆ% − log ˆσ)

= −S(ˆ%) − Tr ˆ%log ∆[ˆσ]

= −S(ˆ%) − Tr ∆[ˆ%] log ∆[ˆσ] + S(∆[ˆ%]) − S(∆[ˆ%])

=S(∆[ˆ%]) − S(ˆ%) + D(∆[ˆ%]||ˆσ)

> S(∆[ˆ%]) −S(ˆ%).

(3.34)

Here, we have used the fact that, given ˆσ ∈ S_inc, ∆[ˆσ] = ˆσ. By minimizing the left-hand-side of the above equation over all incoherent states we arrive at

Cr( ˆ%) =S(∆[ˆ%]) − S(ˆ%). (3.35)

Baumgratz et al in Ref. [25] have shown that REC satisfies all the axioms of a proper measure for the resource theory of coherence. Note that Cr, like any other measure of coherence, is a basis dependent quantity reflecting the fact that coherence is a basis dependent notion. On top of the simplicity of its calculation, REC enjoys the additivity property, namely that, given the product state ˆ%_AB = ˆ%_A⊗ ˆσ_B it holds true that Cr( ˆ%_AB) = Cr( ˆ%_A) +Cr( ˆ%_B). This can also be verified straightforwardly as per below:

Cr( ˆ%_A⊗ ˆσ_B) = S(∆AB[ ˆ%_A⊗ ˆσ_B]) −S(ˆ%A⊗ ˆσ_B)

=S(∆A[ ˆ%A] ⊗ ∆B[ˆσB]) −S(ˆ%A⊗ ˆσB)

=S(∆A[ ˆ%_A]) +S(∆B[ˆσ_B]) −S(ˆ%A) −S(ˆσB)

=Cr( ˆ%_A) +Cr(ˆσ_B).

(3.36)

We will discuss the usefulness of this property in more details within the next section.