Coherence, Quantum Fisher Information, Superradiance and Entanglement are Interconvertible Resources

# Coherence, Quantum Fisher Information, Superradiance and Entanglement are Interconvertible Resources

## Abstract

We demonstrate that quantum Fisher information and superradiance can be formulated as coherence measures in accordance with the resource theory of coherence, thus establishing a direct link between metrological information, superradiance and coherence. We also show that the trivial coherence measure has a metrological interpretation. The arguments are then generalized to show that coherence may be considered as the underlying fundamental resource for any functional of state that is first of all faithful, and second of all concave or linear. Arguments are then presented that show quantum Fisher information and the superradiant quantity are in fact antithetical resources in the sense that if coherence were directed to saturate one quantity, then it must come at the expense of the other. Finally, a key result of the paper is to demonstrate that coherence, quantum Fisher information, superradiant quantity, and entanglement are in fact mutually interconvertible resources under incoherent operations.

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## I Introduction

The study of quantum resources has seen another revival of interest over the last several years due to the recent identification and characterization of a resource theory of coherence (1). While the coherence of quantum systems has always, in some form or another, been recognized as a fundamental aspect of the field, the newly developed resource theory now provides a framework that allows for a more quantitative understanding of the subject. Ever since this development, the resource theory of coherence has seen an ever increasing number of applications over a large number of topics. Coherence has now been studied within the contexts of quantum correlations (3); (4); (5), macroscopic quantumness (6); (7), nonclassical light (8); (9); (10), interferometric experiments (11), error correction (12), quantum estimation (13), and quantum algorithms (14); (15). There are also several different variations of the theory (16), such as a recent proposal for a resource theory of superposition (17) which generalizes the concept of coherence. An extensive overview of the subject can be found at (18).

An area that has also garnered considerable interest concerns the convertibility of coherence into nonclassical correlations such as entanglement (4); (5); (19); (20). Already, an experimental conversion of coherence to quantum correlations and vice versa has been recently reported (21). Given that such quantum correlations often find applications in a variety of scenarios, the study of such interconversion processes allow for greater flexibility when extracting practical advantages out of nonclassical quantum resources. Tantalizingly, the ideas that are being explored also suggests that a more fundamental property may underly many different quantum effects, and that all these disparate notions may be accessed by first generating a single type of resource – coherence.

In this paper, we further explore these ideas. We first demonstrate that for an important class of metrological scenarios, the quantum Fisher information (QFI) can be used to construct a bona fide measure of quantum coherence, thus establishing a direct link between metrological information and the coherence within a system. We will then go on to show that superradiant phenomena may also be attributed to quantum coherence in a similar way. The arguments are then generalized to show that coherence may be considered as the underlying fundamental resource for any functional of state that is first of all faithful, and second of all concave or linear. We also demonstrate that QFI and what we refer to as superradiant quantity are in fact antithetical resources in the sense that if coherence were optimized to saturate one quantity, then it must come at the expense of the other. Finally, a key result of the paper is to demonstrate that coherence, QFI, superradiant quantity, and entanglement are in fact mutually interconvertible resources under incoherent operations. A central theme of these results is the optimal application of non coherence increasing operations, otherwise called incoherent operations, on coherent quantum states.

## Ii Preliminaries

In this section, we review some elementary concepts regarding coherence measures. The notion of coherence that we will employ in this paper will be the one identified in (1), where a set of axioms are identified in order to specify a reasonable measure of quantum coherence. The axioms are as follows:

For a given fixed basis , the set of incoherent states is the set of quantum states with diagonal density matrices with respect to this basis. Incoherent completely positive and trace preserving maps (ICPTP) are maps that map every incoherent state to another incoherent state. Given this, we say that is a measure of quantum coherence if it satisfies following properties: (C1) for any quantum state and equality holds if and only if . (C2a) The measure is non-increasing under a ICPTP map , i.e., . (C2b) Monotonicity for average coherence under selective outcomes of ICPTP: , where and for all with and . (C3) Convexity, i.e. , for any density matrix and with .

For example, it is known that the optimal rate at which you can distil maximally coherent qubits in the infinite copy limit satisfies all the above axioms (2). One may check that a particular operation is incoherent if it always maps a diagonal density matrix to another diagonal density matrix. One important example of such an operation is the CNOT gate. It is clear that its action on classical bits is simply a classical CNOT operation, so it is an incoherent operation. In contrast, a CNOT operation does not fall under the regime of local operations and classical communication (22), which form a set of non entanglement increasing operations. For some given set of basis states , all ICPTP maps are prescribed by some set of Kraus operators (23) of the form such that , and is some function on integers.

## Iii Coherence, QFI and Locally Interacting Hamiltonians

We now consider a standard metrological scenario. One first begins with a signal Hamiltonian of interest, which is denoted . In general, can be a function of , which can be interpreted as the number of particles the Hamiltonian interacts with. This signal Hamiltonian encodes a signal on a probe state, which is a specially prepared quantum state of particles, or possibly more if one were to include ancillary quantum particles. The Hamiltonian generates the dynamics and after some time , a measurement is then performed on the state , the outcome of which is specifically designed in order to obtain the most precise estimate of the value of as possible. The optimal sensitivity is known to be given by the quantum Cramer-Rao bound (24) where is the number of measurements performed and is the QFI of a state with respect to , given by

 F(ρ(τ),HS)=2∑i,j(λi−λj)2λi+λj|⟨i|HS|j⟩|2,

where and are eigenvalues and eigenstates of , respectively. We will be primarily interested in the sensitivity of the state locally at the point , so will be the figure of merit we will consider.

Here, we are interested in studying the relationship between the information obtained in the above described metrological scenario and notions of nonclassicality. A class of Hamiltonians that is of particular interest is the class of locally interacting Hamiltonians or simply local Hamiltonians that are a sum of independent Hamiltonians acting on individual particles, i.e. a Hamiltonians of the form where represents a nontrivial interaction acting only on the th particle that is not proportional to the identity. We will also assume that does not depend on the number , which corresponds to the number of particles the signal Hamiltonian is interacting with. An example of a Hamiltonian of this type is a uniform magnetic field in the direction acting on a collection of spins, where in this case , and are the Pauli operators acting on th particle. We will demonstrate that the metrological information for a Hamiltonian of this type is in fact, contained entirely within the nonclassical notion of coherence identified in (1), and that this relationship can be made quantitative. As coherence is a basis dependent concept, we will adopt the basis which is naturally defined by the eigenvectors of . This defines for us a set of local bases for the th particle where , and is the dimension of the particle. Consequently, we will consider the coherence with respect to this set of local bases. Local bases were also previously studied in (3), which noted their connection with quantum correlations such as discord and entanglement.

We will now demonstrate that a coherence monotone can be constructed using QFI. Let us begin with some intuition by considering an qubit system. Suppose we have initially a maximally coherent qubit , where one may interpret as the ground and excited states of the qubits respectively. Consider the local Hamiltonian of the form . We see that one way to reach Heisenberg limited sensitivity via incoherent operations is to begin with a maximally coherent qubit with copies of the “free” classical state , and then perform CNOT operations. The end result is a GHZ state which achieves Heisenberg limited sensitivity for . The CNOT operation, being an important example of an incoherent operation, is clearly not useful if one were to begin with an incoherent qubit. This motivates us to consider the best possible QFI that is achievable over all incoherent operations. Such an optimazation was also previously considered in (4), within the context of entanglement.

As such, for any signal Hamiltonian of the form , and a pure state probe , let us consider the the maximal QFI reachable via all possible incoherent operations on , i.e. . The incoherent operation here is completely general, with no constraints otherwise. In the previous example for instance, a single particle state is mapped to a particle GHZ state via an incoherent operation. Note the dependence on in . In fact, for any incoherent pure state, we can always achieve Heisenberg scaling via a suitable incoherent operation, as demonstrated by the following Lemma:

###### Lemma 1.

For every coherent pure state and locally interacting Hamiltonian , there always exists an incoherent operation that achieves that scales with . The measurement that achieves this Heisenberg limited scaling can also be performed incoherently.

###### Proof.

Let us first consider . For each , let and be eigenvectors that corresponds to eigenvectors for the maximum and minimum eigenvalues and , respectively. In this 2 dimensional subspace, let us define the Pauli operator .

Let , where are eigenstates of which construct the incoherent basis. Without any loss in generality, we assume that the coefficients are positive real and . We will also assume that and since this is just a relabelling of the basis which can be done using an incoherent unitary. The ‘extra’ particles may be considered ancillary particles that assist during the metrological process.

We now apply an incoherent CNOT type operation that performs the map and then let the state evolve according to the Hamiltonian . Let us now consider only the the first 2 terms of , which under evolves as

 √λ1|1…1⟩+√λ2|2…2⟩→√λ1|1…1⟩+√λ2eiϕτ|2…2⟩

up to an overall phase factor. We have where .

We will choose some basis on the Hilbert space space of particles for such that . Define the following POVM:

 Missing or unrecognized delimiter for \left

where , , and . The quantum operation is then defined as . This operation is incoherent and is effectively an incoherent implementation of 2 measurements: a projection onto the basis followed by a parity measurement on the axis. Suppose we perform the naive protocol where if the measurement outcome is , we keep the parity measurement outcome, and assign a value of zero otherwise. Let us call this measurement .

We can then verify using the error propagating formula that

 ΔM′2∣∣∂τ⟨M′⟩∣∣2=(√λ1+√λ2)221ϕ2

Finally, we observe that and . depends only on , all of which do not contain any dependence on , and neither does the coefficient , which depends only on the initial state. As such, we have

 ΔM′2∣∣∂τ⟨M′⟩∣∣2≤(√λ1+√λ2)221N2ϕmin∼O(1N2)

This proves that for every pure coherent state, Heisenberg limited scaling is reachable using only incoherent operations. ∎

Here, we note that in the above Lemma, there is an important differentiation between , which captures the number of particles is interacting with, and the actual physical number of particles, which can be any arbitrary number so long as it is reachable via an incoherent operation.

###### Definition 1 (Coherence of QFI).

For a general mixed state of , we define via the convex roof construction the Coherence of QFI:

 Missing or unrecognized delimiter for \right

where the minimization is over all pure state decompositions of the form , is a local Hamiltonian of the form where is nontrivial.

We now show that for any given and that is local, always monotonically decreases under incoherent operations and that it is in fact, a valid coherence measure.

###### Theorem 1.

is a coherence measure.

###### Proof.

First, we need to prove that the measure is faithful. We observe that if is incoherent, then it is diagonal w.r.t. , so the QFI is . Resorting to any incoherent operation will not improve the situation as it always maps a diagonal state to another diagonal state so we have . Lemma 1 then demonstrates that if is coherent, then since has to have at least one pure state in its pure state decomposition that is coherent. This proves that is incoherent iff , so the measure is faithful.

The convexity of the measure is guaranteed by the convex roof construction, so it remains to demonstrate the monotonicity of the measure under incoherent operations. Let be the optimal decomposition such that

 CF(ρ)=∑jqjmaxΦ∈ICPTPF(Φ(∣∣ψj⟩⟨ψj∣∣),HS).

We only need to prove the strong monotonicity condition for every ICPTP map represented by Kraus operators . is immediately convex due to the convex roof construction, and convexity together with strong monotonicity immediately implies weak monotonicity. We then have the following chain of inequalities:

 ∑ipiCF(ρi) =∑ipiCF(Ki∑jqj∣∣ψj⟩⟨ψj∣∣K†i/pi) (1) ≤∑i,jrijqjCF(Ki∣∣ψj⟩⟨ψj∣∣K†i/rij) (2) =∑i,jrijqjmaxΦ∈ICPTPF(Φ[Ki∣∣ψj⟩⟨ψj∣∣K†i/rij],HS) (3) =∑i,jrijqjmaxΦij∈ICPTPF(Φij[Ki∣∣ψj⟩⟨ψj∣∣K†i/rij]⊗|i,j⟩⟨i,j|,HS⊗|i,j⟩⟨i,j|) (4) Missing or unrecognized delimiter for \right (5) Missing or unrecognized delimiter for \right (6) =maxΦ∈ICPTPF(∑jqjΦ[∣∣ψj⟩⟨ψj∣∣⊗|j⟩⟨j|],HS⊗∑j|j⟩⟨j|) (7) Missing or unrecognized delimiter for \right (8) =∑jqjmaxΦ∈ICPTPF(Φ[∣∣ψj⟩⟨ψj∣∣],HS) (9) =CF(ρ) (10)

The inequality in the second line is due to the convexity of . In line 5, is the optimal incoherent operation on the state where , which leads to the inequality in line 6 since the overall operation is a special case of all possible incoherent operations on the state . In line 8, we used the convexity of QFI, and line 9 follows because is overall just an incoherent operation on since is an incoherent state. Line 9 is then just the definition of , which completes the proof. ∎

The above demonstrates that for any and , is in fact a valid coherence measure. We observe that the above theorem generalizes to arbitrary signal Hamiltonians, as we can simply set so where in principle can be any arbitrary nontrivial Hamiltonian. Nonetheless, the multipartite case of local Hamiltonians is still interesting due to its connections with multipartite quantum correlations.

Another immediate observation is that for any local Hamiltonian , any metrological information stored in the state must necessarily be extracted from the coherence present within the quantum state. However, even though the above definition is intuitive, and is a valid coherence measure in the strict sense, the measure may saturate really quickly even for very small amounts of coherence, which limits its practical utility as a measure of quantum coherence for general states. We know this because a single maximally coherent qubit can already be converted to a GHZ state via a series of CNOT operations, which is enough to saturate the QFI for the signal Hamiltonian . In order to circumvent this limitation, one may instead consider the following generalization:

###### Definition 2 (Distributed coherence of QFI).

The distributed QFI for a pure state is defined to be

 CMF(|ψ⟩)\coloneqqmaxΦ∈ICPTPM∑i=1F{TrP(1),…,P(i−1),P(i+1),…,P(M)[Φ(|ψ⟩⟨ψ|)],H(i)S}

where is the th local Hamiltonian of the form and are nontrivial. refers to the th partition of particles in the state which is partitioned into collections of particles that separately interacts with the Hamiltonians . The partial trace above is to be interpreted as tracing out every particle except the ones in .

The generalization to mixed states is obtained via the convex roof construction

 CMF(ρ)\coloneqqmin{pi,|ψi⟩}∑ipiCMF(|ψ⟩)

where the minimization is over all pure state decompositions of the form .

The above definition corresponds to a scenario where a quantum state is prepared via an incoherent operation, partitioned into separate subsystems, and then distributed to different parties which locally performs a metrological experiment with the signal Hamiltonian . Equivalently, it can also be interpreted as a single party scenario where a is prepared, then partitioned into separate subsystems where separate metrological experiments are performed.

A similar argument shows that is also a valid coherence measure for every and , where in the case , we retrieve .

###### Theorem 2.

is a coherence measure.

###### Proof.

The fact that iff is incoherent (See proof of Theorem 1) also automatically implies iff is incoherent. Convexity is implied by the convex roof construction. Therefore, we only need to prove strong monotonicity.

The proof of monotonicity also follows from similar arguments as the case . We replicate the arguments here.

We only need to prove that for every ICPTP map represented by Kraus operators . Suppose is the optimal decomposition that achieves . We have the following chain of inequalities:

 ∑ipiCMF(ρi) =∑ipiCMF(Ki∑jqj∣∣ψj⟩⟨ψj∣∣K†i/pi) (11) ≤∑i,jrijqjCMF(Ki∣∣ψj⟩⟨ψj∣∣K†i/rij) (12) =∑i,jrijqjmaxΦij∈ICPTPM∑k=1F{TrP(1),…,P(k−1),P(k+1),…,P(M)[Φij(Ki∣∣ψj⟩⟨ψj∣∣K†i/rij)],H(k)S} (13) =maxΦij∈ICPTP∑i,jrijqjM∑k=1F{TrP(1),…,P(k−1),P(k+1),…,P(M) (14) [Φij(Ki∣∣ψj⟩⟨ψj∣∣K†i/rij)⊗|i,j⟩⟨i,j|],H(k)S⊗|i,j⟩⟨i,j|} (15) =maxΦij∈ICPTPM∑k=1F{TrP(1),…,P(k−1),P(k+1),…,P(M) (16) Missing or unrecognized delimiter for \left (17) ≤maxΦj∈ICPTPM∑k=1F{TrP(1),…,P(k−1),P(k+1),…,P(M) (18) Missing or unrecognized delimiter for \left (19) ≤∑jqjmaxΦj∈ICPTPM∑k=1F{TrP(1),…,P(k−1),P(k+1),…,P(M) (20) [Φj(∣∣ψj⟩⟨ψj∣∣)],H(k)S} (21) =CMF(ρ) (22)

The first inequality is due to the convexity of . The second inequality is due to the fact that the state can be obtained via an incoherent operation on . The last inequality follows from the convexity of QFI, which completes the proof.

We note also the following consequence of Lemma 1, which we can also use to define a genuine coherence measure which satisfies all the properties listed in (1), albeit a really simple one. To this end, we define the following quantity

###### Definition 3 (Asymptotic coherence of QFI).

The Asymptotic coherence of QFI is defined to be the quantity:

 CAF(ρ)\coloneqqmax(limN→∞logCF(ρ)logN−1,0).

where is the number of terms in the signal Hamiltonian and is nontrivial.

From this definition, we can show the following:

###### Theorem 3.

is the trivial coherence measure where iff is coherent, and otherwise.

###### Proof.

First, we show that that is incoherent iff . This is immediately true since no incoherent state can reach the Heisenberg limit even with the help of incoherent operations, so if is incoherent, . On the other hand, from Lemma 1, we see that every pure state can reach the Heisenberg limit which implies that if is coherent, since the pure state decomposition of must contain at least one coherent pure state. This implies that in the limit , , so . This proves that is incoherent iff . Here, we recall that corresponds to the number of particles that interact with the signal Hamiltonian , and not the actual number of physical particles, which can be any arbitrary number so long as it is reachable via some incoherent operation.

We also immediately see that this is just the trivial coherence measure that assigns a value of 1 if a state is coherent, and assigns the value 0 otherwise. It is then easy to verify that the trivial coherence measure satisfies convexity and strong monotonicity, and so is indeed a valid coherence measure in the strict sense. This completes the proof. ∎

The above theorem therefore provides one physical interpretation for the trivial coherence measure via . We see that by considering the limiting case of , the trivial measure corresponds to the fact that every coherent pure state can always achieve Heisenberg limited scaling by applying an appropriate incoherent operation, while for incoherent states the Heisenberg limit is always inaccessible even with the help of arbitrary incoherent operations.

## Iv Supperradiance as coherence monotone

In this section, we demonstrate that the effect chiefly responsible for the supperradiant phenomena can also be attributed to coherence.

We first make some necessary definitions. Consider a system consisting of subsystems with an excited and a ground state denoted and respectively. We note that this does not necessarily imply that state is composed of 2 level systems, just that only the optical transition between 2 states out of a possible levels for each of the particles are addressed. This optical transition corresponds to some optical wavelength . We can define the raising and lowering operators acting on each individual system as and respectively.

In standard florescence, it is assumed that each of these 2 level systems interacts independently with a radiation field of wavelenght , in which case the the total rate of photon emission is simply the sum of the emission rate for a single system, which is given by . This implies the emission rate at most scales with , which is intuitively simply the maximum possible number of excited states at any moment.

In the superradiant regime, is it assumed that the linear dimension of the systems is small with respect to , so there is a collective, coherent interaction with the radiation field rather than independent interactions. In this case, the systems collectively behave like a point like dipole, in which case the emission rate is described by the expectation of the collective operators (25)

 WN∝⟨∑iD(i)+∑jD(j)−⟩=∑i⟨D(i)+D(i)−⟩+∑i≠j⟨D(i)+D(j)−⟩.

We see that the second to last term is sum of single system emissions, and so is responsible for emissions that correspond to standard florescence. The last term is due to the collective behaviour of the the subsystems and the source of supperradiant phenomena, which can potentially scale with . We will refer to this as the superradiant quantity.

In a similar vein as the case for QFI, this motivates us to consider the following quantity:

 CS(|ψ⟩)\coloneqqmaxΦ∈ICPTPN∑i≠jTr[(Φ(|ψ⟩⟨ψ|)D(i)+D(j)−],

which is essentially the maximal superradiant quantity that is achievable via an incoherent operation for the pure state . Here, the incoherent basis of coherence is given by the product of excited/ground states for each particles. In this case, the quantity refers to the maximum number of excited states that are being addressed, just as for the case of QFI, where it is to be interpreted as the number of particles interacting nontrivially with the signal Hamiltonian. We generalize this to general mixed states by applying the convex roof construction as before:

###### Definition 4 (Coherence of superradiance).

The coherence of superradiance is defined as

 CS(ρ)\coloneqqmin{pi,|ψi⟩}∑ipiCS(|ψi⟩)

Interestingly, it turns out that this, too, is a valid coherence measure. Before we prove this proper, we first present the following useful proposition which will be used frequently:

###### Proposition 1.

Suppose is a valid coherence measure over pure states. If we consider the convex roof construction of

 Cconv.(ρ)=min{pi,|ψi⟩}∑ipiC(|ψi⟩),

then is convex, and the strong monotonicity for pure states with implies the strong monotonicity of : with .

###### Proof.

Convexity: Suppose and the optimal decomposition of is that . Note that is one of the possible decompositions of , thus .

Strong monotonicity: Suppose the optimal decomposition of is . Then , where . Note that for all . Thus,

 Cconv.(ρ) =∑μp∗μC(∣∣ψ∗μ⟩) ≥∑μ,jp∗μqμjC⎛⎜ ⎜⎝Kj∣∣ψ∗μ⟩√qμj⎞⎟ ⎟⎠ ≥∑jqjCconv.(ρj),

where the last inequality comes from that is one of the possible decompositions of and . ∎

From the above definition, we can then prove the following theorem:

###### Theorem 4.

The superradiant coherence is a valid coherence measure for every .

###### Proof.

We first note that convexity is guaranteed by the convex roof construction, so we just need to prove the faithfulness property and the strong monotonicity property for to be a valid coherence measure.

To prove faithfulness, we sufficient to demonstrate that it is valid for pure states. It is easy to verify that if some state is incoherent, then for every . Resorting to an incoherent operation will not help, since that will only lead to a mixture of incoherent pure states, and since the superradiant quantity is a linear functional, this implies that when is incoherent.

In order to prove that when is coherent, again, we only need to prove that it is true for pure states due to the convex roof construction. Suppose is some coherent state. Then we are guaranteed that . Without any loss in generality, we will assume that and that and corresponds to the ground and excited states respectively. We note here that all the coefficients may be made positive via an incoherent unitary operation. We then perform the following incoherent transformation of state:

 U|ψ⟩|0⟩=a0|0⟩|1⟩+a1|1⟩0+…

where is an incoherent unitary operation. We can then directly verify that so there always exists one incoherent operation that achieves non-zero superradiant quantity for any coherent state . This implies that when is coherent, which demonstrates that iff is incoherent, and proves the faithfulness property.

We now prove strong monotonicity. Here, we also only need to prove it for pure states and the convex roof construction implies it is also true in general. Let the incoherent operation be the operation with corresponding Kraus operators such that . We note that the map is also a valid incoherent operation as long as is also incoherent for every . Let us assume that is the optimal incoherent operation that achieves the maximal superradiant quantity for . We then have the following chain of inequalities:

 CS(|ψ⟩) =maxΦ∈ICPTPN∑i≠jTr[(Φ(|ψ⟩⟨ψ|)D(i)+D(j)−] ≥N∑i≠jTr[(Ω(|ψ⟩⟨ψ|)D(i)+D(j)−] =N∑i≠j∑k∣∣⟨ψ|K†kKk|ψ⟩∣∣Tr[(Ωk(|ψk⟩⟨ψk|)D(i)+D(j)−] =∑k∣∣⟨ψ|K†kKk|ψ⟩∣∣CS(|ψk⟩)

where the last line is simply the strong monotonicity condition expressed for a pure state. This completes the proof. ∎

The above theorem therefore demonstrates that superradiant phenomena is also closely related to coherence in the sense that the quantum advantage is stored within the coherence of the initial state. In fact, the case of superradiance is suggestive of a much larger class of operational or physical quantities where any quantum advantage may be directly associated with coherence.

###### Theorem 5.

Let be some functional that maps a quantum state to the nonnegative portion of the real line. Then if or, more generally , is zero iff is a incoherent pure state, and is concave or linear, then the convex roof construction

 Cf(ρ)\coloneqqmin{pi,|ψi⟩}maxΦi∈ICPTP∑ipif[Φi(|ψi⟩)]

is a valid coherence measure. The minimization is over all pure state decompositions of the state .

###### Proof.

In order to prove that it is a valid measure, we only need to demonstrate that the above quantity satisfies the strong monotonicity condition for pure states. The faithfulness condition is assumed and convexity comes about naturally due the convex roof construction.

The proof of strong monotonicity is only lightly modified for the proof of Theorem 4. Let the incoherent operation be the operation with corresponding Kraus operators such that . We note that the map is also a valid incoherent operation as long as is also incoherent for every . Let us assume that is the optimal incoherent operation that achieves the maximal value of for . We then have the following chain of inequalities:

 Cf(|ψ⟩) Missing or unrecognized delimiter for \left ≥f(Ω(|ψ⟩⟨ψ|) Missing or unrecognized delimiter for \left Missing or unrecognized delimiter for \left =∑k∣∣⟨ψ|K†kKk|ψ⟩∣∣Cf(|ψk⟩).

The first inequality is due to the maximization over all incoherent operations, of which is simply one possible candidate, and the second inequality is the to the concavity or linearity of . The last line is simply the expression of strong monotonicity for a pure state. The generalization of strong monotonicity then a natural consequence of the convex roof construction. ∎

We therefore see that Theorem 4 is just a special case of Theorem 5 together with the additional arguments demonstrating that the superradiant quantity is faithful when maximized over incoherent operations and pure states. Note that the present proof of Theorem 1, while similar in construction, is not implied by the above theorem as QFI is convex w.r.t. quantum states in general, so the proof of Theorem 1 requires additional properties.

Before proceeding, we provide numerical examples comparing our coherence measures and with the relative entropy of coherence (1). is defined by where is the von Neumann entropy of a density matrix and is the diagonal part of the density matrix . While and requires an optimization over all incoherent operations, and the optimal solution for a general quantum state is unknown, any specific incoherent procedure will provide a lower bound for the measure so we may still proceed by ansatz.

To demonstrate a qualitative behavior of our measures, we consider the set of -qubit Dicke state with excitations. The state is given by

 |m,k⟩=(mk)−12∑P|P(0…0m-k1…1k)⟩

where refers to a particular permutation of the state and the summation is over all possible permutations. We note that is composed of equally weighted terms, which implies . Also, by verifying the majorization condition(29) for two uniformly distributed pure states, we can check that there exist an incoherent operation which maps where is the largest integer satisfying . For the QFI based measure, we will apply this particular incoherent operation to the state , thus obtaining a lower bound to . For the superradiance based measure, we will simply apply the identity operation. To compare the qualitative behavior for increasing , we normalize the coherence measures so that they coincide at . The graph of these three coherence measures as functions of is given by FIG. 1. As the graph shows, the qualitative behavior of the lower bounds are similar to the relative entropy of coherence. 1

## V QFI and superradiance as opposing resources

Here, we provide evidence that optimizing QFI and the superradiant quantity are in fact antithetical problems through the lens of coherence. Previously, it was discussed that the optimal QFI and superradiant quantity that is achievable with respect to incoherent operations are independently valid measures of coherence. Interestingly, the total sum of distributed QFI and the distributed form of the supperradiant quantity is also itself a coherence measure.

We first define the coherence of Superradiant-QFI:

###### Definition 5 (Coherence of Superradiant-QFI).

For pure states, the Coherence of Superradiant-QFI is defined as

 CSF(|ψ⟩)\coloneqqmaxΦ∈ICPTPM∑i=1[14F{TrP(1),…,P(i−1),P(i+1),…,P(M)[Φ(|ψ⟩⟨ψ|)],H(i)S} (23) Missing or unrecognized delimiter for \left (24)

where is the th local Hamiltonian of the form