Optimizing nontrivial quantum observables using coherence

In this paper we consider the quantum resources required to maximize the mean values of any nontrivial quantum observable. We show that the task of maximizing the mean value of an observable is equivalent to maximizing some form of coherence, up to the application of an incoherent operation. For any nontrivial observable, there always exists a set of preferred basis states where the superposition between such states is always useful for optimizing the mean value of a quantum observable. The usefulness of such states is expressed in terms of an infinitely large family of valid coherence measures which is then shown to be efficiently computable via a semidefinite program. We also show that these coherence measures respect a hierarchy that gives the robustness of coherence and the l1 norm of coherence additional operational significance in terms of such optimization tasks.


Introduction
Quantum coherence has long been recognized as a fundamental aspect of quantum mechanics. In comparison, the identification of quantum coherence as a useful and quantifiable resource is a much more recent development. Progress in this area has been greatly accelerated due to the resource theoretical framework for quantum coherence [1][2][3]. Inspired by the resource theory of entanglement [4,5], the notion of what quantum coherence is and how it can be quantified is now axiomatically defined, thus allowing quantum coherence phenomena to be discussed much more unambiguously. Since this development, many coherence measures have been proposed. Some known measures include geometric measures [2], the robustness of coherence [6][7][8], and entanglement based measures [9]. Coherence measures have now been studied in relation to a diverse range of quantum effects such as quantum interference [10], exponential speed-up in quantum algorithms [11,12] and quantum metrology [13,14], nonclassical light [15][16][17], quantum macroscopicity [18,19] and quantum correlations [20][21][22][23][24][25]. An overview of coherence measures and their structure may be found in [26,27]. Also related is the study of coherence witnesses, which concerns the detection, but not necessarily the quantification, of quantum coherences via an observable [6-8, 10, 28].
In this paper, we discuss how to construct a coherence measure from a quantum observable M. The structure of this paper follows: in section 2 we briefly review some essential concepts such as the Kraus and Choi-Jamiolkowski representations of quantum channels, the resource theory of coherence, and semidefinite programming. In section 3, we show that optimizing the mean value of the observable á ñ M for an input state is the same as maximizing the coherence of the input state pertaining to a specific class of bases, up to the application of some incoherent operation. Given any nontrivial observable M, it is therefore always possible to construct a coherence measure for some specific set of bases. We also show that the converse is possible, by identifying observables M and constructing a coherence measure for any given basis. In section 4, we prove that this measure is computable via a semidefinite program. In section 5, we demonstrate that the robustness of coherence and the l 1 norm of coherence establishes the quantum limits of such tasks. The relationship between our proposed measures and coherence witnesses is also discussed. In section 6, we present examples that illustrate the key ideas of our approach and provide several examples of previously known measures that turn out to be special cases of our proposed measures. Finally, in section 7 we summarize and discuss the implications of our results.

Preliminaries
We review some elementary concepts concerning coherence measures, quantum channels, and semidefinite programs.
We first briefly describe the formalism of quantum channels, which we take here to mean the set of all Completely Positive, Trace Preserving (CPTP) maps. There are several equivalent characterizations of quantum maps, but for our purposes, we will be concerned with the Kraus [29] and the Choi-Jamiolkowski representations [30,31]. In the Kraus representation, a quantum operation is represented by a map of the form in order to qualify as a valid quantum operation. In the Choi-Jamiolkowski representation, a quantum map Φ is represented by an operator , the corresponding Choi- . The notion of coherence that we will employ in this paper will be the one identified in [1,2], 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 ñ {| } i , 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 quantum maps that map every incoherent state to another incoherent state. Consider some set of ICPTP maps . Given this, we say that  is a measure of quantum coherence if it satisfies following properties: (C1) (Faithfulness)   r ( ) 0 for any quantum state ρ and equality holds if and only if , for any density matrix ρ and σ with   l 0 1. One may check that a particular operation is incoherent if its Kraus operators always maps a diagonal density matrix to another diagonal density matrix. One important example of such an operation is the CNOT gate. We can also additionally distinguish between the maximal set of ICPTP maps, which we refer to as maximally incoherent operations (MIO) [1] from the set of ICPTP maps whose Kraus operators additionally satisfy , which we refer to as simply incoherent operations (IO) [2]. From this definition, it is clear that Ì IO MIO. We highlight that both MIO and IO are commonly used abbreviations, and that other possible sets of ICPTP maps are also actively being considered (see [26] for examples). In this article, we will typically consider either MIO and IO for the set .
Finally, we review some basic notions regarding semidefinite programs. A semidefinite program is a linear optimization problem over the set of positive matrices X, subject to a set of constraints that can be expressed in the following form: where A and B i are Hermitian matrices and f i is a linear, Hermiticity preserving map (i.e. it maps every Hermitian matrix to another Hermitian matrix) representing the ith constraint. The above is called the primal problem. The optimal solution to the primal problem is always upper bounded by the optimal solution to the dual problem, when they exist. The dual problem may be written as the following optimization problem over all possible Hermitian matrices Y i : In this case, * f i refers to the conjugate map that satisfies for every matrix C and D.
The solutions to the primal and dual problems are usually equal except in the most extreme cases. Nonetheless, this needs to be verified on a case by case basis. A sufficient condition for both primal and dual solution to be equal is called Slater's Theorem, which states that if the set of positive matrices X that satisfies all the constraints f i is nonempty, and if the set of Hermitian matrices Y i that satisfies the strict inequality is also nonempty, then the optimal solutions for both problems, also referred to as the optimal primal value and the optimal dual value, must be equal.

Coherence measures from maximally incoherent operations
In this section, we will discuss how a quantum observable M may be used to construct a coherence measure that satisfies axioms (C1)-(C3) (see section 2). The following theorem introduces a quantity that satisfies the strongly monotonic condition (C2b), which will prove useful when we eventually construct the coherence measure. Proof. We first observe that any incoherent operation represented by some set of incoherent Kraus operators { } K i IO is, by definition, also a maximally incoherent operation. Note that for any set of maximally incoherent IO is also maximally incoherent since it is just a concatenation of the incoherent operation represented by { } K i IO , followed by performing a maximally incoherent operation W i MIO conditioned on the measurement outcome i. Let us assume that We note that the last line is simply the expression for strong monotonicity, which proves the result for the case when  is MIO. Identical arguments apply when considering IO, which completes the proof. , In the above proof, we see that the optimization over MIO yields a valid coherence monotone in within the regime of IO, so drawing a sharp distinction between the two sets of operations is not always necessary.
We note that satisfying strong monotonicity qualifies the quantity as a coherence monotone, but is insufficient to fully qualify it as a coherence measure. In order for that to happen, we need to demonstrate that Tr 0 whenever ρ is a coherent state. It is clear that this is only true for some special cases of M. However, the following theorem shows that even if M does not by itself satisfy the above conditions, it is still possible to construct a valid coherence measure using M. Proof. We begin by observing that the matrix Tr is trace zero. Since M′ is nontrivial (not proportional to the identity operator), it implies that the sum of its positive eigenvalues and negative eigenvalues must be exactly equal. Let l be the vector of eigenvalues of M′ arranged in decreasing order. We recall the Schur-Horn theorem, which states that for every vector , there exists a Hermitian matrix with the same vector of eigenvalues l  , but with diagonal entries 0, , 0 always satisfies this condition. Therefore, there always exist a basis ñ {| } i for M′ where the main diagonals are all zero, such that á ¢ ñ = | | i M i 0 for every ñ |i , which proves the first part of the theorem. See proposition 1for an example of such a basis using mutually unbiased bases. Proposition 1 presents an alternative proof for theexistence of such bases but it is important to note that not every basis that satisfies the condition á ¢ ñ = | | i M i 0 for every ñ |i is necessarily mutually unbiased. Now, we proceed to prove that   r ( ) M is a coherence measure of with respect to the basis ñ {| } i . The strong monotonicity condition is already satisfied due to theorem 1. The convexity of the measure is immediate from the linearity of the trace operation and the definition of   M as a maximization over MIO or IO. Therefore, we only need to establish the faithfulness property of the measure.
In order to prove this, recall that in the basis ñ {| } i , the diagonal elements of M′ is all zero. Therefore, there always exists some projection onto a 2 dimensional space M′ such that the corresponding submatrix has the form * . We can assume without loss of generality that the projection is onto the subspace ñ ñ {| | } 0 , 1 , since at this point, the numerical labelling of the basis is arbitary.
For some coherent quantum state ρ, there is at least one nonzero off-diagonal element. Since basis permutation is an incoherent operation, we can assume the nonzero off-diagonal element is r 01 . In fact, we can assume that it is the only nonzero off-diagonal element as we can freely project onto the subspace spanned by T r can only be positive when ρ is coherent (the basis is specified by the theorem). This establishes that every nontrivial observable M is, in fact, a witness of some form of coherence. One just needs to subtract the constant ( ) M d Tr from the mean value á ñ M to verify the presence of coherence. Second, it establishes that if M is a coherence witness, then it can be interpreted as the lower bound of the bona fide coherence measure   M . Recall that the measure   M quantifies the operational usefulness of a quantum state when one considers MIO or IO type quantum operations and the task is to maximize the mean value of a given observable M. Other examples of coherence measures with operational interpretations in terms of MIO or IO include the relative entropy of coherence, which quantifies the number of maximally coherent qubits you can distill using IO [32], as well as quantities considering how much entanglement and Fisher information can be extracted via MIO or IO [9,14].
Third, theorem 2 defines the preferred incoherent bases where the coherence is useful for optimizing á ñ M and shows that such bases always exist. The following proposition states that the coherence with respect to any basis that is mutually unbiased with respect to the eigenbasis of the observable M will always satisfy the necessary condition in theorem 2.
Tr for every = ¼ i d 1, , , which is the required condition. Note that this can be considered an alternative proof of the first statement in theorem 2. , In theorem 2, we established the existence of a coherence measure   M but the proof is not constructive in the sense that given the observable M, it does not immediately inform us of a procedure to obtain the basis ñ {| } i and corresponding measure   M . Proposition 1 closes this gap. Given any observable M, one may obtain the eigenbasis, find another basis that is mutually unbiased with respect to this eigenbasis, and construct   M . An overview of how to construct mutually unbiased bases can be found in [33]. Note that mutually unbiases bases are not the only kinds of bases satisfying theorem 2.
We now consider the reverse construction. Suppose instead of starting from a given observable M and inferring the basis for the coherence measure, we wish to begin with some basis ñ {| } i and construct an observable M with corresponding measure   M that quantifies the coherence in the basis ñ {| } i . The method to do this also follows from theorem 2, as we can choose any Hermitian matrix to be M so long as the leading diagonals are zero. This is guaranteed to lead to a reasonable measure according to theorem 2. Such a matrix is easy to construct, as any arbitrary Hermitian matrix written in the basis ñ {| } i with its leading diagonal elements replaced with zero will suffice. This is summarized in the form of the following corollary.

A semidefinite program for computing coherence measures
Previously, we have considered both MIO and IO during the construction of our coherence measures. Here, we show that for MIOs, the corresponding coherence measure  M MIO is efficiently computable via a semidefinite program. Many other quantities related to coherence may be phrased as a semidefinite program. For examples, see [6,7,[34][35][36][37].
Let us first define the matrix C . Furthermore, we will assume that . We now prove the following: ñá Ä ñá Ä ñá ñá Tr   subject to  Tr  1 1  Tr  1 1   Tr  2 2 1, , , . Note that all the matrices here are assumed to be written in a basis of the type specified in theorem 2.
Proof. We begin by first noting that the matrix X can be written as the matrix * * The * indicates possible nonzero elements, but they do not appear in the objective function we are trying to optimize, nor do they appear within the linear constraints, so they can be arbitrary so long as X0. The matrix A written in matrix form looks like Tr Tr .
, so X 1 actually represents a valid quantum operation in the Choi-Jamiolkowski representation. This implies ( ) AX Tr has the form r F [ ( ) ] M Tr A A for some valid quantum operation Φ.
All that remains is for us to prove that under the set of constraints for all i=1, K, d and j=1, K, d, Φ must be a maximally incoherent operation. We first note that the number is just the main diagonal elements of the matrix X 2 , so it must be nonnegative since X is positive and X 2 is a principle submatrix of X. We can therefore rewrite the constraint as where l i j , is nonnegative. This necessarily means that every incoherent state ñá under the quantum map represented by X 1 , which defines maximally incoherent operations, and completes the proof. , Given the primal problem in theorem 3, we can also write down the dual problem, which is detailed in the following corollary: Corollary 3.1. The dual to the primal problem in theorem 3 is the following optimization over all possible Hermitian Y A and Y B : Furthermore, the optimal primal value is equal to the optimal dual value.
Proof. The first constraint in the primal problem can be written as The conjugate map can be verified to be the map . The rest of the constraints can be written as In this case the conjugate map is Summing over the variable i, we have The dual program can therefore be written as: The third line of the constraint is actually just , which is equivalent to the contraint that the main diagonal of Y i A is all negative. As such, the program can be further simplified to the following: which is the form that was presented in the corollary. Finally, we just need to check that the primal and dual programs satisfies Slater's conditions. For the primal problem, the optimization is over all MIO's, so the primal feasible set is nonempty (for instance, we can just consider the Choi-Jamiolkowski representation of the identity operation, which also falls under MIO). Furthermore, there exists at least one set of represents the largest eigenvalue of A. As such, Slater's conditions are satisfied and the primal optimal value is equal to the dual optimal value. ,

Relation to robustness and l 1 norm of coherence
It was observed in [6,7] that the robustness of coherence   , which may be interpreted as the minimal amount of quantum noise that can be added to a system before it becomes incoherent, is a coherence measure that is also simultaneously the mean value of an observable. That is, for any state ρ, there always exists some optimal witness W ρ such that . It was also demonstrated that the l 1 norm upper bounds the robustness, so . The coherence weight is another similar example where the measure is given by the mean value of an observable [37]. Note that for both the robustness of coherence and the coherence weight, the optimal observables depend on the state. The following theorem shows that both the robustness and the l 1 norms of coherence are fundamental upper bounds of   M . In [28], it was also observed that when M is a witness that achieves its maximum value for the maximally coherent state, then  M IO is upper bounded by the l 1 norm of coherence under certain normalization conditions. for  k 0 where  is MIO or IO. As such, without any loss in generality, we can assume that M is a traceless matrix where the leading matrix elements are zero, and that its smallest eigenvalue is normalized such that l = -

Examples
In this section, we discuss examples that illustrate our key results. We first consider the simplest system consisting of a single spin. Let ñ ñ {| | } , be the basis vectors corresponding to a spin pointing in the +z and the −z directions respectively. Suppose our measurement observable is a Pauli measurement along the x axis. In this case, our observable M is the Pauli matrix s = ( ) 0 1 1 0 x . Notice that σ x always is always zero along its leading diagonals, as was the case considered in theorem 2. Suppose we perform many Pauli X measurements and the outcome of the measurement is always +1, then we can be certain that the state must be ñ + ñ (| | ) 0 1 1 2 which corresponds to a maximally coherent qubit. The larger the mean values of your measurement, the more confident you are that the state is close to the maximally coherent state, which in turn suggests that the state contains more coherence. Theorems 1 and 2 can be interpreted as a reflection of this confidence, generalized to arbitrary finite dimensional systems. The maximization over MIO and IO, which are both sets of operations that do not increase coherence, are further required in order to make this relationship more quantitative, such that it satisfies the axioms (C1)-(C3) (See section 2).
Furthermore, we also see that the chosen basis ñ ñ {| | } , is mutually unbiased with respect to the eigenbasis of σ x , which again is the case being considered in proposition 1. Observe that the designation of the +z direction is arbitrary. Given the direction +x, we can equivalently define any direction along the y-z plane as our new z axis. Consequently, we can be assured that any basis corresponding to a direction orthogonal to the x axis is mutually unbiased with respect to the eigenbasis of σ x . A measurement along a given axis is therefore related to the amount of coherence along an orthogonal direction. More generally, any qubit observable M can always be written in the form x y z is the standard vector of Pauli matrices and  r is a real three dimensional vector. In this case, the outcomes of the measurement M contains information about the coherence in a basis that is orthogonal to  r in the Bloch sphere. Proposition 1 generalizes this observation to higher dimensions.
We now consider higher dimensional, multipartite scenarios and present numerical examples of our computable measure  M MIO . Let us consider for spin systems the total magnetic moment operator.