Coherence makes quantum systems magical
Two primary facets of quantum technological advancement that holds great promise are quantum communication and quantum computation. For quantum communication, the canonical resource is entanglement. For quantum gate implementation, the resource is ‘magic’ in an auxiliary system. It has already been shown that quantum coherence is the fundamental resource for the creation of entanglement. We argue on the similar spirit that quantum coherence is the fundamental resource when it comes to the creation of magic. This unifies the two strands of modern development in quantum technology under the common underpinning of existence of quantum superposition, quantified by the coherence in quantum theory. We also attempt to obtain magic monotones inspired from coherence monotones and vice versa. We further study the interplay between quantum coherence and magic in a qutrit system and that between quantum entanglement and magic in a qutrit-qubit setting.
What makes quantum technologies so much more powerful than their classical counterparts ? The answer must ultimately lie in the postulates of quantum theory. Especially the linear superposition principle, implying the existence of quantum coherence, can be intuitively thought of as the driving agent behind any quantum advantage. The recent quantification of superposition through the resource theory of quantum coherence [1, 2, 3, 4] has allowed us to formalize this intuition in a more rigorous way. Quantum entanglement, the basic resource behind quantum communication schemes like dense coding , teleportation  or remote state preparation , has been connected with quantum coherence . However, quantum technologies are not limited to communication schemes. One of the principle scientific developments in last few decades has been the emergence of the theory of quantum computation as a more powerful alternative to the paradigm of classical computation. Resources like quantum entanglement and quantum coherence have at various points been shown to lead to quantum advantage vis-a-vis classical computers. For example, to implement Shor’s algorithm, one needs a large amount of entanglement , whereas, to implement Grover’s algorithm, one needs a small amount of entanglement [10, 11]. Similarly, it has recently been demonstrated that in order to implement the Deutsch-Jozsa algorithm , one requires coherence as a resource . Quantum coherence has also been related to the success probability of the Grover search algorithm [14, 15]. However, for a quantum computer to work, we must ultimately be able to implement quantum logic gates. If we insist on implementing only classically simulable gates, which is a reasonable demand if the performances of nascent quantum computers have to be evaluated through classical means, this implies the need for auxiliary quantum states which are outside the stabilizer polytope and in a so called ‘magic’ state. A resource theory for such magic states was recently proposed  and is a topic of active interest [17, 18]. In this article, we ask the following question - how does quantum coherence relate to magic in quantum states. Firstly, starting with a brief recapitulation of resource theories of coherence and magic, we show, using contractive distance based monotones, that the magic generated in a quantum state through incoherent operations  is upper bounded by the amount of coherence initially in the state. Subsequently, we prove that the maximum amount of magic generated through such incoherent operations can, by itself, be shown to be a coherence monotone. The inverse question of constructing magic monotones from coherence monotones is equally important and we provide a partial approach here. We further obtain a full coherence monotone based on the discrete Wigner function representation [19, 20, 21] of quantum states, the latter being useful for providing a calculable measure of magic. Next, we propose the counterparts to various types of incoherent operations in the resource theory of magic states. We subsequently move on to revealing the link between magic and other quantum resources like quantum coherence and entanglement, in small quantum systems. Finally, we outline some possible directions of future work.
2 Resource theories of quantum coherence and magic
2.1 Resource theory of coherence
The resource theory of coherence seeks to quantify the amount of superposition in quantum states with respect to a fixed basis, say . Thus, the set of free states, hence called , consists of purely classical mixtures of eigenkets, i.e., states of the form . The free operations, dubbed , are defined as being those CPTP operations whose every Kraus element maps an incoherent state to another incoherent state. One can easily see that this is a stronger condition than merely requiring the CPTP operations to map every incoherent state to another incoherent state. The necessary conditions that any monotone must satisfy under incoherent operation are thus given by
If , then . Otherwise, .
For any incoherent operation and any state , .
If are Kraus operators corresponding to any incoherent operation such that , then the coherence should not increase under selective measurement, i.e. .
2.2 Resource theory of magic
The main aim behind the resource theory of magic, also known as the resource theory of stabilizer computation, is to quantitatively characterize the extent to which a quantum system, acting as an auxiliary, can help in implementing classically non-simulable gates. The pure free states in this resource theory are the ones reachable via Clifford unitaries acting on any member of the computational basis, say . The total family of free states, denoted as , consists of the convex hull formed by the pure free states. The free operations consist of Clifford unitaries, measurement in the computational basis, composition with other stabilizer states and partial trace, as well as these operations conditioned on measurement results. Magic monotones are relatively less studied until now, although some monotones have been found ranging from distance based monotones  to robustness type monotones  to monotones inspired from the Wigner function representation of states in discrete phase space .
As a succinct reminder, the table below summarizes the primary features for both the resource theories.
|Resource theory||Free states||Free operation|
|Coherence||diagonal incoherent states in the basis||Incoherent operation
|Magic||States inside polytope accessible via Clifford unitary rotation of computational basis||Stabilizer operation
3 Linking resource theories of coherence and magic
3.1 Coherence Quantifiers through Magic Monotones
The resource theories of coherence and magic, as reviewed above, seem quite disjoint. But are they really so ? This is the question we seek to address. Specifically, in this section, we demonstrate how the presence or lack of quantum coherence in systems constrains the amount of magic in the system. In doing so, we reveal that quantum coherence can be quantified by the maximum amount of magic generated through incoherent operation on arbitrary quantum states. In subsequent work, unless otherwise stated, the basis with respect to which quantum coherence is defined, is the computational basis and the pure stabilizer states are the ones obtainable through Clifford unitary rotation of one of the basis elements, say , of the computational basis. We now state our first result -
Result 1. For any distance based coherence quantifier and corresponding magic quantifier , the amount of magic generated through incoherent operations on a quantum state is upper bounded by the amount of coherence originally present in that state.
Proof- lhs equals where we used the fact that any incoherent state in the computational basis is a stabilizer state. ∎
We now propose the following set of coherence monotones corresponding to every distance based magic monotone
and prove the monotonicity conditions below. It is trivial to see that any incoherent state with respect to the adequately chosen basis is a stabilizer state. Monotonicity under CPTP maps is guaranteed for any contractive distance based measure. Therefore we only present the proof for strong monotonicity under selective measurements. The proof is identical in spirit to the one presented in  for entanglement.
Result 2 (Strong Monotonicity). If and where are the Kraus operators corresponding to some incoherent operation, then
Proof- Let us assume that the condition above is false. Then there will exist at least one set of incoherent operations for which
Now, since magic monotones are non-increasing on average under measurements in the computational basis ,
Now, one can write a bipartite incoherent operation such that the Kraus operators for are written as , where are the Kraus operators corresponding to the incoherent operation and is the incoherent unitary . For this operation, the LHS =
This is in contradiction with the result (1) proved earlier , thus completing the proof. ∎
3.2 Coherence monotone inspired from another magic monotone
In the preceding subsection, we showed how to construct coherence monotones from distance based magic monotones. In most cases, these monotones are extremely hard to exactly calculate. There is however, a computable monotone, called sum negativity, already in the literature  in terms of the negativity of the discrete Wigner function representation of a state. We show that the discrete Wigner function representation can also give rise to a coherence monotone. Something similar was possibly attempted in Ref. . However, their paper (sans abstract) was written in Chinese, hence not accessible to us or anybody not familiar with the language. Therefore, we furnish a complete proof for the sake of completeness.
For finite Hilbert space dimension , the expression for characteristic function associated with each point on the phase grid is given by
where and . and are the well known Shift and Boost operators respectively, and is the -th root of unity and is shorthand for . The Wigner function of a quantum state represented by the density matrix , at a phase space point , is given by . Sum of Wigner functions along a line is always positive semidefinite. Now let us propose the following candidate for a coherence monotone
Here is a probability vector whose elements are the sums of Wigner functions along parallel lines in the phase grid and is the statistical distance between probability distributions and .
Clearly, vanishes for incoherent states. Moreover, from the monotonicity of trace distance under CPTP maps, is monotonically decreasing under any CPTP map. The remaining, i.e., strong monotonicity and convexity conditions have been shown in literature  to be equivalent to the equality condition . The LHS of the above condition now reads as where and . This completes the proof of the assertion that is a full coherence monotone.
3.3 Coherence based measure of magic
In the foregoing section, we have seen how to construct coherence monotones from corresponding distance based magic monotones. However, since there are already a plethora of coherence monotones (see, e.g., Ref.  for an overview), it is perhaps more urgent to consider the inverse problem, i.e., the construction of magic monotones from adequately defined coherence monotones.
In this section , we propose a quantifier of magic based on coherence monotones valid for prime dimensions only. In such cases, the stabilizer states can be completely characterized as eigenvectors corresponding to different MUBs possible.
Definition (Magic of Formation) : The magic of formation for an arbitrary mixed state in prime dimension is defined as follows
where k is over all the chosen stabilizer basis directions and is a coherence monotone, for example, the relative entropy of coherence = along k direction. Clearly, for any pure stabilizer state, the corresponding coherence element is zero and hence the magic of formation vanishes. For any mixed state within the stabilizer polytope, the magic of formation is similarly zero.
Invariance under Clifford unitaries- The effect of applying Clifford unitaries is to simply permute the bases with which coherences are to be computed.
Monotonicity under computational basis measurement- The post measurement state can be written in the pure state decomposition in terms of the computational basis. Since coherence in the computational basis for each of these pure state elements is zero, the post measurement magic of formation vanishes.
Since the composition with stabilizer states renders the dimension of the resultant state to be non-prime and partial tracing is not possible for a state with prime dimension, our proposal suffers from the flaw of not being testable under these conditions.
3.4 Hierarchy of Stabilizer Operations
In analogy with the resource theories of quantum coherence or entanglement, we may formulate various generalizations and specializations of stabilizer operations. A tentative hierarchy of such operations, roughly following the corresponding formulation for incoherent operations in Ref. [32, 33] is depicted in Fig. 2.
Genuinely Stabilizer Operations - The most stringent of all the stabilizer operations must be the genuinely stabilizer operations (GSO) similar to genuinely incoherent operations introduced in  for which every stabilizer state is supposed to be a fixed point for the dynamics. In the following proposition- we prove that such an operation is impossible unless it is the trivial identity transformation.
Proposition - There is no non-trivial Genuinely Stabilizer Operation.
Proof-Let us illustrate the proof for . Suppose there is such a CPTP operation which is a Genuinely Stabilizer Operation. This implies is a genuinely incoherent operation with respect to both the eigenbasis of and . Thus the Kraus operators corresponding to this operation are diagonal in both basis as well as basis, which holds true only for the trivial identity operation. ∎.
Incoherent Stabilizer Operations - Stabilizer operations can still generate quantumness in the form of quantum coherence. Thus, if we are to construct a resource theory encompassing both the stabilizer formalism and quantum superposition, it is relevant to consider incoherent stabilizer operations. In the stabilizer protocol, two operations stand out as potentially generators of quantum coherence. One being the Clifford unitary operation, the other being composition with different stabilizer states. The other operations, viz. measurement in computational basis or partial trace, can easily be shown to be incoherent operations as well. Thus, we write down the following subset of these two operations -
Incoherent Clifford Unitary - Defined as those clifford unitaries which do not generate quantum coherence, these now represent permutations of computation basis vectors. For example, in the qubit case, is an incoherent Clifford unitary, but the Hadamard gate H = is not.
Composition with other incoherent states Incoherent states are by definition, within the stabilizer polytope, and composition with other incoherent states keeps quantum coherence fixed . Thus this represents a suitable incoherent stabilizer operation.
Clearly every coherence monotone is a monotone under this formalism.
Proposition - The -norm is a monotone under incoherent stabilizer operations for every .
Proof - Every stabilizer protocol on a state can be expressed as where is a Clifford unitary and is an ancilla stabilizer state. According to the conditons above, we must restrict to the set of incoherent stabilizer states and to the set of incoherent Clifford unitaries. The effect of incoherent unitaries is merely to permute the basis labellings for coherence. The -norm of a state is given by
Now, for an incoherent ancilla state the norm , where we have used Hölder’s inequality. Similarly one can also check using the triangle inequality for the -norm, that partial tracing doesn’t increase . Thus, -norm is indeed a monotone for every under incoherent stabilizer protocols. ∎
Incoherent Stabilizer Preserving Operations- Continuing in the spirit of connecting the two resource theories, one can impose on the set of incoherent operations only the constraint that it does not generate any magic from stabilizer states. For example, the phase rotation is an incoherent operation, which may easily be seen to create magic starting from a stabilizer state.
Stabilizer Preserving Operations - This is the most general type of free operation in the resource theory of magic that one can envisage. One only imposes the constraint that no stabilzer state is mapped to a magic state. In fact, such operations were studied in detail in Ref.  and a family of monotones derived.
4 Concrete results in small quantum systems
In this section, we shift our focus towards linking magic with other quantum resources in low dimensional systems. The smallest dimension for which we have a concrete computable expression for magic is , which is expressed via the sum negativity of discrete Wigner functions. Let us now look at the interplay between quantum coherence and magic in this scenario. Since signature of the connection between magic and contextuality has already been revealed [36, 37], our method of relating magic to other resource theories connects contextuality inter alia with these resources. The nascent resource theoretic formulation of contextuality [38, 39] can shed further light on the results we obtain here.
4.1 Explicit expression for magic in the qutrit case
For the qutrit case, we have the following discrete Wigner distribution corresponding to a qutrit density matrix .
Where density matrix elements , . Now the sum negativity is simply given by
4.2 Effect of coherent and incoherent noise on magic states
As with many other quantum resource theories, the maximally mixed state is a free state in the resource theory of magic while the maximally resourceful state turns out to be a pure state. In the qutrit scenario, the maximally magical pure states come in two different varieties, viz. the Strange states and the Norrell states . It may therefore be interesting to have an answer to the question that which class of states remain more magical under admixture of noise. However, noise can be either coherent or incoherent. As we demonstrate below, depending on the character of the noise, the relative robustness of two types of magical states may be of different nature.
Proposition - Strange states are more robust under mixture with maximally incoherent, i.e. white noise than Norrell states.
Proof- Due to symmetry, it suffices to check for one strange state and one Norrell state, respectively. Let this strange state be and this Norrell state be . Let us consider the strange state (Norrell state) mixed with the maximally mixed state to have the family of states . With the explicit expression for sum negativity given previously, it is easy to check that the sum negativity for the noisy strange state is given by
while the corresponding sum negativity for the noisy Norrell state is given by
Therefore, we see that the strange state remains more robust against admixture with white noise than the Norrell state.∎
Now, let us consider an example of a purely coherent noise, i.e. admixture of a maximally magical pure state with a maximally coherent state .
Proposition - The Norrell state above is more robust under the admixture of aforementioned coherent noise than the strange state above.
Proof- Proceeding similarly as before, the expression for sum negativity of the noisy strange state is now given by
while the corresponding expression for sum negativity of the noisy Norrell state is given by
Thus, throughout the range of the noise parameter , the noisy Norrell state contains more magic than the corresponding noisy strange state, which is demonstrated in Fig. b.
4.3 Relation betweeen quantum coherence, quantum entanglement, and magic
Continuing with our theme of attempting to unearth the relation of coherence and magic in quantum systems, it is a natural question to ask whether we can find a bound for the quantity of magic in terms of coherence in the qutrit scenario. One bound is quite obvious. Every incoherent state lies within the stabilizer polytope, therefore it is easy to see that any quantum state, pure or mixed, is at least as close to a stabilizer state as to an incoherent state. Thus, the magic of a quantum state is upper bounded by the amount of coherence in the system. However, for qutrit pure states, numerical simulation in Fig. a leads us to conjecture the following inequality, which gives a reverse, i.e., lower bound to the magic in terms of quantum coherence.
Proposition - The following condition on quantum coherence, quantified via the -norm, and magic, quantified by the sum negativity, holds for qutrit pure states
It has already been shown that the presence of entanglement in a bipartite state adversely affects the coherence [40, 41] as well as contextuality  in the reduced state. Since magic as a resource in quantum computation has ultimately been ascribed to the contextual nature of quantum mechanics , it is important to quantify the corresponding trade off for entanglement in the joint system and magic in the reduced system. The simplest case is that of a qutrit qubit joint system. In this situation, we conjecture the following trade off relation between bipartite entanglement, quantified by the negativity, of a qutrit qubit joint system , and that of magic, quantified by sum negativity, in the reduced qutrit system .
Proposition - The negativity of entanglement and the sum negativity satisfies the following trade off relation
Although an analytical proof is lacking, the numerical result furnished in Fig. b strongly suggests that the proposition above is true and indeed, almost tight for pure states.
Unification is a common theme in physics. Following the spirit of unification, it is thus a worthwhile effort to bring various nascent resource theories in quantum information theory under one umbrella. In this work, we have indicated the link between the resource theories of coherence and magic. We demonstrated that quantum coherence in a state is ultimately the currency for creation of magic through incoherent operations. We proposed a measure of magic in prime dimensional quantum states through product of coherence quantifiers with respect to mutually unbiased bases and, furthermore, we derived another full coherence monotone from the discrete Wigner function representation of quantum states in discrete phase space. We also proposed several sub-classes of quantum operations as free operations if coherence and magic are to be simultaneously considered as resources. We also investigated the link between coherence and magic in the concrete scenario of a qutrit system. We believe the concepts and results in this paper should spur more detailed investigations into the link between these two different resource theories. More generally, the method utilized to prove (2) is almost identical to the corresponding proof in Ref. , which indicates that a more general result along these lines can be proved in other situations as well. Our work suggests that various resources useful for quantum technology are ultimately the manifestation of the superposition principle in the quantum world, quantified through the resource theory of coherence.
CM thanks Department of Atomic Energy, Govt of India for financial support through fellowship. We also acknowledge Uttam Singh, Samyadeb Bhattacharya, and Victor Veitch for useful inputs. We thank Francesco Albarelli for pointing out an error in an earlier version of the manuscript.
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