Structure of the Resource Theory of Quantum Coherence

Abstract

Quantum coherence is an essential feature of quantum mechanics which is responsible for the departure between classical and quantum world. The recently established resource theory of quantum coherence studies possible quantum technological applications of quantum coherence, and limitations which arise if one is lacking the ability to establish superpositions. An important open problem in this context is a simple characterization for incoherent operations, constituted by all possible transformations allowed within the resource theory of coherence. In this work, we contribute to such a characterization by proving several upper bounds on the maximum number of incoherent Kraus operators in a general incoherent operation. For a single qubit, we show that the number of incoherent Kraus operators is not more than 5, and it remains an open question if this number can be reduced to 4. The presented results are also relevant for quantum thermodynamics, as we demonstrate by introducing the class of Gibbs-preserving strictly incoherent operations, and solving the corresponding mixedstate conversion problem for a single qubit.Quantum resource theories [1, 2] provide a strong framework for studying fundamental properties of quantum systems and their applications for quantum technology. The basis of any quantum resource theory is the definition of free states and free operations. Free states are quantum states which can be prepared at no additional cost, while free operations capture those physical transformations which can be implemented without consumption of resources. Having identified these two main features, one can study the basic properties of the corresponding theory, such as possibility of state conversion, resource distillation, and quantification. An important example is the resource theory of entanglement, where free states are separable states, and free operations are local operations and classical communication [3,4].In the resource theory of quantum coherence [5-9], free states are identified as incoherent statesi.e., states which are diagonal in a fixed specified basis {|i }. The choice of this basis depends on the particular problem under study, and in many relevant scenarios such a basis is naturally singled out by the unavoidable decoherence [10]. The definition of free operations within the theory of coherence is not unique, and several approaches have been discussed in the literature, based on different physical (or mathematical) considerations [8]. Two important frameworks are known as incoherent [6] and strictly incoherent operations [7,11], which will be denoted by IO and SIO, respectively. The characterizing feature of IO is the fact that they admit an incoherent Kraus decomposition, i.e., they can be written as [6] where each of the Kraus operators K j cannot create coherence individually, K j |m ∼ |n for suitable integers n and m. This approach is motivated by the fact that any quantum operation can be interpreted as a selective measurement in which outcome j occurs with probability p j = Tr[K j ρK ...

Quantum coherence and state conversion: theory and experiment

“…The state conversion problem is important in any resource theory, since it determines the value of states for protocols using the resource under study. The result presented here are a significant generalization of recent results on single-shot coherence theory [10,[23][24][25][26] and single-shot resource theories in general [27], and include necessary conditions on the existence of stochastic conversions, which we generalized to higher dimensions.…”

Optimal distillation of quantum coherence with reduced waste of resources

“…[11]. A relevant subset of IO is constituted by strictly incoherent operations (SIOs), which are completely positive trace-preserving maps whose Kraus operators K i satisfy both [21][22][23][24][25]. We will demonstrate that, although SIO have a very limited coherence distillation power when mixed input states are concerned [17], they nonetheless suffice for our distillation protocol with pure input states.…”

Analytically Computable Symmetric Quantum Correlations

“…[26][27][28][29][30][31][32][33][34][35][36][37][38][39] The same is true for its counterpart quantity, [40][41][42] that is, the measurementinduced nonlocality (MIN) which characterizes the global disturbance in a composite state caused by a local nondisturbing measurement on one subsystem. In particular, with the development of the resource theory of quantum coherence, [53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69][70] it was shown that the coherence of a system could be converted through incoherent operations into the SQD, instead of the asymmetric QD, of a composite system. [42][43][44][45][46][47][48][49] However, neither the QD nor the MIN of a bipartite system is symmetric in the sense that different results will be obtained (especially for some zero-discord states) if the two subsystems are swapped, which implies that the QD and MIN cannot completely quantify all of the quantum correlations present in a state.…”