Operational one-to-one mapping between coherence and entanglement measures

Zhu, Huangjun; Ma, Zhihao; Cao, Zhu; Fei, Shao-Ming; Vedral, Vlatko

doi:10.1103/physreva.96.032316

Cited by 145 publications

(175 citation statements)

References 81 publications

(243 reference statements)

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“…Alternatively, the form of the achievable region can be proven using results in [18][19][20]. We also note that for pure states the conversion problem has been completely solved for SIO [7] and IO [45].…”

Section: Summary Of Resultsmentioning

confidence: 98%

“…This extension has found several applications in remote quantum protocols, including the tasks of quantum state merging [42,43] and assisted coherence distillation [38,41], which has also been demonstrated experimentally very recently [44]. Coherence in multipartite systems has also been studied with respect to other types of nonclassicality such as entanglement [21,23,30,45] and quantum discord [46,47].…”

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confidence: 91%

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Structure of the Resource Theory of Quantum Coherence

et al. 2017

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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 ...

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Section: Summary Of Resultsmentioning

confidence: 98%

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confidence: 91%

Structure of the Resource Theory of Quantum Coherence

et al. 2017

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“…Note that these two classes, incoherent operations and strictly incoherent operations, are distinct as sets of CPTP maps, although it is known that they induce the same possible state transformations of a given state into a target state for qubits [27] and for pure states in arbitrary dimension [28] (correcting the earlier erroneous proof of the claim by [19]; the SIO part of the pure state transformations is due to [17]). However, for the distillation of pure coherence at rate C r (ρ) [17], IO are needed and it remains unknown whether SIO can attain the same rate.…”

Section: Proposition 3 (Chitambar/hsieh [25])mentioning

confidence: 91%

“…It may still be the case that there is in general a difference between MIO-simulation cost and MIO-coherence generating capacity, but deciding this possibility is beyond the scope of the present investigation. We only note that Theorem 1 gives us a singleletter formula for the MIO-coherence generating capacity, namely C MIO gen (T ) = P r (T ) = sup ρ on A⊗C C r (T ⊗id)ρ −C r (ρ), (28) the complete relative entropy coherence power of T . The supremum is over all auxiliary systems C and mixed states ρ on A ⊗ C. Indeed, the upper bound of Eq.…”

Section: Discussionmentioning

confidence: 99%

Resource theory of coherence: Beyond states

et al. 2017

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We generalize the recently proposed resource theory of coherence (or superposition) [Baumgratz, Cramer & Plenio, Phys. Rev. Lett. 113:140401; Winter & Yang, Phys. Rev. Lett. 116:120404] to the setting where not only the free ("incoherent") resources, but also the objects manipulated, are quantum operations rather than states.In particular, we discuss an information theoretic notion of coherence capacity of a quantum channel, and prove a single-letter formula for it in the case of unitaries. Then we move to the coherence cost of simulating a channel, and prove achievability results for unitaries and general channels acting on a d-dimensional system; we show that a maximally coherent state of rank d is always sufficient as a resource if incoherent operations are allowed, and rank d 2 for "strictly incoherent" operations. We also show lower bounds on the simulation cost of channels that allow us to conclude that there exists bound coherence in operations, i.e. maps with non-zero cost of implementing them but zero coherence capacity; this is in contrast to states, which do not exhibit bound coherence.

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“…However, for SIO and IO, while K GIO n is mainly responsible for coherence-destruction, the permutation operator P n and relabeling operator R n enable the transfers between different incoherent basis states. In fact, the above analysis implies the reason why GIO or SIO is equally powerful as other seemingly more powerful operations (such as IO or MIO) on many occasions, a phenomenon emerged in many recent relevant works [20,69,70].…”

Section: Observation 3 the Kraus Operators Of Sio And Io Can Be Obtamentioning

confidence: 96%

Interpreting quantum coherence through a quantum measurement process

et al. 2017

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Recently, there has been a renewed interest in the quantification of coherence or other coherencelike concepts within the framework of quantum resource theory. However, rigorously defined or not, the notion of coherence or decoherence has already been used by the community for decades since the advent of quantum theory. Intuitively, the definitions of coherence and decoherence should be the two sides of the same coin. Therefore, a natural question is raised: how can the conventional decoherence processes, such as the von Neumann-Lüders (projective) measurement postulation or partially dephasing channels, fit into the bigger picture of the recently established theoretic framework? Here we show that the state collapse rules of the von Neumann or Lüders-type measurements, as special cases of genuinely incoherent operations (GIO), are consistent with the resource theories of quantum coherence. New hierarchical measures of coherence are proposed for the Lüders-type measurement and their relationship with measurement-dependent discord is addressed. Moreover, utilizing the fixed point theory for C * -algebra, we prove that GIO indeed represent a particular type of partially dephasing (phase-damping) channels which have a matrix representation based on the Schur product. By virtue of the Stinespring's dilation theorem, the physical realizations of incoherent operations are investigated in detail and we find that GIO in fact constitute the core of strictly incoherent operations (SIO) and generally incoherent operations (IO) and the unspeakable notion of coherence induced by GIO can be transferred to the theories of speakable coherence by the corresponding permutation or relabeling operators.

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Operational one-to-one mapping between coherence and entanglement measures

Cited by 145 publications

References 81 publications

Structure of the Resource Theory of Quantum Coherence

Structure of the Resource Theory of Quantum Coherence

Resource theory of coherence: Beyond states

Interpreting quantum coherence through a quantum measurement process

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