## Abstract

Manipulation and quantification of quantum resources are fundamental problems in quantum physics. In the asymptotic limit, coherence distillation and dilution have been proposed by manipulating infinite identical copies of states. In the nonasymptotic setting, finite data-size effects emerge, and the practically relevant problem of coherence manipulation using finite resources has been left open. This Letter establishes the one-shot theory of coherence dilution, which involves converting maximally coherent states into an arbitrary quantum state using maximally incoherent operations, dephasing-covariant incoherent operations, incoherent operations, or strictly incoherent operations. We introduce several coherence monotones with concrete operational interpretations that estimate the one-shot coherence cost - the minimum amount of maximally coherent states needed for faithful coherence dilution. Furthermore, we derive the asymptotic coherence dilution results with maximally incoherent operations, incoherent operations, and strictly incoherent operations as special cases. Our result can be applied in the analyses of quantum information processing tasks that exploit coherence as resources, such as quantum key distribution and random number generation.

Original language | English (US) |
---|---|

Article number | 070403 |

Journal | Physical review letters |

Volume | 120 |

Issue number | 7 |

DOIs | |

State | Published - Feb 13 2018 |

Externally published | Yes |

## ASJC Scopus subject areas

- Physics and Astronomy(all)

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*Physical review letters*,

*120*(7), Article 070403. https://doi.org/10.1103/PhysRevLett.120.070403

**One-Shot Coherence Dilution.**/ Zhao, Qi; Liu, Yunchao; Yuan, Xiao et al.

In: Physical review letters, Vol. 120, No. 7, 070403, 13.02.2018.

Research output: Contribution to journal › Article › peer-review

*Physical review letters*, vol. 120, no. 7, 070403. https://doi.org/10.1103/PhysRevLett.120.070403

**One-Shot Coherence Dilution**. In: Physical review letters. 2018 ; Vol. 120, No. 7.

}

TY - JOUR

T1 - One-Shot Coherence Dilution

AU - Zhao, Qi

AU - Liu, Yunchao

AU - Yuan, Xiao

AU - Chitambar, Eric

AU - Ma, Xiongfeng

N1 - Funding Information: Zhao Qi 1 Liu Yunchao 1 Yuan Xiao 1 ,* Chitambar Eric 2 ,† Ma Xiongfeng 1 ,‡ Center for Quantum Information, 1 Institute for Interdisciplinary Information Sciences, Tsinghua University , Beijing 100084, China Department of Physics and Astronomy, 2 Southern Illinois University , Carbondale, Illinois 62901, USA * yxbdwl@gmail.com † echitamb@siu.edu ‡ xma@tsinghua.edu.cn 13 February 2018 16 February 2018 120 7 070403 28 November 2017 29 June 2017 © 2018 American Physical Society 2018 American Physical Society Manipulation and quantification of quantum resources are fundamental problems in quantum physics. In the asymptotic limit, coherence distillation and dilution have been proposed by manipulating infinite identical copies of states. In the nonasymptotic setting, finite data-size effects emerge, and the practically relevant problem of coherence manipulation using finite resources has been left open. This Letter establishes the one-shot theory of coherence dilution, which involves converting maximally coherent states into an arbitrary quantum state using maximally incoherent operations, dephasing-covariant incoherent operations, incoherent operations, or strictly incoherent operations. We introduce several coherence monotones with concrete operational interpretations that estimate the one-shot coherence cost—the minimum amount of maximally coherent states needed for faithful coherence dilution. Furthermore, we derive the asymptotic coherence dilution results with maximally incoherent operations, incoherent operations, and strictly incoherent operations as special cases. Our result can be applied in the analyses of quantum information processing tasks that exploit coherence as resources, such as quantum key distribution and random number generation. National Key R&D Program of China 2017YFA0303900 2017YFA0304004 National Natural Science Foundation of China 10.13039/501100001809 11674193 National Science Foundation 10.13039/100000001 1352326 Quantum coherence is a fundamental property that can emerge within any quantum system. With respect to some physically preferable reference frame [1–3] , such as the energy levels of an atom or a selected measurement basis, coherence empowers the ability of many quantum information tasks, including cryptography [4] , metrology [5] , and randomness generation [6,7] . Furthermore, coherence is a widespread resource playing important roles in biological systems [8,9] and small-scale thermodynamics [10–14] . Various efforts have been devoted to building a resource framework of coherence [15–17] . In general, a resource theory is defined by a set of free states and a corresponding set of free operations that preserve the free states. States that are not free are said to possess resource, and various measures can be constructed to quantify the amount of resource in a given state. For example, in the resource theory of entanglement [18–20] , free states and free operations are defined by separable states, local operation and classical communication (LOCC), respectively. Entanglement measures include the relative entropy of entanglement [21] and entanglement of formation [18] . In the resource theory of coherence [15,16] , free or incoherent states are those that are diagonal in a priori fixed computational basis; free or incoherent operations are some specified classes of physically realizable operations that act invariantly on the set of incoherent states. Different definitions of incoherent operations have been studied due to different motivations. In this work, we focus on the maximally incoherent operation (MIO) proposed by Åberg [15] , the dephasing-covariant incoherent operation (DIO) proposed independently by Chitambar and Gour [22] and Marvian and Spekkens [23] , the incoherent operation (IO) proposed by Baumgratz et al. [16] , and the strictly incoherent operation (SIO) proposed by Winter and Yang [24] . Coherence measures include the relative entropy of coherence [16] , coherence of formation [6,15] , robustness of coherence [25] , etc. We refer to Ref. [17] for a comprehensive review of recent developments of the resource theory of coherence. Investigating state transformations via free operations is of paramount importance in a resource theory. In particular, many efforts have been devoted to understand the interconversion between a given state ρ and copies of a canonical unit resource | Ψ ⟩ [26] via free operations. Specifically, the dilution problem is to convert unit resource | Ψ ⟩ to the target state ρ , and the distillation problem is the reverse process. In the asymptotic case, where infinite copies of ρ and | Ψ ⟩ are provided, the dilution rate (or coherence cost) and the distillable rate describe the maximal proportion of ρ and | Ψ ⟩ that can be obtained on average, respectively. In entanglement theory, the well-known distillable entanglement [27] and entanglement cost [28] of a state measure its optimal rate of asymptotic distillation and dilution, respectively. The asymptotic distillation and dilution of coherence under IO and SIO have been investigated by Winter and Yang [24] who proved that the distillable coherence is given by the relative entropy of coherence and that the coherence cost is given by the coherence of formation. The processes of asymptotic distillation and dilution are studied under two crucial assumptions: (i) a source is available that prepares independent and identically distributed (IID) copies of the same state and (ii) an unbounded number of copies of this states can be generated. These assumptions overlook possible correlations between different state preparations and they become unreasonable when only a finite supply of states are available. In order to relax the two assumptions, it is necessary to consider the most general scenario, i.e., the one-shot scenario, where the conversion is from a general initial state to a general final state. Such a scenario reflects realistic experimental setups where we only manipulate finite and correlated states. In many quantum information tasks, such as quantum key distribution [29] , device independent processing [30,31] , thermodynamics [10–14,32–35] , quantum channel capacity [36–39] , and general resource theory [40] , some analyses have already been conducted in the one-shot scenario. In particular, one-shot entanglement distillation and dilution have been investigated under LOCC [41–43] , as well as nonentangling maps and operations that generate negligible amount of entanglement [44] . In thermodynamics, conversion under thermal operations is known only for qubit states [11] . For coherence, the necessary and sufficient conditions for single-copy state transformations are known only for pure states [24,45,46] and single qubit mixed states [22,47] . Generally, one-shot coherence distillation and dilution of general quantum states have been left as open problems [17,24] . In this Letter, we consider one-shot coherence dilution under four widely accepted incoherent operations: MIO, DIO, IO, and SIO. We first review the coherence framework by Åberg [15] and Baumgratz et al. [16] . Then, we introduce several coherence monotones for different incoherent operations. In addition, we define the one-shot coherence cost in the dilution process, and explicitly show that the optimal one-shot coherence cost is characterized by the introduced coherence monotones. Moreover, when applying our results to the asymptotic IID scenario, we obtain the coherence cost under MIO and show that it is equal to the relative entropy of coherence. Similarly, we also derive the asymptotic coherence cost under IO and SIO and show that it equals to the coherence of formation, which is consistent with the results in [24] . Our main results are summarized in Table I . We introduce and discuss the results in details below and provide the proofs in the Supplemental Material [48] . I 10.1103/PhysRevLett.120.070403.t1 TABLE I. Coherence dilution in the one-shot and asymptotic scenarios with MIO, DIO, IO, and SIO. The two columns “one-shot” and “asymptotic” denote the coherence measures in the one-shot and asymptotic scenarios, respectively. Operation One-shot Asymptotic MIO C MIO ϵ ≈ C max ϵ C MIO ∞ = C r DIO C DIO ϵ ≈ C Δ , max ϵ C DIO ∞ = C r IO C IO ϵ = C 0 ϵ C IO ∞ = C f SIO C SIO ϵ = C 0 ϵ C SIO ∞ = C f Framework .—Considering a computational basis I = { | i ⟩ } i = 0 d - 1 in a d -dimensional Hilbert space H d , incoherent states are defined as δ = ∑ i = 0 d - 1 p i | i ⟩ ⟨ i | , where { p i } is a probability distribution. We denote the set of incoherent states as I . The MIOs introduced in Ref. [15] are physical or completely positive trace preserving (CPTP) maps Λ such that Λ ( δ ) ∈ I , ∀ δ ∈ I . A CPTP map Λ is called a dephasing-covariant incoherent operation (DIO) if Λ [ Δ ( ρ ) ] = Δ [ Λ ( ρ ) ] for all ρ [22,23] . Here, Δ ( ρ ) = ∑ i | i ⟩ ⟨ i | ⟨ i | ρ | i ⟩ is the dephasing channel, and clearly DIO is a subset of MIO. Another subset of MIO are the incoherent operations (IOs) [16] which are CPTP maps that admit a Kraus operator representation Λ ( ρ ) = ∑ n K n ρ K n † with the { K n } being “incoherent-preserving” operators, that is K n δ K n † / p n ∈ I for all n and all δ ∈ I . Here p n = Tr [ K n ρ K n † ] is the probability of obtaining the n th outcome. In general, when Λ ( ρ ) = ∑ n K n ρ K n † is an incoherent operation, the Kraus operator can always be represented as K n = ∑ i c i | f ( i ) ⟩ ⟨ i | , where f is a function on the index set and c i ∈ [ 0 , 1 ] [24] . Finally, strictly incoherent operations (SIOs) are CPTP maps admitting a Kraus operator representation Λ ( ρ ) = ∑ n K n ρ K n † such that both { K n } and { K n † } are incoherent-preserving operators [24] . The relations among different incoherent operations are shown in Fig. 1 . 1 10.1103/PhysRevLett.120.070403.f1 FIG. 1. Comparison among different incoherent operations. The largest set O contains all possible physical operations. Associated with each of these operational classes is a family of monotone functions. A real-valued function C ( ρ ) is called a MIO (DIO) monotone if C ( ρ ) ≥ C [ Λ ( ρ ) ] whenever Λ is MIO (DIO). Since IO and SIO are defined in terms of Kraus operator representations, it is natural to modify the monotonicity condition to average postmeasurement values. That is, C ( ρ ) is called an IO (SIO) monotone if C ( ρ ) ≥ ∑ n p n C ( K n ρ n K n † / p n ) whenever { K n } ( { K n } and also { K n † } ) are incoherent-preserving Kraus operators. Following the notions of monotonicity defined above, a coherence measure for a class of operations O is defined as a real valued function C ( ρ ) that satisfies the following requirements: (C1) C ( ρ ) ≥ 0 with equality if and only if ρ ∈ I ; (C2) C ( ρ ) is a monotone for operational class O ; (C3) Convexity: coherence cannot increase under mixing states, i.e., C ( ∑ n p n ρ n ) ≤ ∑ n p n C ( ρ n ) . When a function satisfies conditions (C1) and (C2), we call it a coherence monotone for operational class O . Although a coherence monotone may not satisfy (C3), it can still play important roles in tasks that process coherence. Coherence monotones .—In the following, we first introduce three coherence monotones of quantum states defined on Hilbert space H d . To do so, we make use of the generalized quantum α -Rényi divergence, D ˜ α ( ρ ∥ σ ) = ( 1 / α - 1 ) log 2 { Tr [ ( σ ( 1 − α / 2 α ) ρ σ ( 1 − α / 2 α ) ) α ] } , where α ∈ ( 0 , 1 ) ∪ ( 1 , ∞ ) [49] . For α = 0 , 1 , ∞ , the Rényi divergence is defined by taking the limit of α → 0 , 1 , ∞ , respectively. Then, quantum coherence measures can be defined by C α ( ρ ) = min δ ∈ S D ˜ α ( ρ ∥ δ ) , where S ⊂ D ( H d ) and D ( H d ) denotes the set of density matrices in H d [50] . First we let S = I . In the limit of α → 1 , we recover the relative entropy of coherence C r ( ρ ) = min δ ∈ I S ( ρ ∥ δ ) . Here, S ( ρ ∥ δ ) = Tr [ ρ log 2 ρ ] - Tr [ ρ log 2 δ ] is the quantum relative entropy and the minimization is over all incoherent states. When α → ∞ , we have the max-relative entropy D max ( ρ ∥ σ ) = lim α → ∞ D ˜ α ( ρ ∥ σ ) = log 2 min { λ | ρ ≤ λ σ } . Then we introduce our first coherence monotone by C max ( ρ ) = min δ ∈ I D max ( ρ ∥ δ ) . (1) With the properties of the max-relative entropy D max ( ρ ∥ σ ) [51] , we can verify that C max ( ρ ) satisfies (C1)–(C2). In addition, we show that it is quasi-convex, i.e., C max ( ∑ i p i ρ i ) ≤ max i C max ( ρ i ) . Theorem 1.— The quantity C max ( ρ ) is a coherence monotone under MIO [47] and it is quasiconvex. We next consider a different set, A ρ = { 1 t [ ( 1 + t ) Δ ( ρ ) - ρ ] | t > 0 , ( 1 + t ) Δ ( ρ ) - ρ ≥ 0 } , and let A ¯ ρ denote its closure. In particular, Δ ( ρ ) ∈ A ¯ ρ by taking the limit t → ∞ . Analogous to Eq. (1) , we define C Δ , max = min σ ∈ A ¯ ρ D max ( ρ ∥ σ ) . The quantity C Δ , max was originally introduced in Ref. [47] and shown to have the simplified form C Δ , max ( ρ ) = log 2 min { λ | ρ ≤ λ Δ ( ρ ) } . (2) Theorem 2.— C Δ , max is a coherence monotone under DIO [47] and it is quasiconvex. Alternatively, another way of defining coherence measure is via the convex-roof construction. For instance, the coherence of formation C f ( ρ ) can be defined by C f ( ρ ) ≡ min { p j , | ψ j ⟩ } ∑ j p j S ( Δ ( | ψ j ⟩ ⟨ ψ j | ) ) [6,15] . Here, S ( ρ ) = - Tr [ ρ log 2 ρ ] is the Von-Neumann entropy, and the minimization is over all possible pure-state decompositions of ρ = ∑ j p j | ψ j ⟩ ⟨ ψ j | . Now, suppose | ψ j ⟩ = ∑ i = 0 d - 1 a i j | i ⟩ and denote T j to be the number of nonzero elements in { a 0 j , … , a d - 1 , j } . We introduce our third coherence monotone C 0 ( ρ ) , C 0 ( ρ ) = min { p j , | ψ j ⟩ } max j log 2 T j . (3) Under this convex-roof construction, we show that C 0 ( ρ ) is a coherence monotone. Nevertheless, it does not satisfy the convexity requirement (C3). Theorem 3.— C 0 ( ρ ) is a coherence monotone under IO and it violates the convexity requirement (C3). One-shot dilution .—With the three coherence monotones, we are now ready to consider the process of coherence dilution which converts maximally coherent states into a target state. The canonical maximally coherent state of dimension M is given by | Ψ M ⟩ = ( 1 / M ) ∑ i = 0 M - 1 | i ⟩ [16] . One-shot coherence cost measures the minimal length M such that | Ψ M ⟩ can be converted into a target state ρ via incoherent operations within some finite error. Based on different definitions of incoherent operations, we define different one-shot coherence costs. Definition 1.— Let O ∈ { MIO , DIO , IO , SIO } denote a class of incoherent operations. Then for a given state ρ and ϵ ≥ 0 , the one-shot coherence cost under O is defined by C O ϵ ( ρ ) = min Λ ∈ O { log 2 M | F [ ρ , Λ ( Ψ M ) ] ≥ 1 - ϵ } , (4) where F ( ρ , σ ) = ( Tr [ ρ σ ρ ] ) 2 is the fidelity measure between two states ρ and σ . In the definition, there is a “smoothing” parameter ϵ in the dilution process, which is also known as the failure probability in cryptography [52] . Instead of obtaining the exact state ρ , we allow the final state to be deviated no more than ϵ from ρ , where the deviation is measured in fidelity. Specifically, when ϵ = 0 , it becomes the case of perfect state conversion. As shown in Fig. 1 , since SIO ⊂ IO ⊂ MIO and SIO ⊂ DIO ⊂ MIO , we have C MIO ϵ ( ρ ) ≤ C IO ϵ ( ρ ) ≤ C SIO ϵ ( ρ ) and C MIO ϵ ( ρ ) ≤ C DIO ϵ ( ρ ) ≤ C SIO ϵ ( ρ ) in general. However, IO and DIO are incomparable operations and we thus cannot derive a relationship between C IO ϵ ( ρ ) and C DIO ϵ ( ρ ) directly from the definitions. Nevertheless, the hierarchy C DIO ϵ ( ρ ) ≤ C IO ϵ ( ρ ) can be established in the asymptotic case according to the following theorems. To characterize coherence cost with certain error ϵ , we apply a smoothing to a general coherence measure C ( ρ ) by minimizing over states ρ ′ that satisfy F ( ρ , ρ ′ ) ≥ 1 - ϵ , C ϵ ( ρ ) = min ρ ′ : F ( ρ , ρ ′ ) ≥ 1 - ϵ C ( ρ ′ ) . (5) We show that the one-shot coherence cost under MIO is bounded by the smoothed coherence measure C max ϵ ( ρ ) , Theorem 4.— For any state ρ and ϵ ≥ 0 C max ϵ ( ρ ) ≤ C MIO ϵ ( ρ ) ≤ C max ϵ ( ρ ) + 1. (6) Similarly, the one-shot coherence cost using DIO is bounded by the smoothed coherence measure C Δ , max ϵ ( ρ ) , Theorem 5.— For any state ρ and ϵ ≥ 0 , C Δ , max ϵ ( ρ ) ≤ C DIO ϵ ( ρ ) ≤ C Δ , max ϵ ( ρ ) + 1 . (7) Finally, the one-shot coherence cost under IO and SIO is exactly characterized by the smoothed coherence measure C 0 ϵ ( ρ ) , Theorem 6.— For any state ρ and ϵ ≥ 0 C IO ϵ ( ρ ) = C SIO ϵ ( ρ ) = C 0 ϵ ( ρ ) . (8) The main proof idea of the theorems is to firstly prove the lower bound of the one-shot coherence cost by exploiting the monotonicity property of coherence measures. Then, the next step is to explicitly construct an incoherent operation such that this lower bound is saturated. We leave the detailed proofs and the explicit transformations in the Supplemental Material [48] . Asymptotic case .—Our one-shot coherence cost results hold for any state and any smooth parameter ϵ . As a special case, we can consider the asymptotic coherence dilution with an infinitely large number of IID target states. We define the regularized coherence cost by taking the limit n → ∞ and ϵ → 0 + : C O ∞ ( ρ ) = lim ϵ → 0 + lim n → ∞ 1 n C O ϵ ( ρ ⊗ n ) , (9) where O ∈ { MIO , IO , SIO , DIO } . Following the results of one-shot coherence dilution, we obtain the coherence cost in the asymptotic case. Theorem 7.— For any state ρ , the asymptotic coherence cost under MIO is quantified by C MIO ∞ ( ρ ) = C r ( ρ ) . (10) Combining this with the work of Winter and Yang [24] , we see that both the asymptotic coherence cost under MIO and the distillable coherence under IO is given by the relative entropy of coherence C r ( ρ ) . Since MIO is more powerful than IO, it follows that the distillable coherence under MIO is also characterized by C r ( ρ ) . One can also see this by noting that the converse proof for distillable coherence given in Ref. [24] also holds for MIO. Thus, coherence is asymptotically reversible under MIO. This is a slight strengthening of the general result presented in Ref. [53] which implies reversibility by MIO when an asymptotically small amount of coherence can be generated. Interestingly, we find that C DIO ∞ ( ρ ) = C r ( ρ ) , which is rather surprising since C MIO 0 ( ρ ) ≤ C DIO 0 ( ρ ) , with the inequality being strict in many cases. Yet, evidently MIO and DIO yield the same coherence dilution rate in the asymptotic case, i.e., C r ( ρ ) = lim ϵ → 0 + lim n → ∞ 1 n C MIO ϵ ( ρ ⊗ n ) = lim ϵ → 0 + lim n → ∞ 1 n C DIO ϵ ( ρ ⊗ n ) . (11) The proof of this fact will be presented in a separate paper as it employs techniques quite different from the ones used in this work. Theorem 8.— For any state ρ , the asymptotic coherence cost under IO and SIO is quantified by C IO ∞ ( ρ ) = C SIO ∞ ( ρ ) = C f ( ρ ) . (12) The asymptotic coherence dilution under IO and SIO has been investigated by Winter and Yang [24] . The coherence cost is characterized by the coherence of formation C f ( ρ ) , which is consistent with our result. Note that, although the problems of one-shot and asymptotic coherence dilutions are similar, the methods are different. Our method holds for any state and any ϵ while the method by Winter and Yang holds only for infinite copies of the same state and the limit with ϵ → 0 + . Furthermore, the definition in Eq. (9) can also be generalized to C O ∞ , ϵ ( ρ ) = lim n → ∞ 1 n C O ϵ ( ρ ⊗ n ) ϵ ∈ ( 0 , 1 ) . (13) By applying the property of the quantum asymptotic equipartition [29,54,55] , we may also obtain the same results in Theorem 7 and 8. It would be interesting to propose in future work, the generalized asymptotic coherence dilution with a finite smooth parameter ϵ and relate it to the corresponding coherence monotone. Discussion .—Our work solves the open problem of one-shot coherence dilution [17] and derives the conventional coherence dilution formula in the asymptotic limit. Our results also indicate that coherence is asymptotically reversible under MIO and DIO. According to recent investigations of the resource theory of coherence [17] , our results also shed light on the role of coherence as a resource in quantum information processing tasks like random number generation [6,7] and cryptography [4] . In the asymptotic scenario, the distillable coherence and coherence cost are additive, i.e., C f ( ρ 1 ⊗ ρ 2 ) = C f ( ρ 1 ) + C f ( ρ 2 ) and C r ( ρ 1 ⊗ ρ 2 ) = C r ( ρ 1 ) + C r ( ρ 2 ) [24] . In contrast, the proposed one-shot coherence monotones do not satisfy the additivity property in general. For example, one can show that C 0 ϵ ( ρ ⊗ n ) ≠ C 0 ϵ ( ρ ) + C 0 ϵ ( ρ ⊗ n - 1 ) when n → ∞ and ϵ → 0 + . Furthermore, given ϵ ≥ ϵ 1 + ϵ 2 , we can derive that C 0 ϵ ( ρ 1 ⊗ ρ 2 ) ≤ C 0 ϵ 1 ( ρ 1 ) + C 0 ϵ 2 ( ρ 2 ) . The inequality also holds for C max ϵ and C Δ , max ϵ with proofs shown in Sec. VIII of the Supplemental Material [48] . An interesting future direction is to study general and tight additive inequalities for these coherence monotones. Another interesting perspective is to quantify the one-shot coherence distillation under different incoherent operations. The essential problem is to find the conversion from a general mixed state to the maximally coherent state. Some interesting results have been obtained [56] but the general results still remain to be solved. Furthermore, due to the strong similarity, we also expect that our result can shed light on the one-shot coherence conversion under thermal operations in the thermodynamic scenario, which has been partially solved only for qubits recently [11] . We acknowledge F. Buscemi, T. Peng, and A. Winter for the insightful discussions. This work was supported by the National Key R&D Program of China Grants No. 2017YFA0303900 and No. 2017YFA0304004, the National Natural Science Foundation of China Grant No. 11674193, and the National Science Foundation (NSF) Early CAREER Award No. 1352326. Q. Z., Y. L., and X. Y. contributed equally to this Letter. [1] 1 Y. Aharonov and L. Susskind , Phys. Rev. 155 , 1428 ( 1967 ). PHRVAO 0031-899X 10.1103/PhysRev.155.1428 [2] 2 A. Kitaev , D. Mayers , and J. Preskill , Phys. Rev. A 69 , 052326 ( 2004 ). PLRAAN 1050-2947 10.1103/PhysRevA.69.052326 [3] 3 S. D. Bartlett , T. Rudolph , and R. W. Spekkens , Rev. Mod. Phys. 79 , 555 ( 2007 ). RMPHAT 0034-6861 10.1103/RevModPhys.79.555 [4] 4 P. J. Coles , E. M. Metodiev , and N. Lütkenhaus , Nat. Commun. 7 , 11712 ( 2016 ). NCAOBW 2041-1723 10.1038/ncomms11712 [5] 5 V. Giovannetti , S. Lloyd , and L. Maccone , Nat. Photonics 5 , 222 ( 2011 ). NPAHBY 1749-4885 10.1038/nphoton.2011.35 [6] 6 X. Yuan , H. Zhou , Z. Cao , and X. Ma , Phys. Rev. A 92 , 022124 ( 2015 ). PLRAAN 1050-2947 10.1103/PhysRevA.92.022124 [7] 7 J. Ma , X. Yuan , A. Hakande , and X. Ma , arXiv:1704.06915 . [8] 8 M. B. Plenio and S. F. Huelga , New J. Phys. 10 , 113019 ( 2008 ). NJOPFM 1367-2630 10.1088/1367-2630/10/11/113019 [9] 9 P. Rebentrost , M. Mohseni , and A. Aspuru-Guzik , J. Phys. Chem. B 113 , 9942 ( 2009 ). JPCBFK 1520-6106 10.1021/jp901724d [10] 10 J. Åberg , Phys. Rev. Lett. 113 , 150402 ( 2014 ). PRLTAO 0031-9007 10.1103/PhysRevLett.113.150402 [11] 11 P. Ćwikliński , M. Studziński , M. Horodecki , and J. Oppenheim , Phys. Rev. Lett. 115 , 210403 ( 2015 ). PRLTAO 0031-9007 10.1103/PhysRevLett.115.210403 [12] 12 M. Lostaglio , K. Korzekwa , D. Jennings , and T. Rudolph , Phys. Rev. X 5 , 021001 ( 2015 ). PRXHAE 2160-3308 10.1103/PhysRevX.5.021001 [13] 13 M. Lostaglio , D. Jennings , and T. Rudolph , Nat. Commun. 6 , 6383 ( 2015 ). NCAOBW 2041-1723 10.1038/ncomms7383 [14] 14 V. Narasimhachar and G. Gour , Nat. Commun. 6 , 7689 ( 2015 ). NCAOBW 2041-1723 10.1038/ncomms8689 [15] 15 J. Aberg , arXiv:quant-ph/0612146 . [16] 16 T. Baumgratz , M. Cramer , and M. B. Plenio , Phys. Rev. Lett. 113 , 140401 ( 2014 ). PRLTAO 0031-9007 10.1103/PhysRevLett.113.140401 [17] 17 A. Streltsov , G. Adesso , and M. B. Plenio , Rev. Mod. Phys. 89 , 041003 ( 2017 ). RMPHAT 0034-6861 10.1103/RevModPhys.89.041003 [18] 18 C. H. Bennett , D. P. DiVincenzo , J. A. Smolin , and W. K. Wootters , Phys. Rev. A 54 , 3824 ( 1996 ). PLRAAN 1050-2947 10.1103/PhysRevA.54.3824 [19] 19 M. B. Plenio and S. Virmani , arXiv:quant-ph/0504163 . [20] 20 R. Horodecki , P. Horodecki , M. Horodecki , and K. Horodecki , Rev. Mod. Phys. 81 , 865 ( 2009 ). RMPHAT 0034-6861 10.1103/RevModPhys.81.865 [21] 21 V. Vedral , M. B. Plenio , M. A. Rippin , and P. L. Knight , Phys. Rev. Lett. 78 , 2275 ( 1997 ). PRLTAO 0031-9007 10.1103/PhysRevLett.78.2275 [22] 22 E. Chitambar and G. Gour , Phys. Rev. Lett. 117 , 030401 ( 2016 ). PRLTAO 0031-9007 10.1103/PhysRevLett.117.030401 [23] 23 I. Marvian and R. W. Spekkens , Phys. Rev. A 94 , 052324 ( 2016 ). PLRAAN 2469-9926 10.1103/PhysRevA.94.052324 [24] 24 A. Winter and D. Yang , Phys. Rev. Lett. 116 , 120404 ( 2016 ). PRLTAO 0031-9007 10.1103/PhysRevLett.116.120404 [25] 25 C. Napoli , T. R. Bromley , M. Cianciaruso , M. Piani , N. Johnston , and G. Adesso , Phys. Rev. Lett. 116 , 150502 ( 2016 ). PRLTAO 0031-9007 10.1103/PhysRevLett.116.150502 [26] The unit resource | Ψ ⟩ is an EPR pair or a maximally coherent state in the resource theories of entanglement and coherence, respectively. [27] 27 E. M. Rains , IEEE Trans. Inf. Theory 47 , 2921 ( 2001 ). IETTAW 0018-9448 10.1109/18.959270 [28] 28 P. M. Hayden , M. Horodecki , and B. M. Terhal , J. Phys. A 34 , 6891 ( 2001 ). JPHAC5 0305-4470 10.1088/0305-4470/34/35/314 [29] 29 M. Tomamichel , arXiv:1203.2142 . [30] 30 U. Vazirani and T. Vidick , Phil. Trans. R. Soc. A 370 , 3432 ( 2012 ). PTRMAD 1364-503X 10.1098/rsta.2011.0336 [31] 31 C. A. Miller and Y. Shi , in Proceedings of the Forty-sixth Annual ACM Symposium on Theory of Computing , STOC ’14 ( ACM , New York, NY, USA, 2014 ), pp. 417–426 . [32] 32 J. Åberg , Nat. Commun. 4 , 1925 ( 2013 ). NCAOBW 2041-1723 10.1038/ncomms2712 [33] 33 P. Skrzypczyk , A. J. Short , and S. Popescu , Nat. Commun. 5 , 4185 ( 2014 ). NCAOBW 2041-1723 10.1038/ncomms5185 [34] 34 M. Horodecki and J. Oppenheim , Nat. Commun. 4 , 2059 ( 2013 ). NCAOBW 2041-1723 10.1038/ncomms3059 [35] 35 F. Brandão , M. Horodecki , N. Ng , J. Oppenheim , and S. Wehner , Proc. Natl. Acad. Sci. U.S.A. 112 , 3275 ( 2015 ). PNASA6 0027-8424 10.1073/pnas.1411728112 [36] 36 R. Renner , S. Wolf , and J. Wullschleger , in 2006 IEEE International Symposium on Information Theory, Seattle, WA, USA ( IEEE , 2006 ), pp. 1424–1427 . [37] 37 M. Mosonyi and N. Datta , J. Math. Phys. 50 , 072104 ( 2009 ). JMAPAQ 0022-2488 10.1063/1.3167288 [38] 38 F. Buscemi and N. Datta , IEEE Trans. Inf. Theory 56 , 1447 ( 2010 ). IETTAW 0018-9448 10.1109/TIT.2009.2039166 [39] 39 L. Wang and R. Renner , Phys. Rev. Lett. 108 , 200501 ( 2012 ). PRLTAO 0031-9007 10.1103/PhysRevLett.108.200501 [40] 40 G. Gour , Phys. Rev. A 95 , 062314 ( 2017 ). PLRAAN 2469-9926 10.1103/PhysRevA.95.062314 [41] 41 N. Datta , IEEE Trans. Inf. Theory 55 , 2816 ( 2009 ). IETTAW 0018-9448 10.1109/TIT.2009.2018325 [42] 42 F. Buscemi and N. Datta , J. Math. Phys. 51 , 102201 ( 2010 ). JMAPAQ 0022-2488 10.1063/1.3483717 [43] 43 F. Buscemi and N. Datta , Phys. Rev. Lett. 106 , 130503 ( 2011 ). PRLTAO 0031-9007 10.1103/PhysRevLett.106.130503 [44] 44 F. G. S. L. Brandao and N. Datta , IEEE Trans. Inf. Theory 57 , 1754 ( 2011 ). IETTAW 0018-9448 10.1109/TIT.2011.2104531 [45] 45 S. Du , Z. Bai , and Y. Guo , Phys. Rev. A 91 , 052120 ( 2015 ). PLRAAN 1050-2947 10.1103/PhysRevA.91.052120 [46] 46 H. Zhu , Z. Ma , Z. Cao , S.-M. Fei , and V. Vedral , Phys. Rev. A 96 , 032316 ( 2017 ). PLRAAN 2469-9926 10.1103/PhysRevA.96.032316 [47] 47 E. Chitambar and G. Gour , Phys. Rev. A 94 , 052336 ( 2016 ). PLRAAN 2469-9926 10.1103/PhysRevA.94.052336 [48] 48 See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.120.070403 for detailed mathematical derivation. [49] 49 M. Mller-Lennert , F. Dupuis , O. Szehr , S. Fehr , and M. Tomamichel , J. Math. Phys. 54 , 122203 ( 2013 ). JMAPAQ 0022-2488 10.1063/1.4838856 [50] 50 A. E. Rastegin , Phys. Rev. A 93 , 032136 ( 2016 ). PLRAAN 2469-9926 10.1103/PhysRevA.93.032136 [51] 51 N. Datta , IEEE Trans. Inf. Theory 55 , 2816 ( 2009 ). IETTAW 0018-9448 10.1109/TIT.2009.2018325 [52] 52 R. Renner , Ph.D. Thesis, ETH Zurich , 2005 . [53] 53 F. G. S. L. Brandão and G. Gour , Phys. Rev. Lett. 115 , 070503 ( 2015 ). PRLTAO 0031-9007 10.1103/PhysRevLett.115.070503 [54] 54 M. Tomamichel , R. Colbeck , and R. Renner , IEEE Trans. Inf. Theory 55 , 5840 ( 2009 ). IETTAW 0018-9448 10.1109/TIT.2009.2032797 [55] 55 F. Dupuis , O. Fawzi , and R. Renner , arXiv:1607.01796 . [56] 56 B. Regula , K. Fang , X. Wang , and G. Adesso , arXiv:1711.10512 . Publisher Copyright: © 2018 American Physical Society.

PY - 2018/2/13

Y1 - 2018/2/13

N2 - Manipulation and quantification of quantum resources are fundamental problems in quantum physics. In the asymptotic limit, coherence distillation and dilution have been proposed by manipulating infinite identical copies of states. In the nonasymptotic setting, finite data-size effects emerge, and the practically relevant problem of coherence manipulation using finite resources has been left open. This Letter establishes the one-shot theory of coherence dilution, which involves converting maximally coherent states into an arbitrary quantum state using maximally incoherent operations, dephasing-covariant incoherent operations, incoherent operations, or strictly incoherent operations. We introduce several coherence monotones with concrete operational interpretations that estimate the one-shot coherence cost - the minimum amount of maximally coherent states needed for faithful coherence dilution. Furthermore, we derive the asymptotic coherence dilution results with maximally incoherent operations, incoherent operations, and strictly incoherent operations as special cases. Our result can be applied in the analyses of quantum information processing tasks that exploit coherence as resources, such as quantum key distribution and random number generation.

AB - Manipulation and quantification of quantum resources are fundamental problems in quantum physics. In the asymptotic limit, coherence distillation and dilution have been proposed by manipulating infinite identical copies of states. In the nonasymptotic setting, finite data-size effects emerge, and the practically relevant problem of coherence manipulation using finite resources has been left open. This Letter establishes the one-shot theory of coherence dilution, which involves converting maximally coherent states into an arbitrary quantum state using maximally incoherent operations, dephasing-covariant incoherent operations, incoherent operations, or strictly incoherent operations. We introduce several coherence monotones with concrete operational interpretations that estimate the one-shot coherence cost - the minimum amount of maximally coherent states needed for faithful coherence dilution. Furthermore, we derive the asymptotic coherence dilution results with maximally incoherent operations, incoherent operations, and strictly incoherent operations as special cases. Our result can be applied in the analyses of quantum information processing tasks that exploit coherence as resources, such as quantum key distribution and random number generation.

UR - http://www.scopus.com/inward/record.url?scp=85042168812&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85042168812&partnerID=8YFLogxK

U2 - 10.1103/PhysRevLett.120.070403

DO - 10.1103/PhysRevLett.120.070403

M3 - Article

C2 - 29542962

AN - SCOPUS:85042168812

SN - 0031-9007

VL - 120

JO - Physical review letters

JF - Physical review letters

IS - 7

M1 - 070403

ER -