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Quantum coherence is an essential ingredient in quantum information processing and plays an important role in quantum computation. Therefore, it is a hot issue about how to quantify the coherence of quantum states in theoretical framework. The coherence effect of a state is usually described by the off-diagonal elements of its density matrix with respect to a particular reference basis. Recently, based on the established notions from quantitative theory of entanglement, a resource theory of coherence quantification has been proposed[1,2]. In the theory framework, a proper measure of coherence should satisfy three criteria: the coherence should be zero for all incoherent state; the coherence should not increase under mixing quantum states; the coherence should not increase under incoherent operations. Then, a number of coherence measures have been suggested, such as l1 norm of coherence and the relative entropy of coherence[2]. Wigner function is known as an important tool to study the non-classical property of quantum states for continuous-variable quantum systems. It has been generalized to finite-dimensional Hilbert spaces, and named as discrete Wigner function[9-16]. The magic property of quantum states, which promotes stabilizer computation to universal quantum computation, can be generally measured by the absolute sum of the negative items (negativity sum) in the discrete Wigner function of the observed quantum states. In this paper we investigate quantum coherence from the view of discrete Wigner function. From the definition of the discrete Wigner function of the quantum systems with odd prime dimensions, for a given density matrix we analyze in phase space the performance of its diagonal and off-diagonal items. We find that, the discrete Wigner function of a quantum state contains two aspects: the true quantum coherence and the classical mixture, where the part of classical mixture can be excluded by only considering the discrete Wigner function of the diagonal items of the density matrix. Thus, we propose a possible measure method for quantum coherence from the discrete Wigner function of the off-diagonal items of the density matrix. We show that the proposed measure method satisfies the criteria (C1) and (C2) of coherence measure perfectly. For the criteria (C3), we give a numerical proof in three-dimensional quantum system. Meanwhile, we compare the proposed coherence measure with l1 norm coherence, and get an inequality relationship between them. Finally, an inequality is obtained to discuss the relation between quantum coherence and the negativity sum of discrete Wigner function, which shows that the quantum coherence is only necessary but not sufficient for quantum computation speed-up.
[1] Aberg J 2006 arXiv:quant-ph/0612146v1
[2] Baumgratz T, Cramer M, Plenio M B 2014 Phys. Rev. Lett. 113 140401
[3] Girolami D 2014 Phys. Rev. Lett. 113 170401
[4] Streltsov A, Singh U, Dhar H S, Bera M N, Adesso G 2015 Phys. Rev. Lett. 115 020403
[5] Yuan X, Zhou H Y, Cao Z, Ma X F 2015 Phys. Rev. A 92 022124
[6] Shao L H, Xi Z J, Fan H, Li Y M 2015 Phys. Rev. A 91 042120
[7] Xi Z J, Li Y M, Fan H 2015 Sci. Rep. 5 10922
[8] Yao Y, Xiao X, Ge L, Sun C P 2015 Phys. Rev. A 92 022112
[9] Wootters W K 1987 Ann. Phys. 176 1
[10] Gibbons K S, Hoffman M J, Wootters W K 2004 Phys. Rev. A 70 062101
[11] Cormick C, Galvao E F, Gottesman D, Paz J P, Pittenger A O 2006 Phys. Rev. A 73 012301
[12] Galvao E F 2005 Phys. Rev. A 71 042302
[13] Buot F A 1974 Phys. Rev. B 10 3700
[14] Gross D 2006 J. Math. Phys. 47 122107
[15] Baron T 2009 EPL 88 10002
[16] Zhu H J 2016 Phys. Rev. Lett. 116 040501
[17] Veitch V, Ferrie C, Gross D, Emerson J 2012 New J. Phys. 14 113011
[18] Veitch V, Mousavian S A H, Gottesman D, Emerson J 2014 New J. Phys. 16 013009
[19] Galvao E F 2005 Phys. Rev. A 71 042302
[20] Mari A, Eisert J 2012 Phys. Rev. Lett. 109 230503
[21] Pashayan H, Wallman J J, Bartlett S D 2015 Phys. Rev. Lett. 115 070501
[22] Zhang Z M 2015 Quantum Optics (Beijing: Science Press) pp111-116 (in Chinese) [张智明 2015 量子光学 (北京: 科学出版社) 第111-116页]
[23] Vedral V, Plenio M B 1998 Phys. Rev. A 57 1619
[24] Plenio M B, Virmani S 2007 Quantum Inf. Comput. 7 1
[25] Vedral V, Plenio M B, Rippin M A, Knight P L 1997 Phys. Rev. Lett. 78 2275
[26] Lee C W, Jeong H 2011 Phys. Rev. Lett. 106 220401
[27] Cormick C, Paz J P 2006 Phys. Rev. A 74 062315
[28] Thew R T, Nemoto K, White A G, Munro W J 2002 Phys. Rev. A 66 012303
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[1] Aberg J 2006 arXiv:quant-ph/0612146v1
[2] Baumgratz T, Cramer M, Plenio M B 2014 Phys. Rev. Lett. 113 140401
[3] Girolami D 2014 Phys. Rev. Lett. 113 170401
[4] Streltsov A, Singh U, Dhar H S, Bera M N, Adesso G 2015 Phys. Rev. Lett. 115 020403
[5] Yuan X, Zhou H Y, Cao Z, Ma X F 2015 Phys. Rev. A 92 022124
[6] Shao L H, Xi Z J, Fan H, Li Y M 2015 Phys. Rev. A 91 042120
[7] Xi Z J, Li Y M, Fan H 2015 Sci. Rep. 5 10922
[8] Yao Y, Xiao X, Ge L, Sun C P 2015 Phys. Rev. A 92 022112
[9] Wootters W K 1987 Ann. Phys. 176 1
[10] Gibbons K S, Hoffman M J, Wootters W K 2004 Phys. Rev. A 70 062101
[11] Cormick C, Galvao E F, Gottesman D, Paz J P, Pittenger A O 2006 Phys. Rev. A 73 012301
[12] Galvao E F 2005 Phys. Rev. A 71 042302
[13] Buot F A 1974 Phys. Rev. B 10 3700
[14] Gross D 2006 J. Math. Phys. 47 122107
[15] Baron T 2009 EPL 88 10002
[16] Zhu H J 2016 Phys. Rev. Lett. 116 040501
[17] Veitch V, Ferrie C, Gross D, Emerson J 2012 New J. Phys. 14 113011
[18] Veitch V, Mousavian S A H, Gottesman D, Emerson J 2014 New J. Phys. 16 013009
[19] Galvao E F 2005 Phys. Rev. A 71 042302
[20] Mari A, Eisert J 2012 Phys. Rev. Lett. 109 230503
[21] Pashayan H, Wallman J J, Bartlett S D 2015 Phys. Rev. Lett. 115 070501
[22] Zhang Z M 2015 Quantum Optics (Beijing: Science Press) pp111-116 (in Chinese) [张智明 2015 量子光学 (北京: 科学出版社) 第111-116页]
[23] Vedral V, Plenio M B 1998 Phys. Rev. A 57 1619
[24] Plenio M B, Virmani S 2007 Quantum Inf. Comput. 7 1
[25] Vedral V, Plenio M B, Rippin M A, Knight P L 1997 Phys. Rev. Lett. 78 2275
[26] Lee C W, Jeong H 2011 Phys. Rev. Lett. 106 220401
[27] Cormick C, Paz J P 2006 Phys. Rev. A 74 062315
[28] Thew R T, Nemoto K, White A G, Munro W J 2002 Phys. Rev. A 66 012303
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