突然看到这篇文章。对重测序的RIL来进行QTL定位比较感兴趣。作个记号 Plant Cell Rep. 2013 Jan;32(1):103-16. doi: 10.1007/s00299-012-1345-6. Epub 2012 Oct 12. Identification of QTLs associated with tissue culture response through sequencing-based genotyping of RILs derived from 93-11 × Nipponbare in rice (Oryza sativa). Li S, Yan S, Wang AH, Zou G, Huang X, Han B, Qian Q, Tao Y. SourceThe College of Agriculture and Biotechnology, Zhejiang University, 388 Yuhangtang Road, Hangzhou, 310058, China, sujuanli2001@163.com . Abstract KEY MESSAGE : The performance of callus induction and callus differentiation was evaluated by 9 indices for 140 RILs; 2 major QTLs associated with plant regeneration were identified. In order to investigate the genetic mechanisms of tissue culture response, 140 recombinant inbred lines (RILs) derived from 93-11 (Oryza sativa ssp. indica) × Nipponbare (Oryza sativa ssp. japonica) and a high quality genetic map based on the SNPs generated from deep sequencing of the RIL genomes, were used to identify the quantitative trait loci (QTLs) associated with in vitro tissue culture response (TCR) from mature seed in rice. The performance of callus induction was evaluated by indices of induced-callus color (ICC), induced-callus size (ICS), induced-callus friability (ICF) and callus induction rate (CIR), respectively, and the performance of callus differentiation was evaluated by indices of callus proliferation ability (CPA), callus browning tendency (CBT), callus greening ability (CGA), the average number of regenerated shoots per callus (NRS) and regeneration rate (%, RR), respectively. A total of 25 QTLs, 2 each for ICC, ICS, ICF, CIR and CBA, 3 for CPA, 4 each for CGA, NRS and RR, respectively, were detected and located on 8 rice chromosomes. Significant correlations were observed among the traits of CGA, NRS and RR, and QTLs identified for these three indices were co-located on chromosomes 3 and 7, and the additive effects came from both Nipponbare and 93-11, respectively. The results obtained from this study provide guidance for further fine mapping and gene cloning of the major QTL of TCR and the knowledge of the genes underlying the traits investigated would be very helpful for revealing the molecular bases of tissue culture response. 。。。。。。。。。。。。。。。。。。。。。。。 Mapping 49 quantitative trait loci at high resolution through sequencing-based genotyping of rice recombinant inbred lines. Wang L, Wang A, Huang X, Zhao Q, Dong G, Qian Q, Sang T, Han B. Theor Appl Genet . 2011 Feb;122(2):327-40. doi: 10.1007/s00122-010-1449-8. Epub 2010 Sep 28.
arXiv:0710.0956v3 28 Feb 2008 Second Law of Thermodynamics with Discrete Quantum Feedback Control Takahiro Sagawa1 and Masahito Ueda1,2 1Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan 2ERATO Macroscopic Quantum Control Project, JST, 2-11-16 Yayoi, Bunkyo-ku, Tokyo 113-8656, Japan (Dated: February 28, 2008) A new thermodynamic inequality is derived which leads to the maximum work that can be extracted from multi-heat baths with the assistance of discrete quantum feedback control. The maximum work is determined by the free-energy difference and a generalized mutual information content between the thermodynamic system and the feedback controller. This maximum work can exceed that in conventional thermodynamics and, in the case of a heat cycle with two heat baths, the heat efficiency can be greater than that of the Carnot cycle. The consistency of our results with the second law of thermodynamics is ensured by the fact that work is needed for information processing of the feedback controller. PACS numbers: 03.67.-a,05.70.Ln,05.30.-d,03.65.Ta Among a large number of studies conducted on the relationship between thermodynamics and information processing , particularly provoking is the work by Szilard who argued that positive work Wext can be extracted from an isothermal cycle if Maxwell’s demon plays the role of a feedback controller . It is now well understood that the role of the demon does not contradict the second law of thermodynamics, because the initialization of the demon’s memory entails heat dissipation . We note that, in the case of an isothermal process, the second law of thermodynamics can be expressed as Wext ≤ −FS, (1) where FS is the difference in the Helmholtz free energy between the initial and final thermodynamic equilibrium states. In a different context, quantum feedback control has attracted considerable attention for controlling and stabilizing a quantum system . It can be applied, for example, to squeezing an electromagnetic field , spin squeezing , and stabilizing macroscopic coherence . While the theoretical framework of quantum feedback control as a stochastic dynamic system is well developed, the possible thermodynamic gain of quantum feedback control has yet to be fully understood. In this Letter, we derive a new thermodynamic inequality which sets the fundamental limit on the work that can be extracted from multi-heat baths with discrete quantum feedback control , consisting of quantum measurement and a mechanical operation depending on the measurement outcome. The maximum work is characterized by a generalized mutual information content between the thermodynamic system and the feedback controller. We shall refer to this as the QC-mutual information content, where QC indicates that the measured system is quantal and that the measurement outcome is classical. The QC-mutual information content reduces to the classical mutual information content in the case of classical measurement. In the absence of feedback control, the new inequality (12) reduces to the Clausius inequality. In the case of an isothermal process, its upper bound exceeds that of inequality (1) by an amount proportional to the QC-mutual information content. We consider a thermodynamic process for system S which can contact heat baths B1, B2, · · · , Bn at respective temperatures T1, T2, · · · , Tn. We assume that system S is in thermodynamic equilibrium in the initial and final states. For simplicity, we also assume that the initial and final temperature of S is given by T ≡ (kB )−1. This can be realized by contacting S with, for example, B1 in the preparation of the initial state and during equilibration to the final state; in this case T = T1. We do not, however, assume that the system is in thermodynamic equilibrium between the initial and final states. We assume that system S and heat baths Bm are as a whole isolated and that they only come into contact with some external mechanical systems and the feedback controller. Apart from the feedback controller, the total Hamiltonian can be written as ˆH (t) = ˆH S(t) + Xn m=1 ( ˆH SBm(t) + ˆHBm), (2) where ˆH SBm(t) is the interaction Hamiltonian between system S and heat bath Bm. The Hamiltonian ˆH S(t) describes a mechanical operation on S through such external parameters as an applied magnetic field or volume of the gas, and the Hamiltonian ˆH SBm(t) describes, for example, the attachment (detachment) of an adiabatic wall or Bm to (from) S. We consider a time evolution from ti to tf , assume ˆH SBm(ti) = ˆH SBm(tf ) = 0 for all m, and write ˆH S(ti) = ˆH S i and ˆH S(tf ) = ˆH S f . The time 2 evolution of the total system with discrete quantum feedback control can be divided into the following five stages: Stage 1 (Initial state) At time ti, the initial state of S and that of Bm are in thermodynamic equilibrium at temperatures T and Tm, respectively. We assume that the density operator of the entire state is given by the canonical distribution ˆi = exp(− ˆHS i ) ZS i ⊗ exp(− 1 ˆH B1 ) ZB1 ⊗· · ·⊗ exp(− n ˆH Bn) ZBn , (3) where m ≡ (kBTm)−1 (m = 1, 2, · · · , n), ZS i ≡ tr{exp(− ˆH S i )}, and ZBm ≡ tr{exp(− m ˆH Bm)}. We denote the Helmholtz free energy of system S as FS i ≡ −kBT lnZS i . Stage 2 (Unitary evolution) From ti to t1, the entire system undergoes unitary evolution ˆUi = Texp R t1 ti ˆH (t)dt/i~ . Stage 3 (Measurement ) From t1 to t2, the feedback controller performs quantum measurement on S described by measurement operators { ˆMk} and obtains each outcome k with probability pk. Let X be the set of outcomes k’s, and {ˆDk} be POVM as defined by ˆD k ≡ ˆM † k ˆM k; we then have pk = tr(ˆDk ˆ). We denote the pre-measurement density operator of the entire system as ˆ1, the post-measurement density operator with outcome k as ˆ(k) 2 ≡ ˆMk ˆ ˆM † k/pk, and define ˆ2 ≡ P k pk ˆ(k) 2 . Note that our scheme can be applied not only to a quantum measurement, but also to a classical measurement which can be described by setting = 0 for all k. Stage 4 (Feedback control ) From t2 to t3, the feedback controller performs a mechanical operation on S depending on outcome k. Let ˆUk be the corresponding unitary operator on the entire system, and ˆ(k) 3 ≡ ˆUk ˆ(k) 2 ˆU † k be the density operator of the entire system at t3 corresponding to outcome k. We define ˆ3 ≡ P k pk ˆ(k) 3 . Note that the feedback control is characterized by { ˆM k} and {ˆUk}. Stage 5 (Equilibration and final state) From t3 to tf , the entire system evolves according to unitary operator ˆU f which is independent of outcome k. We assume that by tf system S and heat bath Bm will have reached thermodynamic equilibrium at temperatures T and Tm, respectively. We denote as ˆf the density operator of the final state of the entire system, which is related to the initial state as ˆf = E(ˆi) ≡ X k ˆUf ˆU k ˆMkˆUi ˆiˆU † i ˆM † k ˆU † k ˆU † f . (4) We emphasize that ˆf need not equal the rigorous canonical distribution ˆcan f , as given by ˆcan f = exp(− ˆH S f ) ZS f ⊗ exp(− 1 ˆH B1 ) ZB1 ⊗· · ·⊗ exp(− n ˆHBn) ZBn , (5) where ZS f ≡ tr{exp(− ˆHS f )}. We only assume that the final state is in thermodynamic equilibrium from a macroscopic point of view . We will proceed to our main analysis. The difference in the von Neumann entropy between the initial and final states can be bounded from the foregoing analysis as follows: S(ˆi) − S(ˆf ) =S(ˆ1) − S(ˆ3) ≤S(ˆ1) − X k pkS(ˆ(k) 3 ) =S(ˆ1) − X k pkS(ˆ(k) 2 ) =S(ˆ1) + X k tr q ˆD k1 q ˆD k ln p ˆD k ˆ1 p ˆD k pk ! =S(ˆ1) + H({pk}) + X k tr( q ˆD k1 q ˆD k ln q ˆDk 1 q ˆD k), (6) where S(ˆ) ≡ −tr(ˆ ln ˆ) is the von Neumann entropy and H({pk}) ≡ − P k∈X pk ln pk is the Shannon information content. Note that in deriving the inequality (6), we used the convexity of the von Neumann entropy, i.e. S( P k pk ˆ(k) 3 ) ≥ P k pkS(ˆ(k) 3 ). Defining notations ˜H (ˆ1,X) ≡ − P k tr( p ˆD k ˆ1 p ˆD k ln p ˆD k ˆ1 p ˆD k) and I(ˆ1 :X) ≡ S(ˆ1) + H({pk}) − ˜H (ˆ1,X), (7) we obtain S(ˆi) − S(ˆf ) ≤ I(ˆ1 :X). (8) We refer to I(ˆ1 : X) as the QC-mutual information content which describes the information about the measured system that has been obtained by measurement. As shown later, I(ˆ1 :X) satisfies 0 ≤ I(ˆ1 :X) ≤ H({pk}). (9) We note that I(ˆ1 : X) = 0 holds for all state ˆ1 if and only if ˆDk is proportional to the identity operator for all k, which means that we cannot obtain any information about the system by this measurement. On the other hand, I(ˆ1 : X) = H({pk}) holds if and only if ˆDk is the projection operator satisfying = 0 for all k, which means that the measurement on state ˆ1 is classical and error-free. In the case of classical measurement (i.e. = 0 for all k), I(ˆ1 : X) reduces to the classical mutual information content. In fact, we can write I(ˆ1 : X) in this case as I(ˆ1 : X) = − P i qi ln qi − P k,i qip(k|i) ln p(k|i), where ˆ1 ≡ P i qi| iih i| is the spectrum decomposition of the measured state, and p(k|i) ≡ h i|ˆDk| ii can be interpreted as the conditional probability of obtaining outcome k under the condition that the measured state is | ii. 3 I(ˆ1 : X) can be written as I(ˆ1 : X) = ({ˆ(k) 2 }) − Smeas, where ({ˆ(k) 2 }) ≡ S(ˆ2) − P k∈X pkS(ˆ(k) 2 ) is the Holevo quantity which sets the Holevo bound , and Smeas ≡ S(ˆ2) − S(ˆ1) is the difference in the von Neumann entropy between the pre-measurement and post-measurement states. If Smeas = 0 holds, that is, if the measurement process does not disturb the measured system, then I(ˆ1 :X) reduces to the Holevo quantity; in this case, the upper bound of the entropy reduction with discrete quantum feedback control is given by the distinguishability of post-measurement states {ˆ(k) 2 }. Nielsen et al. have derived inequality S(ˆi) − S(ˆf ) ≤ S(ˆi, E) , where S(ˆi, E) is the entropy exchange which depends on entire process E, including the feedback process. In contrast, our inequality (8) is bounded by I(ˆ1 :X) which does not depend on the feedback process, but only depends on pre-measurement state ˆ1 and POVM {ˆDk}, namely, on the information gain by the measurement alone. It follows from inequality (8) and Klein’s inequality that S(ˆi) ≤ −tr(ˆf ln ˆcan f ) + I(ˆ1 :X). (10) Substituting Eqs. (3) and (5) into inequality (10), we have (ES i −ES f )+ Xn m=1 T Tm (EBm i −EBm f )≤FS i −FS f +kBT I(ˆ1 :X), (11) where ES i ≡ tr( ˆH S i ˆi), ES f ≡ tr( ˆH S f f ), EBm i ≡ tr( ˆH Bm ˆi), and EBm f ≡ tr( ˆH Bm ˆf ). Defining the difference in the internal energy between the initial and final states of system S as US ≡ ES f − ES i , the heat exchange between system S and heat bath Bm as Qm ≡ EBm i − EBm f , and the difference in the Helmholtz free energy of system S as FS ≡ FS f − FS i , we obtain − US + Xn m=1 T Tm Qm ≤ −FS + kBT I(ˆ1 :X). (12) This is the main result of this Letter. Inequality (12) represents the second law of thermodynamics in the presence of a discrete quantum feedback control, where the effect of the feedback control is described by the last term. For a thermodynamic heat cycle in which I(ˆ1 : X) = 0, US = 0, and FS = 0 hold, inequality (12) reduces to the Clausius inequality Xn m=1 Qm Tm ≤ 0. (13) The equality in (12) holds if and only if ˆ(k) 3 is independent of measurement outcome k (i.e. the feedback control is perfect), and ˆf coincides with ˆcan f . We will discuss two important cases for inequality. Let us first consider a situation in which the system undergoes an isothermal process in contact with single heat bath B at temperature T . In this case, (12) reduces to Wext ≤ −FS + kBT I(ˆ1 :X), (14) Pwhere the first law of thermodynamics, Wext = n m=1 Qm − US, is used. Inequality (14) implies that we can extract work greater than −FS from a single heat bath with feedback control, but that we cannot extract work larger than −FS+kBT I(ˆ1 :X). If we do not get any information, (14) reduces to (1). On the other hand, in the case of classical and error-free measurement, (14) becomes Wext ≤ −FS + kBTH({pk}). The upper bound of inequality (14) can be achieved with the Szilard engine which is described as follows. A molecule is initially in thermal equilibrium in a box in contact with a heat bath at temperature T . We quasi-statically partition the box into two smaller boxes of equal volume, and perform a measurement on the system to find out in which box the molecule is. When the molecule is found in the right one, we remove the left one and move the right one to the left position, which is the feedback control. We then expand the box quasistatically and isothermally so that the final state of the entire system returns to the initial state from a macroscopic point of view. During the entire process, we obtain ln 2 of information and extract kBT ln 2 of work from the system. We next consider a heat cycle which contacts two heat baths: BH at temperature TH and BL at TL with TH TL. We assume that ˆHS i = ˆH S f , US = 0, and FS = 0. Noting that Wext = QH + QL, we can obtain Wext ≤ 1 − TL TH QH + kBTLI(ˆ1 :X). (15) Without a feedback control, (15) shows that the upper bound for the efficiency of heat cycles is given by that of the Carnot cycle: Wext/QH ≤ 1 − TL/TH. With feedback control, (15) implies that the upper bound for the efficiency of heat cycles becomes larger than that of the Carnot cycle. The upper bound of (15) can be achieved by performing a Szilard-type operation during the isothermal process of the one-molecule Carnot cycle; if we perform the measurement and feedback with ln 2 of information in the same scheme as the Szilard engine during the isothermal process at temperature TH, the work that can be extracted is given by Wext = (1 − TL/TH)(QH − kBTH ln 2) + kBTH ln 2 = (1 − TL/TH)QH + kBTL ln 2. Note that we can reach the same bound by performing the Szilard-type operation during the isothermal process at temperature TL. We now prove inequality (9). For simplicity of notation, we consider a quantum system denoted as Q in general, instead of S and Bm’s. The measured state 4 of system Q is written as ˆ, and POVM as {ˆDk}k∈X. We introduce auxiliary system R which is spanned by orthonormal basis {| ki}k∈X, and define two states ˆ1 and ˆ2 of Q + R as ˆ1 ≡ P k √ˆˆDk√ˆ ⊗ | kih k| and ˆ2 ≡ P k p ˆD k ˆ p ˆD k ⊗ | kih k|. It can be shown that tr(√ˆˆD k√ˆ) = tr( p ˆD k ˆ p ˆD k) = pk, trR(ˆ1) = ˆ, and trQ(ˆ1) = P k pk| kih k| ≡ ˆR. Defining ˆ(k) 1 ≡ √ˆˆDk√ˆ/pk, ˆ(k) 2 ≡ p ˆD k ˆ p ˆD k/pk and ˆ′ P ≡ k pkˆ(k) 2 , we have S(ˆ2) = X k pkS q ˆD k ˆ q ˆD k ⊗ | kih k|/pk + H({pk}) = X k pkS(ˆ(k) 2 ) + H({pk}) = ˜H (ˆ,X). (16) Since S(ˆL†ˆL) = S(ˆL ˆL †) holds for any linear operator ˆ L, we have S(ˆ 2) = P k pkS(ˆ(k) 2 ) + H({pk}) = P k pkS(ˆ(k) 1 ) + H({pk}) = S(ˆ1). Therefore ˜H (ˆ,X) = S(ˆ1) ≤ S(ˆ) + S(ˆR) = S(ˆ) + H({pk}), (17) which implies I(ˆ:X) ≥ 0. The equality in (17) holds for all ˆ if and only if ˆ1 can be written as tensor product ˆ ⊗ ˆR for all ˆ: that is, ˆDk is proportional to the identity operator for all k. We will next show that I(ˆ : X) ≤ H({pk}). We make spectral decompositions as ˆ = P i qi| iih i| and ˆ′ = P j rj | ′ j ih ′ j |, where rj = P i qidij , and define dij ≡ P k |h i| p ˆD k| ′ j i|2, where P i dij = 1 for all j and P j dij = 1 for all i. It follows from the convexity of −x ln x that S(ˆ) = − P i qi ln qi ≤ − P j rj ln rj = S(ˆ′). Therefore, H({pk}) − I(ˆ:X) = ˜H (ˆ,X) − S(ˆ) = H({pk}) + X k pkS(ˆ(k) 2 ) − S(ˆ) ≥ H({pk}) + X k pkS(ˆ(k) 2 ) − S(ˆ′) ≥ 0. (18) It can be shown that the left-hand side is equal to zero for all ˆ if and only if ˆDk is a projection operator satisfying = 0 for all k. Our results do not contradict the second law of thermodynamics, because there exists an energy cost for information processing of the feedback controller . Our results are independent of the state of the feedback controller, be it in thermodynamic equilibrium or not, because the feedback control is solely characterized by { ˆMk} and { ˆUk}. In conclusion, we have extended the second law of thermodynamics to a situation in which a general thermodynamic process is accompanied by discrete quantum feedback control. 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