Alkaline lamprophyre (camptonite) from Bayyaram area, NE ... and Arvind. Qu… · lamprophyre category12. But the Bay-yaram lamprophyre shows ultrapotassic nature. According to their

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the basic principle of polymorphism in microsatellite markers. Since the poly-morphism detected through SSR markers is length polymorphism (simple sequence length polymorphism, SSLP), more the variation in the length amplified, more will be the polymorphism. Segment with more number of repeats carries better chances of producing polymorphism than the shorter ones. Moreover, larger varia-tions are easily detected in gel electro-phoresis. Thus, SSRs with 20–30 repeats are found to be more useful in detecting genetic variations between the two species. The study reveals that enormous genetic variations still exist between G. max and G. soja. It also highlights some useful properties of SSR markers that would help in studying genetic variability

in soybean genotypes. Further, the poly-morphic markers identified in this study would be highly useful in mapping of QTL for yield and other important agro-nomic traits in soybean.

1. Saghai-Maroof, M. A., Soliman, K. M., Jergensen, R. A. and Allard, R. W., Proc. Natl. Acad. Sci. USA, 1984, 81, 8014–8018.

2. Hwang, T. et al., DNA Res., 2009, 16, 213–225.

3. Kumar, V. et al., Indian J. Genet., 2011, 71, 372–376.

4. Concibido, V. C. et al., Theor. Appl. Genet., 2003, 106, 575–582.

Received 23 January 2015; revised accepted 3 September 2015

YASHPAL1 D. R. RATHOD1

JYOTI DEVI2 ANIL KUMAR1

KEYA MUKHERJEE1 DEEPIKA CHERUKU1

SUBHASH CHANDRA1 S. K. LAL1

AKSHAY TALUKDAR1,* 1Division of Genetics, ICAR-Indian Agricultural Research Institute, New Delhi 110 012, India 2ICAR-Indian Vegetable Research Institute, Varanasi 221 005, India *For correspondence. e-mail: [email protected]

Alkaline lamprophyre (camptonite) from Bayyaram area, NE margin of the Eastern Dharwar Craton, southern India The widespread occurrence of lampro-phyres is known, since more than a few decades, from the various parts of Cudappah Igneous Provice (CIP)/Praka-sham Alkaline Province (PAP) of the Eastern Dharwar Craton (EDC), southern India (Figure 1 a). On inspecting the dis-tribution of lamprophyres in three cra-tons, i.e. EDC, Aravalli–Bundelkhand and Bastar–Bhandar Craton1, it is evident that the EDC alone hosts the maximum number and variety of lamprophyres2. The present correspondence reports a lamprophyre dyke near the Bayyaram area (8009:1735) at the northeastern margin of the EDC (Figure 1 a). It also addresses the petrology, geochemistry and significance of this occurrence. The study area mainly consists of granitoids of Peninsular Gneissic Com-plex (PGC-II) of the EDC. Regionally, the area is bounded by two Proterozoic sedimentary basins, i.e. Pakhal basin to the east and Cuddapah basin to the south (Figure 1 a). The lamprophyre of the study area has been intruded within granitoids of the EDC. The dyke shows NW–SE trend having 15–20 m length and 1 m width approximately (Figure 1 b). Megascopically, the dyke is mesocratic to melanocratic, fine-grained and shows porphyritic texture. Phenocrysts are uniformly distributed in the fine-grained

Figure 1. a, Geological map of the Bayyaram area. (Inset) Map showing location of lampro-phyre–lamproite–kimberlite fields in the Eastern Dharwar Craton (EDC) of South India (GMNK, Gulbarga, Maddur, Narayanpet Kotakonda Kimberlites; RG, Raichur Gawal Kimberlites; TB, Tungabhadra kimberlites; WLCL, Wajrakarur, Lattavaram, Chigicherla and Kalyanadurg kim-berlites; RLF, Ramadugu lamproite field; KLF, Krishna lamproite field; ESKPPKR, Elchuru (1), Settupalle (2), Kommalapadu (3), Purimetla (4), Pusupugullu (5), Kellampalle (6), Ravipadu (7) lamprophyres in Prakasam district and Polayapalle (8), Bayyaram (9) Lamprophyres in Kham-man district). b, Field photograph showing outcrop pattern of lamprophyre dyke.

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matrix. The dominant phenocrysts are pyroxene and olivine set in brownish-green matrix. Ocellar structures of 1–3 mm are also observed in the lamprophyre. Petrographically, it displays the pres-ence of (i) relict porphyritic, panidomor-phic texture (although original minerals are pseudomorphed) of multiple genera-tions of mafic phenocrysts (mainly oli-vine and clinopyroxene; Figure 2 a–c), (ii) leucocratic (carbonate-rich) ocelli, and (iii) volatile mineralogical composi-tion (dominated by amphibole, carbon-ate, chlorite, epidote and serpentine), which constitute important evidence for identification as lamprophyres. As these rocks contain clinopyroxene and olivine as the phenocrysts and plagioclase is the main feldspar in the groundmass, they belong to the alkaline lamprophyre cate-gory in general (Figure 3) and campton-ite in particular. Sphene and apatite (needle-shaped) are present as accessory phase (Figure 2 d). A few ocelli are rounded or elliptical (up to 4 mm size), which are dominantly leucocratic and contain calcite (recrytallized; Figure 2 c). The ocellar textures are interpreted as late-stage melts which are formed when liquid immiscibility is produced between a silicate melt, and a melt relatively rich in H2O and CO2 (refs 3, 4), and are con-sidered to be diagnostic of alkaline lam-prophyres5,6. Whole rock major and trace element analyses were carried out at the Chemi-cal laboratory, Geological Survey of India (GSI), Hyderabad, India. X-ray fluorescence spectrometry was used to analyse major oxides, whereas ICP-MS was used to determine trace and rare earth element (REE) concentration. The precision is <5% for all analysed ele-ments when reported at 100X detection limit. Several standards were run along with the studied samples to check accu-racy and precision. Tables 1 and 2 present whole-rock chemical data. Stan-dardized CIPW norms and Mg# for all samples were automatically computed using the IgROCS computer program7. The ferric/ferrous iron-ratio used for CIPW norm calculation was taken from Middlemost8. The petrographic studies were carried out using LEICA DM RX fitted with a camera, at the Petrology laboratory, GSI, Hyderabad. The rock is characterized by low SiO2, generally high MgO, medium Al2O3 and high K2O (K2O/Na2O varies from 7 to 9.22), also having high FeO + MgO, MgO/FeO and

Figure 2. Representative photomicrographs of Bayyaram lamprophyre showing. a, Panido-morphic texture with olivine phenocryst; pseudomorphed by serpentine, carbonate and chlorite (XPL-10X). b, Glomeroporphyritic texture represented by cumulates of olivine and clinopyrox-ene in fine-grained matrix (XPL-10X). c, Glomeroporphyritic texture represented by cumulates of clinopyroxene and presence of carbonate ocelli in the fine-grained matrix (XPL-4X). d, Fine-grained matrix containing biotite, chlorite, needle-shaped apatite, shpene and opaque (PPL-20X). Ol, Olivine; Ser, Serpentine; Cal, Calcite; Cpx, Clinopyroxene; Plg, Plagioclase; Chl, Chlorite; Bt, Biotite; Apt, Apatite, Sph, Shpene; Opq, Opaque.

Figure 3. a, Plot of SiO2 versus (Na2O + K2O) for laprophyres (after ref. 20). CAL, Calc-alkaline lamprophyres; AL, Alkaline lamprophyres; UML, Ultrabasic lamprophyres; LL, Lamproites; ALK, Alkaline rocks; TH, Tholeiites. b, Empirical diagram for distinguishing sho-shonitic (calc-alkaline) and alkaline lamprophyres using the normative parameters5. c, MgO ver-sus SiO2 discrimination plot for various alkaline mafic potassic ultrapotassic rocks (field adapted from ref. 21) showing that samples plot in the alkaline field. d, Sm versus Ce/Yb diagram show-ing Bayyaram lamprophyre plot in the alkaline field (after ref. 12).

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K2O/Na2O ratios resembling the ultra-basic lamprophyre9,10. The lamprophyre shows low SiO2 content–varying from 39.23 to ~40.96 wt%; Na2O + K2O from 5.53 to ~6.56 wt% and K2O/Na2O from 7 to 9.22 wt%. The alkaline nature of these lamprophyric samples is also evident from their normative compositions since all of them exhibit nepheline normative nature (Table 1). Overall, these samples are predominantly olivine and diopside normative indicating strongly undersatu-rated nature4. Low orthoclase normative content is another characteristic feature of alkaline lamprophyres11. The major oxide composition reveals that Bayyaram lamprophyres are silica undersaturated, ultrabasic, high potassic and range from relatively primitive to more evolved compositions. Fractional crystallization of plagioclase and clinopyroxene had a strong influence on magma evolution. It is a well-established fact that the Prakasham province lamprophyres are all uniformly potassic (K2O > Na2O) and that they do not belong to ultrabasic lamprophyre category12. But the Bay-yaram lamprophyre shows ultrapotassic nature. According to their potassic nature, Bayyaram lamprophyres should be categorized as calc-alkaline lampro-phyres, but in the bivariant diagram they are plotted typically within alkaline lamprophyre field (Figure 3 c and d). The basic contradiction of lamprophyres being alkaline and still coming under the potassic type is a rare phenomenon (Table 1). The Bayyaram lamprophyres behave like alkaline lamprophyres and are also marked by distinct deviations because of their potassic nature, which otherwise is a strong geochemical trait of calc-alkaline lamprophyres. The chondrite-normalized REE patterns of the studied rocks confirm crystalliza-tion from a LREE-enriched magma. The multi-element spidergrams involving HFSE indicate that their source regions show subduction-related characteristics and samples plot in overlapping field be-tween subduction zone and within plate field with more affinity towards subduc-tion-related source (Figure 4 a and b). The Sc, Cr, Ni and Co content is simi-lar in the range of the primary magma responsible for lamprophyres13 (Table 2). High concentrations of LREE (Table 3) and relatively high concentrations of compatible elements such as Ni and Cr strongly suggest that the Bayyaram lam-prophyre magma was produced by a

small degree of partial melting of peri-dotite mantle at greater depths in the garnet stability fields3,14. A slight nega-tive Hf anomaly in the multi-element plots also lends support to the derivation of the Bayyaram lamprophyre magma

from within the garnet stability field since Dgarnet=melt of Zr > Hf. The presence of negative Ta–Nb anomalies suggests the involvement of subduction-related process in the origin of the studied rocks15–18. Negative Sr and Eu anomalies

Table 1. Representative major oxide analysis of Bayyaram lamprophyre

Oxides (wt%) BL-1A BL-1B BL-1C BL-1D

SiO2 40.68 40.96 39.23 39.66 TiO2 1.57 1.67 1.57 1.67 Al2O3 11.83 12.24 11.75 12 Fe2O3 12.4829 11.1816 11.4829 10.69 FeO 0 0 0 0 MnO 0.36 0.34 0.22 0.21 MgO 7.97 7.69 7.87 7.49 CaO 10.58 10.85 10.36 10.55 Na2O 0.74 0.82 0.58 0.72 K2O 5.65 5.74 5.35 5.14 P2O5 1.21 1.38 1.21 1.32 LOI 4.98 5.67 7.02 7.65 Total 98.0529 98.5416 96.2429 97.1 K2O/Na2O 7.63 7 9.22 7.13 Mg* 61.63 63.37 63.29 63.80 Or 1.81 3.60 3.02 8.66 Ab 0 0 0 0 An 13.32 13.87 15.39 16.17 Ne 3.68 4.08 2.99 3.72 Lc 27.01 26.08 25.57 20.08 Di 28.57 28.27 27.29 26.87 Ol 15.13 13.40 15.20 13.72 Mt 4.17 3.74 3.98 3.71 Il 3.23 3.44 3.36 3.57 Ap 3.04 3.47 3.15 3.45

Figure 4. Discrimination diagrams for deciphering tectonic setting for the CIP lamprophyres. a, Al2O3 (wt%) versus TiO2 (wt%) discrimination plot for distinguishing within-plate and arc-related basalts22. b, K2O (wt%) versus TiO2 (wt%) plot for distinguishing within-plate and sub-duction-related K-rich mafic lavas (after ref. 23). c, Chondite normalized rare earth element pat-terns for the Bayyaram lamprophyre24. d, Primordial mantle normalized multi-element pattern for the Bayyaram lamprophyre25.

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normally indicate plagioclase fractiona-tion (Figure 4 c and d). The observed negative Sr anomaly is interpreted to in-dicate source characteristics that could be related either to the presence of resid-ual clinopyroxene or to depletion of mantle source in Sr during a previous phase of melts extraction19. Based on combined petrography and geochemistry, these lamprophyres are considered to belong to the alkaline lamprophyre category in general and camptonite in particular. The strong cor-relation between various major and trace elements coupled with high abundance of incompatible and compatible trace ele-ments imply that alteration and crustal contamination have had no or limited ef-fect on the whole-rock geochemistry of

the Bayyaram lamprophyre and that oli-vine fractionation played an important role in their evolution.

1. Chalapathi Rao, N. V., J. Geol. Soc. India, 2008, 72, 57–82.

2. Madhavan, V., David, K., Mallikharjuna Rao, J., Chalapathi Rao, N. V. and Srini-vas, M., J. Geol. Soc. India, 1998, 52, 621–642.

3. Ferguson, J. and Currie, K. L., J. Petrol., 1971, 12, 561–585.

4. Srivastava, R. K. and Chalapathi Rao, N. V., Min.. Petrol., 2007, 89, 189–215.

5. Rock, N. M. S., Earth Sci. Rev., 1977, 13, 123–169.

6. Foley, S. F., Lithos, 1984, 17, 127–137. 7. Verma, S. P. and Gómez, M. A., Epi-

sodes-36, 2013, 2, 115–124. 8. Middlemost, E. A. K., Chem. Geol.,

1989, 77, 19–26.

9. Sarkar, A., Paul, D. K., Balasubrah-manyam, M. N. and Sengupta, N. R., J. Geol. Soc. India, 1980, 21, 188–193.

10. Paul, D. K. and Polls, P. J., Geol. Mag., 1981, 188(4), 393–399.

11. Rock, N. M. S., Lamprophyres, Blackie and Sons Ltd., Glasgow, 1991, p. 287.

12. David, K., Unpub. Ph D thesis, Osmania University, Hyderabad, 1997.

13. Rock, N. M. S., Lamprophyres, Blackie, Glasgow, 1990, p. 285.

14. Hirschmann, M. M., Ghiorso, M. S. and Stolper, E. M., J. Petrol., 1999, 40, 297–313.

15. Foley, S. F., Venturelli, G., Green, D. H. and Toscani, L., Earth Sci. Rev., 1987, 24, 81–134.

16. Peacock, S. M., Science, 1990, 248, 329–337.

17. Saunders, A. D., Storey, M., Kent, R. and Norry, M. J., Geol. Soc. London, Spec. Publ., 1992, 68, 41–60.

18. Kent, R., In Magmatism in Relation to Diverse Tectonic Settings (eds Sriva-stava, R. K. and Chandra, R.), A.A. Balkema, Rotterdam, 1995, pp. 17–41.

19. Chalapathi Rao, N. V., Gibson, S. A., Pyle, D. M. and Dickin, A. P., J. Petrol., 2004, 45, 907–948.

20. Rock, N. M. S., Geol. Soc. Spec. Publ., 1987, 30, 191–226.

21. Lefebvre, N., Kopylova, M. and Kivi, K., Precambr. Res., 2005, 138, 57–87.

22. Muller, D., Morris, B. J. and Ferrand, M. J., Lithos, 1993, 30, 123–137.

23. Thorpe, R. S., Earth Sci., 1987, 77, 361–366.

24. Sun, S. S. and McDonough, W. F., Geol. Soc. London, Spec. Publ., 1989, 42, 313–345.

25. Wood, D. A., Joron, J. L., Treuil, M., Norry, M. and Tarney, J., Contrib. Min-eral. Petrol., 1979, 70, 319–339.

ACKNOWLEDGEMENTS. We thank the Director General, Geological Survey of India (GSI); Deputy Director General, Southern Region, GSI, Hyderabad and Director, Publica-tion Division, GSI, Hyderabad for their permis-sion to publish this manuscript. We also thank Dr C. H. Chalapathi Rao for valuable com-ments. We are grateful to Dr K. R. Randive (RTMNU), Dr G. Suresh, Dr Rajkumar Mesh-ram, Dr M. L. Dora and S. N. Mahapatro (GSI) and Dr K. V. S. Subrahamaniyam and Dr S. Sawant (NGRI) for their valuable sug-gestions that helped improve the manuscript.

Received 20 July 2015; revised accepted 23 August 2015

TUSHAR M. MESHRAM* DEVASHEESH SHUKLA

KALLOLA K. BEHERA

Geological Survey of India, Hyderabad 560 068, India *For correspondence. e-mail: [email protected]

Table 2. Representative trace element analysis of Bayyaram lamprophyre (ppm)

BL-1A BL-1B BL-1C BL-1D

Sc 27.6 25 28 24.7 V 175 187.8 175.3 189 Cr 1015 890 1016 903 Co 26 32 24 31 Ni 195 206 194 201 Cu 36 16 35 17 Zn 0.5 0.5 0.5 0.5 Ga 13 8.9 13 9.0 Rb 132.8 137.5 133 138 Sr 618 677 620 676 Y 47 27 46.7 27.3 Zr 744 752 744.5 751.6 Nb 24 27 25 27 Ba 1983 2213 1980 2205 Hf 10.70 11.65 10.69 11.63 Ta 1.12 1.20 1.11 1.21 Pb 5 5.9 5.3 6 Th 67 75 66.6 74.8 U 7.43 7.96 7.39 8.58

Table 3. Representative rare earth element analysis of Bayyaram lamprophyre (ppm)

BL-1A BL-1B BL-1C BL-1D

La 221.86 195.09 222.29 195.11 Ce 405.94 371.93 406.60 372.18 Pr 49.87 44.76 50.21 44.48 Nd 164.83 150.28 164.08 150.59 Sm 23.30 21.32 23.29 21.33 Eu 5.09 4.47 5.09 4.47 Gd 16.93 14.28 16.94 14.15 Dy 10.97 7.63 11.00 7.64 Er 4.85 2.92 4.84 2.93 Yb 3.35 2.30 3.36 2.29 Tb 2.26 1.83 2.25 1.82 Ho 1.97 1.24 1.98 1.22 Tm 0.65 0.43 0.66 0.42 Lu 0.49 0.34 0.50 0.35 REE 912.36 818.83 913.10 818.99

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Preface The subject of Quantum Measurements has seen an explosive growth both in interest, and in importance, over the recent decades. While internationally there are fre-quent meetings devoted entirely to it, in India, so far, there have not been any except perhaps for a small dis-cussion meeting in 2009 of about 20 participants on the ‘Many Worlds Interpretation’ held at the Poornaprajna Institute of Scientific Research (PPISR), Bangalore. To make a beginning, the Centre for Quantum Information and Quantum Computing (CQIQC) at the Indian Institute of Science, Bengaluru, decided to hold a discussion meet-ing entirely focused on Quantum Measurements. CQIQC is a Department of Science and Technology (Govt of India) sponsored project which got completed in July 2015. Its primary objective was to bring both theo-retical and experimental activities in Quantum Informa-tion, Quantum Computing as well as Foundations of Quantum Mechanics of the Institute under one roof. It ran a very successful series of seminars and colloquia. It organized an international conference in January 2013 as well as a satellite meeting for the 13th Asian Quantum Information Science Conference in September 2013. Additionally, it conducted two summer schools in 2012 and 2013. The discussion meeting on Quantum Measurements took place during 22–24 October 2014. International and national experts as well as many students from all over the country participated in an intense and informed dis-cussion on practically all important aspects of the subject. What was novel to this meeting was that both experimen-tal and theoretical sides of these aspects were repre-sented. The broad range of topics discussed were: (i) varieties of quantum measurements that included Weak measurements, Weak-value measurements, Protective measurements, Arthurs–Kelly measurements of mutually incompatible observables, Quantum non-demolition measurements, and, Ancilla-based quantum measure-ments; (ii) various aspects of uncertainty relations and beyond that included in-depth discussions of the so called error-disturbance relations, entropic uncertainty relations, etc.; (iii) the classical-quantum divide with emphasis on the so called Macrorealism, and, finally, (iv) Decoher-ence. This special section consists of selected talks from the discussion meeting (unfortunately we did not get all the manuscripts we would have liked to see in this issue). The topics that were chosen for discussion at the meet-ing closely mirrored the exciting and revolutionary de-velopments in uncovering the physical interpretation of quantum theory. Even after Heisenberg and Schrödinger had formally completed the development of quantum me-chanics, with impressive empirical successes to follow, such an interpretation of the theory remained obscure. Though Bohr’s correspondence principle helped some-

what, it was not adequate to provide a consistent picture. A watershed development in 1927, two years after the formal completions in 1925 by Heisenberg and Schröd-inger, was Heisenberg’s uncertainty relations, and Bohr’s recognition of the centrality of Measurements in arriving at a consistent physical interpretation of quantum theory. Heisenberg’s initial paper on the uncertainty relations had several lacunae, the most important, as presciently clari-fied by Bohr, were his confusing the recoil experienced by the particle with uncertainty, and the absence of the dual aspects of waves and particles in his analysis. Bohr pointed out that by accurately measuring the momentum of the scattered photon, one could completely account for the recoil with complete certainty. Bohr further pointed out that every stage of the measurement has to be care-fully scrutinized and that a complete picture can only be obtained by employing both the wave and the particle aspects of quantum systems. Despite these criticisms Bohr accepted the essential correctness of the uncertainty principle. Heisenberg acknowledges by adding a lengthy addendum to his paper. Bohr presented a systematic elaboration of these ideas in the now famous Como meet-ing in 1927 which can rightly be seen as the birth of Bohr’s Complementarity principle, as well as that of the theory of quantum measurements. These ideas played a central role in the famous Bohr–Einstein dialogues on the meaning of quantum theory. The Heisenberg analysis of his famous ‘microscope’ gedanken experiment was essentially semi-classical. The central conclusion of that analysis p q h for the product of ‘errors’ in the simultaneous measurement of the two incompatible observables p, q is of a fundamen-tally different nature than the more familiar pq /2. While the former was in the context of a single measure-ment, the latter refers to the statistical outcome of an ensemble of measurements when the incompatible observables are measured separately on distinct suben-sembles as elaborated by von Neumann in his classic The Mathematical Foundations of Quantum Mechanics. The second form, being formulated in terms of variances, cannot in general capture either the correct notion of errors when incompatible observables are simultaneously measured. Nor can it capture the notion of a disturbance on the measurement of one observable arising out of the measurement of another. In a seminal development pio-neered by him, Ozawa (page 2006) has shown that the error–disturbance relation attributable to Heisenberg can indeed be violated and has shown how to go ‘beyond’ the Heisenberg relations. Apart from their significance to the foundational aspects of quantum theory, his results have deep ramifications in practice too. The notions of quan-tum limits to achievable sensitivities in experiments based on the Heisenberg relations will all have to be revised.

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These new versions of the error-disturbance relations (EDR) have been verified experimentally. Hasegawa (page 1972), has, in this special section described their experiments in detail. While Hasegawa’s experiments are done with neutrons, Edamatsu (whose manuscript could not be obtained for this special section) uses weak meas-urements on single photons. Both of them found that the original EDRs can be violated and that their new forms are vindicated. Hasegawa also discusses other experi-ments central to foundations of quantum mechanics such as neutron-optical weak-value measurements, the Quan-tum Cheshire-cat and experiments on contextuality in quantum mechanics. While the uncertainty relations originally due to Heisenberg, and as extended by Ozawa and others essen-tially involve the second moments of the probability distributions for outcomes, the so called entropic uncer-tainty relations characterize the distributions themselves, and consequently can be much more powerful. The es-sence of uncertainty relations being the incompatibility of certain observables, one can expect these forms of uncer-tainty relations to provide stronger characterizations of incompatibility. Prabha Mandayam and Srinivas (page 1997) discuss a class of such measures. They also discuss (page 2044) how entropic methods can be used to prove that disturbances associated with measurements on iden-tically prepared, distinct, ensembles cannot be reduced arbitrarily. Entropic uncertainty relations, in the context of unsharp measurements of incompatible observables, were also the focus of the contribution by Karthik et al. (page 2061). They give an overview of such measure-ments along with results on how presence of quantum memory can significantly strengthen the entropic uncer-tainty relations. An explicit model for such unsharp simultaneous meas-urement of position and momentum was given long ago (1965) by Arthurs and Kelly, who essentially extended the well known von Neumann Measurement Model to this class of measurements. Unfortunately, they chose to publish their results in the not so accessible Bell System Technical Journal! Roy (page 2029) has given a detailed review of the Arthurs–Kelly measurements. As applica-tions he shows how bounds on von Neumann entropy, various forms of Wigner distributions, noiseless tracking of conjugate variables, remote tomography, etc. can all be obtained. It would be interesting to revisit many results presented in this issue on incompatible observables and their measurements in the specific context of Arthurs–Kelly model. Another aspect of the measurement to which Bohr stuck to rather vehemantly was with regard to the nature of the measuring apparatus. He maintained that nothing short of a fully classical description of the apparatus would make any sense. This is in sharp contrast to the stand adopted by von Neumann, who treated the measur-ing apparatus also as a quantum system. Proponents of

his view argue that since the constituents of every appara-tus are also the very same elementary systems whose suc-cessful description requires quantum theory, apparatuses too should be describable quantum mechanically. The issue is technically complex and the verdict is still not out. In the meanwhile there have been intense discussions about the so called classical–quantum divide and the means of bridging the same. One of the most interesting ideas put forward in this connection is that of Macroreal-ism by Anthony Leggett and Anupam Garg. Garg (page 1958) gives a lucid description of Macrorealism, and an assessment of the experimental tests of this concept. In their original work, Leggett and Garg had formulated cer-tain inequalities which have formed the basis for an experimental vindication of Macrorealism. A key ingre-dient to all such experimental tests is the realization of the so called Non-invasive measurements. Dipankar Home (page 1980) discusses these aspects in detail. Lik-ening these Leggett–Garg inequalities to temporal analogs of the Bell’s inequalities, he dwells on the use of the so called Negative Result Measurements as means of realizing non-invasive measurements. Besides providing a Wigner-form of the Legget–Garg inequalities, he also discusses the ramifications concerning unsharp measurements, as well as connections to quantum key distributions. Non-invasive measurements have also been discussed in the experimental work of Mahesh et al. (page 1987) They advance the general theme of ancilla-based quan-tum measurements, and claim to have realized non-invasive measurements as part of these general strategies. In an earlier work they too had provided experimental tests of the Leggett–Garg inequalities. All these are based on NMR-techniques. Additionally, they have used their protocols to perform various aspects of tomography as well as what they call ‘quantum noise engineering’. It remains to be seen as to how non-invasive many of the proposals really are. While the non-invasive measurements alluded to so far aim to minimize the disturbance of a state due to meas-urements performed on it, for quite sometime there has been interest in another class of measurement which can also be considered non-invasive in a sense. These are the so called Quantum Non-Demolition Measurements (QND), developed in the context of detection of gravita-tional waves. Unnikrishnan (page 2052) has given a review of this class of measurements. These are also sometimes called Back-action Evading measurements. For example, a measurement of position could induce an uncertainty in momentum which can feed back into the accuracy of position measurement. QND measurements are designed to avoid such a back reaction. After review-ing the basic concepts and formalism, he discusses appli-cations to quantum optics, gravitational wave detection, as well as to spin-magnetometry. According to the standard lore of quantum measure-ments, projective measurements on a single member of

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an ensemble of unknown states has no statistical significance. This is due to the fact that the outcomes of such measurements are random. Aharonov and Vaidman proposed a remarkable measurement scheme called ‘Pro-tective Measurements’ by them. In their ideal version, they too are non-invasive. According to this, a single measurement on a single copy can reveal the expectation value of the measured observable without affecting the state. Then by repeating the process for a complete set of observables, the state can be determined. Qureshi and Hari Dass (page 2023) have discussed this interesting measurement scheme and its limitations, particularly the impossiblity of determining the state with certainty owing to entanglements that are inevitable in any practical implementation. They also point out some pragmatic merits of protective measurements. Aharonov, Vaidman and Albert also proposed another remarkable class of measurements called Weak Meas-urements. A way to think about these within the von Neumann models is to consider the apparatus as a very broad superposition of pointer states, in contrast to the projective (also called Strong) measurements where the apparatus is prepared in a pointer state. The novelty of these classes of measurements is that each weak meas-urement has almost negligible disturbance on the state. But the price one pays is that the ensemble sizes required for comparable levels of errors are extremely large, else the errors are too big. Another equally interesting variant is the so called Weak-value measurement; this consists of a weak measurement followed by what Aharonov and coworkers call a Post-selection implemented through a projective measurement and selecting only prescribed outcomes. The outcomes of weak-value measurements are suitably normalized matrix elements of the observ-able. These are in general complex, and separate weak measurements are necessary for their real and imaginary parts. The fact that weak values can be larger than the largest eigenvalue has been much sensationalized, start-ing from the provocative title ‘How the result of a meas-urement of a component of the spin of a spin-1/2 particle can turn out to be 100’ by the authors themselves. As such there is nothing in quantum theory that requires these normalized matrix elements to be bounded by the eigenvalues of the observables. One of the spin-offs of these measurements is that small signals can be ampli-fied. The fact that their disturbance of the state is negligi-ble has also been cited as the reason for considering weak measurements to be ideal candidates for non-invasive measurements. Over the last few years weak measurements and weak-value measurements (some people refer to the latter as also Weak measurements, which is a very confusing no-menclature) have dominated both experimental as well as theoretical discussions of quantum measurements, and the discussions at the meeting reflected this trend. Pragya Shukla (page 2039) focuses on situations when the out-

come of a weak-measurement (she really has weak-value measurement in her mind) lies outside the range of eigen-values, which she calls superweak and interprets this as a ‘supershift’ on the measuring device owing to a ‘coherent superposition of waves’. She has likened them to the os-cillations in a band-limited function faster than the maximum frequency over arbitrarily large intervals. An-other interesting feature discussed by her is the universal-ity in the distribution of weak-values. Satya Sainadh et al. (page 2002) theoretically analyse elastic scattering in resonance fluoroscence as a manifestation of weak-value amplification. They make the interesting point that such amplifications can be quite generally understood from the Wigner–-Weisskopf theory of spontaneous emission. Patrick Das Gupta (page 1946), in a presentation some-what disconnected from the concerns of quantum meas-urements, discusses the effect of mutual gravitational interaction between ultra-cold gas atoms on the dynamics of Bose–Einstein condensates. Hasegawa also presents an interesting experimental application of weak-value measurements to the so called Quantum Cheshire Cat wherein he claims to have sepa-rated spin and momentum of neutrons along different paths of a neutron interferometer. Tanay Roy et al. (page 2069) present high precision and highly controllable experiments on various aspects of weak measurements using superconducting qubits. They are able to smoothly go over from regimes of strong (projective) measure-ments to those of weak measurements. In the former they can exhibit the so called Quantum Jumps, while in the latter they can exhibit the characteristic stochastic trajec-tories. They achieve all this in real time. Superconducting qubits promise to be among the best candidates for high-precision experimental probes of quantum measurements. It has been found useful to enlarge the notion of quan-tum evolutions to include measurements also. Apoorva Patel and Parveen Kumar (page 2017) discuss an evolu-tionary formalism to describe both projective and weak measurements. They raise several interesting questions regarding the role of the Born interpretation and also on the desirability to design suitable experiments to throw light on such evolutions. Debmalya Das and Arvind (page 1939) explore the tomographic aspects of weak measurements. They wish to exploit the apparent non-invasiveness of weak measurements to recycle the state. They claim that under certain circumstances their method can outperform state determinations via projective meas-urements. Hari Dass (page 1965) presents three results on weak measurements. The three are (i) a demonstration that repeated weak measurements on a single copy cannot provide any information on the state of the copy, and, they are as invasive as projective measurements; (ii) that weak measurements perform no better than strong meas-urements as candidates for non-invasive measurements in testing the macrorealism inequalities of Leggett and Garg, and, (iii) weak-value measurements are optimal in

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CURRENT SCIENCE, VOL. 109, NO. 11, 10 DECEMBER 2015 1938

the sense of Wooters and Fields when the post-selected states are mutually unbiased with respect to the eigen-states of the measured observable. As part of (i) he explicitly constructs the stochastic trajectory during a weak measurement. It is important to connect this to the experimental results of Tanay Roy et al. as well as the evolutionary formalism of Apoorva Patel and Parveen Kumar. Finally, we turn to one of the most vexing aspects of quantum measurements, namely, the nature of the appara-tus. As already mentioned, Bohr’s take on this was or-thogonal to that of von Neumann and to most of current ideas. He insisted that the apparatus had to be classical. In the opposite view point the apparatus is still a quantum system, albeit highly complex, and the hope is that this complexity leads to an effectively classical behaviour. Though reasonable sounding, to this day there is no com-pletely satisfactory derivation, or even approximate deri-vation of this ‘hope’. No matter which of the many measurement schemes one considers, there is always a

point at which the strong or projective measurements have to be invoked. This always leads to an entangled su-perposition of all possible system-apparatus correlated states, and not the single outcome associated with a suc-cessful measurement. The present thinking on this is that an apparatus–environment interaction decoheres this superposition to give rise to the seemingly classical out-come. The pointer states are also picked out by this inter-action as the basis in which the mixed density matrix becomes diagonal. A big challenge is to vindicate this decoherence picture in a more systematic manner. Sushanta Dattagupta (page 1951) discusses many aspects of decoherence as well as coherence in quantum systems from a condensed matter perspective. In conclusion, we are grateful to Current Science for bringing out this special section and for their patience and energetic support at every stage.

N. D. Hari Dass e-mail: [email protected]

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*For correspondence. (e-mail: [email protected])

Quantum state estimation using weak measurements Debmalya Das* and Arvind Department of Physical Sciences, Indian Institute of Science Education and Research Mohali, Manauli 140 306, India

We explore the possibility of using ‘weak measure-ments’ without ‘weak value’ for quantum state esti-mation. Since for weak measurements the disturbance caused during each measurement is small, we can rescue and recycle the state, unlike for the case of pro-jective measurements. We use this property of weak measurements and design schemes for quantum state estimation for qubits and for Gaussian states. We show, via numerical simulations, that under certain circumstances, our method can outperform the esti-mation by projective measurements. It turns out that ensemble size plays an important role and the scheme based on recycling works better for small ensembles. Keywords: Fidelity, Gaussian state, projective meas-urement, qubit, state estimation, weak measurement.

Introduction

THE quantum superposition principle and the wave func-tion collapse set apart the quantum description of the world from its classical counterpart. As a consequence, in the standard paradigm, the outcome of a single measure-ment cannot be predicted with certainty and we can only assign probabilities to different outcomes. As the meas-urement process disturbs the system and in a projective measurement the state of the system collapses we cannot re-use the state for any further measurement. This neces-sitates the use of an ensemble of identically prepared states to interpret quantum measurements. Ideally, in the large ensemble limit the ensemble average tends to the ex-pectation value of the observable. The question that arises at this point is: what if we are provided with a small en-semble of states and asked to make the best use of it? Projective measurements require a large coupling be-tween the system and the measuring device. However, if the coupling is made small, we inflict a very small distur-bance to the system at the expense of extracting a corre-spondingly small amount of information1. Such measurements are known as weak or unsharp measure-ments. Such measurements have been introduced in vari-ous forms in the past2–8. The coupling strength can be tuned to suit the situation and the state can subjected to

further measurements to extract more information. Whenever we have a small ensemble, each member can be ‘weakly’ measured more than once with a possibility of extracting more information. It is true that all quantum measurements (projective, non-projective, weak, etc.) can be seen as Positive Opera-tor Valued Measures (POVM). Still it is important to know the details and workings of a measurement scheme. A POVM can also be interpreted as a projective meas-urement on a larger Hilbert space1,9,10. For a finite en-semble the upper bound on the amount of information extractable is available11. There is always a cost of infor-mation extraction from quantum systems in terms of the disturbance caused and that too it has also been explored for the case of weak measurements12–15. A recent work suggests some new possibilities that weak measurements can offer with respect to Heisen-berg’s uncertainty relation and the disturbance caused to the state16. Oreshkov and Brun17, wrote down a weak measurement POVM and showed that any generalized measurement can be decomposed into a sequence of weak measurements, without using an ancilla. Lundeen et al. recently came up with a method employing weak values to directly measure the wave function of a quantum sys-tem in a pure state18 and followed it up with a method to measure any general state19. For some further develop-ments in this regard see ref. 20. Unsharp measurements have also been used to make sequential measurements on a single qubit6. Other examples of quantum state tomo-graphy with weak measurements can be found in refs 21–23. An approach to perform quantum state tomography using weak measurement POVMs was introduced by Hofmann24. In this paper, we present some of our results on state estimation by ‘weak’ measurements and a more detailed account for the qubit case is available in our recent paper25. We study the case of a single qubit and show by explicit simulations how under certain circumstances the weak measurement-based state estimation scheme can beat the one based on projective measurements.

Weak and unsharp measurements

The measuring apparatus plays a crucial role in quantum measurements; on the one hand it interacts with the quantum

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system and on the other hand it has classical properties where the outcomes can be read out and recorded. A use-ful model of this process is available due to von Neu-mann. Although originally this model was constructed for strong (projective) measurements26, it has wider applica-tions and can also be applied to weak measurements2–8,27.

von Neumann’s measurement model for discrete basis

Consider the measurement of an observable A of a quan-tum system with eigenvectors {|aj} and eigenvalues {aj}, j = 1,…, n. Imagine an apparatus with continuous pointer positions described by a variable q and its conjugate vari-able p such that [q, p] = i. The initial state of the measur-ing device has an initial spread of q with its Gaussian quantum state |in centred around zero given by

1/4 2

in| d exp | ,2 4

qq q

(1)

where = 1/(q)2 and we have taken = 1. The system and the measuring device are made to interact by means of a Hamiltonian H = g (t – t)A p, (2) where p is the momentum conjugate to the variable q, and g is the coupling strength. The Hamiltonian is so chosen that the system and the device get a kick and interact momentarily at t = t. Let the initial state |in of the system be written in terms of the eigenstates |a1, |a2, …, |an of the operator A.

in1

| | .n

i ii

c a

(3)

The joint evolution of the system and the measuring device under the coupling Hamiltonian gives an entan-gled state for t > t

in inexp d | |i H t

1/4 2

1

( )d exp | | .2 4

ni

i ii

q gaqc a q

(4)

The above state consists of a series of Gaussians centred at ga1, ga2, …, gan for the pointer entangled with corre-sponding eigenstates |a1, |a2, …, |an of the system. At this stage we invoke the fact that the apparatus is classi-cal, because only one of the pointer positions actually shows up. This requires the collapse of the wave function which is brought in as something from outside for the

classical apparatus! Thus the process is completed with the meter showing only one of the gais and the system state collapses into the corresponding eigenstate |ai. The above analysis holds good only if the Gaussians are well separated or distinct. In contrast, when the Gaus-sians overlap, which can happen if the coupling strength g is small or the initial spread in the pointer state given by 1/ is large, the scenario changes2,5,28. This is called the weak or unsharp measurement regime. Weak meas-urements have been employed in developing recipes for the violation of Bell inequalities28 and Leggett-Garg inequalities29. These have also been recently used to study super-quantum discord30.

Weak values and post-selection

In the treatment of weak measurement given by Aharo-nov, Albert and Vaidmann (AAV), first a subsequent pro-jective measurement of a second observable B is carried out, followed by a post-selection of the output state into one of the eigenstates of the second observable, say |bj. The weak value of the observable A, which was measured in the weak regime, is then defined as

in

in

| |.

|j

wj

b AA

b

(5)

When the post-selected state bj is nearly orthogonal to the initial state |in eq. (5) tells us that the weak value becomes very large, so large that it can lie outside the allowed range of the eigenvalue spectrum2,27. The interpretation of weak values is a current topic of research in quantum information theory. Weak values can be complex and the real and the complex parts can be interpreted in terms of the displacements in the position and momentum spaces respectively, of the measuring device31. Weak values have been used to reinterpret the flow of time in quantum mechanics32 and in the direct measurement of the photon wavefunction18 and in the amplification of small signals33–35. Another interesting application of weak values is in connection with quantum Chesire cat experiments36,37. There have been criticisms of the method of post-selection as well, namely that the process of post-selection leads to throwing away data and can lead to suboptimal use of information from a meas-urement. For discussions on the same see refs 38–40. However, we take a different approach in our work, where we do not do any post-selection, i.e. we consider weak measurements without weak values.

Effect of weak and strong measurement on a qubit

How exactly do we carry out the weak measurement? How much is the effect of a weak measurement on the

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CURRENT SCIENCE, VOL. 109, NO. 11, 10 DECEMBER 2015 1941

system? If we carry out weak measurements on all the members of an identically prepared ensemble, what hap-pens to such an ensemble? We illustrate these points by taking an example. Consider a measurement of z (z component of spin) of a qubit in a fixed quantum state. Following the general prescription given in eq. (2) we write the interaction Hamiltonian as H = g (t – t)z p, (6) assuming the initial state of the pointer to be the same as that given in eq. (1). The qubit is taken to be in a pure state given by

in| cos |0 sin |1 ,2 2

(7)

where |0 and |1 are the eigenstates of z with eigen val-ues +1 and –1 respectively. The combined state of the system and the pointer after the interaction is given by taking a special case of eq. (4)

1/4 2

out( )| d cos exp |0 |

2 2 4q gq q

1/4 2( )d sin exp |1 | .

2 2 4q gq q

(8)

At this stage the apparatus and the system are in an en-tangled state. An observation of the apparatus will lead to values whose distribution is determined by the above state. It is clear from eq. (8) that the distribution of values of the apparatus is a Gaussian centred around +g for the system input state |0 and is a Gaussian centred around −g for the system input state |1. The width of the Gaus-sian in each case is given by 1/. By tuning the parameter = g we can change the nature of the measurement in terms of its strength. In our work we have taken g = 1 so that we have = . For large values of we have a

Figure 1. The schematic diagram of our prescription involving two weak measurements of coupling strengths 1 and 2, allowing state re-cycling, followed by a projective measurement.

projective measurement, where the pointer distributions are well separated for the states |0 and |1. Therefore, each reading of the pointer tells us exactly what the state of the system is after the measurement. By repeatedly measuring the same observable we can calculate the ex-pectation value of the observable. The state collapses completely in each measurement and there is no question of re-using these states. However, when the value of is small we have two Gaussians that overlap. From an ob-servation of the pointer we do not learn with certainty as to what value to assign to the system spin z component. The pointer positions are weakly correlated with the eigenstates of z. The state is only partially affected and there is a possibility of re-using the state. The effect of the weak measurement in this case can be explicitly cal-culated and it turns out that there is very little change in the state of the system. The final state of the system can be calculated by taking the state in eq. (4) and then taking a partial trace over the apparatus’s degrees of freedom giving us the final mixed state corresponding to the sys-tem alone

8

8

1 cos (1 )sin1 .2 (1 )sin 1 cosf

(9)

Since is small we can conclude that the disturbance caused to the system is also small. Furthermore, the disturbance can be controlled by changing .

Quantum state estimation of a single qubit

We now turn to the question of using weak measurements with state recycling for the problem of state estimation of a single qubit.

The scheme

In our prescription, we consider a finite size ensemble of pure or mixed states of a qubit. On every member of the ensemble we carry out a z measurement whose strength is defined by the parameter 1. We record the meter read-ing in each case and keep the modified states after measurements to obtain a changed ensemble. This new ensemble is now used to measure x in the same way but with a coupling strength 2. Finally the resultant ensem-ble is used to carry out projective measurement of y on its members. The first two measurements are weak and the last measurement is strong or projective. To avoid statistical errors the results are averaged over many runs. The entire process is summarized in Figure 1. For both the weak measurements, consider a regime in which is neither too large to make the measurement projective, nor too small, as is done in traditional weak measurements. For such values of , the two Gaussians, representing the

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