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Discussion Meeting on Quantum Measurements Centre for Quantum Information and Quantum Computation Indian Institute of Science, Bangalore 22-24 October 2014

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Discussion Meeting on Quantum Measurements

Centre for Quantum Information and Quantum Computation
Indian Institute of Science, Bangalore
22-24 October 2014

Organizing Committee

N.D. Hari Dass Dipankar Home Anil Kumar
CMI, Chennai (Chairman) Bose Institute, Kolkata IISc, Bangalore (Convener)

  • IISER, Pune
  • T.S. Mahesh

Archan S. Majumdar Kota Murali
SNBNCBS, Kolkata IBM, Bangalore

  • IISc, Bangalore
  • Apoorva Patel

Arun K. Pati R. Simon
HRI, Allahabad IMSc, Chennai
Urbasi Sinha RRI, Bangalore Rajamani Vijayaraghavan TIFR, Mumbai

Talks

  • 9:30-10:20
  • 22 October 2014

Quantum measurement theory and the uncertainty principle

Masanao Ozawa
Graduate School of Information Science, Nagoya University, Chikusa-ku, Nagoya, Japan

Heisenberg’s uncertainty principle was originally formulated in 1927 as a quantitative relation between the “mean error” of a measurement of one observable and the “discontinuous change” (or the disturbance) thereby caused on another observable, typically explained through the gamma ray microscope thought experiment. Heisenberg derived this relation under an additional assumption on quantum measurements that is consistent with the so-called repeatability hypothesis, which was postulated in von Neumann’s measurement theory and supported by Schroedinger and contemporaries. However, the repeatability hypothesis has been abandoned in the modern quantum measurement theory, originated by Davies and Lewis in the 1970’s. Later, the universal validity of Heisenberg’s uncertainty principle was questioned typically in a debate on the sensitivity limit to gravitational wave detectors in the 1980’s. A universally valid form of the error-disturbance relation was derived in the modern framework for general quantum measurements by the present speaker in 2003. Since then we have experienced a considerable progress in theoretical and experimental studies of universally valid reformulation of Heisenberg’s uncertainty principle. This talk will give an up-to-date survey on the historical background and the recent progress and in particular discuss debates on the quantum mechanical generalizations of the notion of root-mean-square error and disturbance.

  • 10:20-11:10
  • 22 October 2014

Fundamental phenomena in quantum mechanics studied with neutrons:
Error-disturbance uncertainty relation and quantum Cheshire-cat

Yuji Hasegawa
Atominstitut, TU-Wien, Vienna, Austria

Interferometer and polarimeter experiments with neutrons matter-wave serve as almost ideal tools to investigate foundation of quantum mechanics. The former device explicitly exhibits quantum interference between spatially separated beams and is used for studies, e.g., of intra-partite entanglement, i.e., entanglements between degrees of freedom. In contrast, interference effects between two spin eingenstates are exposed in the latter apparatus, where quantum manipulation with extremely high precision is accomplished. Exploiting both strategies, alternative theories of quantum mechanics, Kochen-Specker theorem and a crypto-contextual model suggested by Leggett are studied. Recently, an experimental test of the error-disturbance uncertainty relation by Heisenberg is carried out, which confirmed clearly the violation of the original relation and support a new relation proposed by Ozawa. In addition, new neutron optical experiments studying weak-values are developed: a new counter-intuitive phenomenon, called a quantum Cheshire-cat, is observed in a neutron interferometer experiment. In my talk, I am going to give an overview of neutron optical approach to investigations of fundamental aspect of quantum mechanics.

  • 11:30-12:20
  • 22 October 2014

A disturbance tradeoff principle for quantum measurements

M.D. Srinivas
Centre for Policy Studies, Chennai

We demonstrate a fundamental principle of disturbance trade-off for quantum measurements, along the lines of the celebrated uncertainty principle: The disturbances associated with measurements performed on distinct yet identically prepared ensembles of systems in a pure state cannot all be made arbitrarily small. Indeed, we show that the average of the disturbances associated with a set of projective measurements is strictly greater than zero whenever the associated observables do not have a common eigenvector. For such measurements, we have an equivalence between disturbance tradeoff measured in terms of fidelity and the entropic uncertainty principle formulated in terms of the Tsallis entropy (T2). We also consider the disturbance tradeoff for POV observables, where the disturbance tradeoff differs significantly from the uncertainty tradeoff. Specifically, for the class of Lu¨ders instruments, we show that the disturbance tradeoff for POV observables can be zero even when the corresponding uncertainty tradeoff is strictly greater than zero.

  • 12:20-13:10
  • 22 October 2014

Experimental test of error-disturbance uncertainty relations by weak measurement

Keiichi Edamatsu
Research Institute of Electrical Communication, Tohoku University, Aoba-ku, Sendai, Japan
In collaboration with: So-Young Baek, Fumihiro Kaneda, Masanao Ozawa

We experimentally test the error-disturbance uncertainty relation (EDR) exploiting a weak measurement for a single-photon polarization qubit. The test is carried out by linear optical devices that realize variable measurement strength from weak measurement to projection measurement. The results show that Heisenberg EDR is violated throughout the range of our measurement strength and yet validates Ozawa and Branciard EDRs. A correct understanding and experimental confirmation of the error-disturbance uncertainty relation will not only foster an insight into fundamental limitations of measurements but also advance the precision measurement technology in quantum information processing.

  • 14:30-15:00
  • 22 October 2014

Weak measurements: Typical weak and superweak values

Pragya Shukla
Department of Physics, Indian Institute of Technology, Kharagpur

Weak values, resulting from the action of an operator on a pre-selected state when measured after post-selection by a different state, can lie outside the spectrum of eigenvalues of the operator: they can be ‘super-weak’. This phenomenon can be quantified by averaging over an ensemble of the two states, and calculating the probability distribution of the weak values. Our main results indicate an unanticipated universality in the distribution of weak and super-weak values. If there are many eigenvalues, distributed within a finite range, this distribution takes a simple universal generalized Lorentzian form, and the ’super-weak probability’, of weak values outside the spectrum, can be as large as 0.293 (almost 30% chance of a weak value being superweak). By contrast, the familiar expectation values always lie within the spectral range, and their distribution, although approximately Gaussian for many eigenvalues, is not universal.
We also explore the connection between large shifts of a pointer in a weak measurement and fast oscillations in functions associated with pre- and post-selected states. Our results reveal how the supershifts emerge as the uncertainty in the initial pointer position increases. This happens if the initial pointer wavefunction is Gaussian or Lorentzian but not if it is an exponential.
[1] M.V. Berry and P. Shukla, J. Phys. A: Math. & Theo. 43 (2010) 354024. [2] M.V. Berry and P. Shukla, J. Phys. A: Math. & Theo. 45 (2012) 015301. [3] M.V. Berry, M.R. Dennis, B. McRoberts and P. Shukla, J. Phys. A: Math. & Theo. 44 (2011) 20530. [4] M.V. Berry, N. Brunner, S. Popescu and P. Shukla, J. Phys. A: Math. & Theo. 44 (2011) 492001.

  • 15:00-15:50
  • 22 October 2014

Protective measurements: Probing single quantum systems

Tabish Qureshi
Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi

Protective measurements, introduced by Aharonov, Anandan and Vaidman, are described and critically analyzed. Several subtleties involved in the process are discussed. Implications for the reality of the quantum state is critically examined. Some proposals for experimental realization of protective measurements are discussed.

  • 16:10-16:40
  • 22 October 2014

Beating entropic uncertainty bound using unsharp measurements

A.R. Usha Devi
Department of Physics, Bangalore University, Bangalore

One of the peculiar aspect of quantum theory is that the outcomes of measurements of non-commuting observables cannot be predicted accurately. This feature is captured quantitatively in terms of uncertainty relations, which give upper bounds on the precision with which the measurement outcomes of incompatible observables can be predicted. Massen-Uffink entropic uncertainty relation [1] constrains the sum of entropies associated with the measurements of a pair of non-commuting observables. An extended entropic uncertainty relation proposed by Berta et al. [2], brought out that it is possible to beat the entropic uncertainty bound when the state of the system is entangled with a quantum memory. In this talk, we discuss the entropic uncertainty bound in the presence of a quantum memory, when unsharp POVM measurements are employed. We show that when the POVMs are jointly measurable [3], beating the entropic uncertainty bound implies violation of an entropic steering inequality [4]. Very recently, it has been established that non-steerability and joint measurability of POVMs imply one another [5]. As a consequence, it is clearly seen that the reduction in the entropic uncertainty bound (which is in turn connected with steerability) is possible iff the POVMs are not jointly measurable. We also discuss an extended entropic uncertainty relation, wherein we have shown [6] that there is a reduction in the uncertainty bound, if the measurement outcomes of non-commuting observables are conditioned by sequential projective measurements of temporally separated observables on a single quantum system. We identify that it is not possible to beat the uncertainty bound if the sequential measurements are done using compatible (jointly measurable) POVMs.
[1] H. Maassen and J.B.M. Uffink, Phys. Rev. Lett. 60 (1988) 1103. [2] M. Berta, M. Christandl, R. Colbeck, J.M. Renes and R. Renner, Nature Physics 6 (2010) 659. [3] P. Busch, Phys. Rev. D 33 (1986) 2253. [4] J. Schneeloch, C.J. Broadbent, S.P. Walborn, E.G. Cavalcanti and J.C. Howell, Phys. Rev. A 87 (2013) 062103. [5] M.T. Quintino, T. Vertesi, and N. Brunner, arXiv:1406.6976; R. Uola, T. Moroder, and O. Guhne, arXiv:1407.2224. [6] H.S. Karthik, A.R. Usha Devi, J. Prabhu Tej and A.K. Rajagopal, arXiv: 1310.5079.

  • 16:40-17:10
  • 22 October 2014

A class of distance-based incompatibility measures for quantum measurements

Prabha Mandayam
Department of Physics, Indian Institute of Technology Madras, Chennai

We propose a class of incompatibility measures for quantum observables based on quantifying the effect of a measurement of one observable on the statistics of the outcomes of another. For finite dimensional systems, we obtain a tight upper bound on the incompatibility of any set of observables and show that the bound is attained when the observables are totally nondegenerate and associated with mutually unbiased bases. In the process, we also establish an important connection between the incompatibility of a pair of observables and the maximal disturbances due to their measurements. Finally, we indicate how these measures of incompatibility and disturbance can be extended to the more general class of non-projective measurements. In particular, we obtain a non-trivial upper bound on the incompatibility of one Lu¨ders instrument with another.
Reference: P. Mandayam and M.D. Srinivas, Phys. Rev. A 89 (2014) 062112.

  • 17:10-17:40
  • 22 October 2014

Probing gravity in quantum systems with protective measurement

Patrick Dasgupta
Department of Physics and Astrophysics, University of Delhi, Delhi

Anandan in 1993 had presented an interesting quantum mechanical analysis of a test particle interacting gravitationally with a bound state formed by superposing two spatially separated energy eigenstates [1]. Mapping the problem mathematically to a Stern-Gerlach type system, he had argued that it is possible to protect the bound system from becoming entangled with the states of the test particle. I revisit the problem and critically assess the conditions required for such a protective measurement. It is seen that even a feeble interaction like gravitation leads to a phase difference between the spatially separated energy eigenstates. Then, I study the case of a superposed BoseEinstein condensate (BEC) of cold atoms interacting gravitationally with a test particle, and consider the possibility of measuring the phase difference induced by gravitation. BEC of bosonic atoms trapped in honeycomb optical lattice are described by a nonlinear Dirac equation in the long wavelength limit [2]. In this context, the effect of a test particles gravity on the wavefunction components associated with inequivalent sites A and B of the optical sublattice, making up the hexagonal structure, is investigated.
[1] J. Anandan, Found. Phys. Lett. 6 (1993) 505532. [2] Patrick Das Gupta, Samiran Raj and Debapriya Chaudhuri, arXiv:1012.0976, and references therein.

  • 9:30-10:20
  • 23 October 2014

Three results on weak measurements

N.D. Hari Dass
Centre for Quantum Information and Quantum Computing, Indian Institute of Science, Bangalore
Chennai Mathematical Institute, Chennai

In this talk I shall present three important results on weak measurements that I have obtained recently. They are: i) repeated mea surements on a single copy can not provide any information on it and further that in the limit of very large such measurements, weak measurements have exactly the same characterstics as strong measurements, ii) the apparent non-invasiveness of weak measurements is no more advantageous than strong measurements in the specific context of Leggett-Garg measurements when errors are properly taken into account and finally, iii) weak value measurements are optimal, in the precise sense of Wootters and Fields, when the post-selected states are mutually unbiased with respect to the eigenstates of the observable whose weak values are being measured.

  • 10:20-11:10
  • 23 October 2014

Quantum measurement with superconducting circuits

Rajamani Vijayaraghavan
Department of Condensed Matter Physics and Materials Science, Tata Institute of Fundamental Research, Mumbai

The past three decades have seen a tremendous rise in experiments probing the quantum nature of various atomic, photonic and solid-state systems. Though primarily driven by the promise of powerful quantum computing machines, these developments are allowing us to revisit some of the very fundamental ideas in quantum mechanics by providing a near ideal experimental platform. One such architecture is based on constructing quantum two level systems or qubits with superconducting electrical circuits operating at a temperature of approximately 10 mK. A near ideal quantum measurement setup is then implemented by coupling these qubits to electromagnetic cavities and probing them with microwave signals. I will describe this so called circuit Quantum Electro Dynamics (cQED) architecture and how one can control the rate of quantum measurement by controlling the microwave power. In particular, I will focus on superconducting parametric amplifiers which allow us to monitor the measurement signals with high efficiency. I will discuss different quantum measurement regimes and how one can use such measurements to implement quantum feedback to stabilize various quantum states. Finally I will discuss their application to quantum error correction which is crucial for the development of practical quantum computing systems.

  • 11:30-12:20
  • 23 October 2014

Experimental tests of macrorealism: An assessment

Anupam Garg
Department of Physics and Astronomy, Northwestern University, Evanston, IL, USA

I will discuss the expeimental test of Leggett and Garg’s macrorealistic inequality by Palacios-Laloy et al. [Nature Physics 6 (2010) 442-447]. The key question, of course, is what new bearing this experiment has on the issue of macrorealism, but additional goals include understanding the purported macrovariable, the quantity being measured, the measurement method, and additional assumptions that might be needed in applying the inequality to this particular setup.

  • 12:20-13:10
  • 23 October 2014

Some recent studies related to the Leggett-Garg inequality

Dipankar Home
Department of Physics, Bose Institute, Kolkata

After an introduction to the basics of the Leggett-Garg inequality (LGI), its underlying motivation and its various forms, a brief indication will be given of the various directions of studies being pursued by our group concerning various aspects of LGI, including a few interesting features of LGI in the context of unsharp or imprecise measurements. The remainder of the talk will be mainly concentrated on discussing our recently studied application of LGI in revealing non-classicality of the harmonic oscillator coherent state and probing its macrolimit, along with results relevant to realizable experimental study.

  • 14:30-15:00
  • 23 October 2014

An evolutionary formalism for weak measurements

Apoorva Patel
Centre for High Energy Physics, Indian Institute of Science, Bangalore

Unitary evolution and projective measurement are the conventional axioms of quantum mechanics. Even though projective measurement yields one of the eigenstates of the measured operator as the outcome, there is no theory that predicts which eigenstate will be observed in which experimental run. There exists only an ensemble description, which predicts probabilities of various outcomes over many experimental runs. We propose a dynamical evolution equation for the projective collapse of the quantum state in individual experimental runs. In case of gradual weak measurements, its predictions for ensemble evolution are different from those of Born’s rule. It is an open question whether or not suitably designed experiments can observe this alternate evolution.

  • 15:00-15:50
  • 23 October 2014

Model study of coherence and decoherence

Sushanta Dattagupta
Visva-Bharati, Santiniketan, West Bengal

One of the critical issues of quantum information processing, in general and quantum computing, in particular, is the maintenance of Coherence. Unfortunately however, small systems (which are ideal for quantum applications) are invariably under the influence of the environment that leads to De-coherence, The environmental effects manifest as noise, dissipation, dephasing etc. I will present a few examples in which these issues can be studied effectively. The examples are in the context of Tunneling Systems, Landau Diamagnetism, the Aharonov-Bohm effect and Telegraphic Noise in Qubits. The last case will be dealt with in some detail, wherein we will discuss how coherence can be partially salvaged even though the quantum device is strongly under the impact of a classical noise source.

  • 16:10-17:00
  • 23 October 2014

Quantum non-demolition experiments: Concepts, theory and practice

C.S. Unnikrishnan
Department of High Energy Physics, Tata Institute of Fundamental Research, Mumbai

The focus of this talk is quantum non-demolition (QND) experiments in the general sense of a measurement on a quantum system without uncontrolled change of the uncertainty dispersion associated with the state of the system, for the observable being measured. The fundamental ideas, relevant theory and the conditions and scope for applicability will be discussed. There are several types and flavors of such measurements, but the most common are the ones called the ’back-action’ evading measurements. Precision measurements have indeed gained from developing QND measurements and some examples from quantum optics, gravitational wave detectors and spin-magnetometry will be discussed.
The distinction between QND measurements and some other types of commonly discussed quantum-specific measurements, like interaction-free measurements, Zeno-measurements, weak measurements etc. will be brought out, briefly.

  • 17:00-17:30
  • 23 October 2014

Can weak value amplification be singled out in atomic resonance fluorescence process by pre- and post-selection of system states?

Andal Narayanan
Light and Matter Physics, Raman Research Institute, Bangalore In collaboration with: U. Satya Sainadh, R. Vathsan, S.N. Sandhya

In Quantum Mechanics, a weak measurement refers to a measurement process wherein the system which is being measured is only slightly disturbed by the measuring apparatus (pointer). A pointer shift during weak measurement, after post-selection of the system in a chosen final state through a strong projective measurement, is called the weak value. For some pre- and post-selected states, the pointer shift may correspond to a measurement result which is well outside the eigenspectrum of the measured observable: a process known as weak value amplification. In a recent paper [I. Shomroni et al., Phys. Rev. Lett. 111 (2013) 023604] an experimental demonstration of weak value amplification was presented for an atomic V system in coherent interaction with electromagnetic fields. The weak value amplification was shown as an increase of spontaneous emission lifetime of the atom in the excited state, for particular pre- and post-selected system states. In this talk, we present the result of our density matrix calculations for this very system, which shows that, the increase in lifetime has more to do with the coherent nature of atom-photon interaction than the weak measurement process employed. As an important corollary to this study, we address the more fundamental question of whether decoherence processes induced by system-bath interaction can naturally lead to final states which can be interpreted as weak value amplified states.

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    Quantum Physics in Consciousness Studies Dirk K. F. Meijer and Simon Raggett Review/Literature compilation: The Quantum Mind Extended* Contents: Introduction in Quantum aspects of brain function, p 1-19 Quantum approaches to neurobiology: state of art, p 19-31 David Bohm: Wholeness and the implicate order, p 30-39 Henry Stapp: Attention, intention and quantum coherence, p 39-44 Roger Penrose: Consciousness and the geometry of universe, p 45-54 Stuart Hameroff: Objective reduction in brain tubules, p 55-68 Hiroomi Umezawa and Herbert Frohlich: Quantum brain dynamics, p 69-72 Mari Jibu & Kunio Yasue: Quantum field concepts, p 72-76 Johnjoe McFadden: Electromagnetic fields, p77-80 Gustav Bernroider: Ion channel coherence, p 80-85 Chris King: Cosmology, consciousness and chaos theory, p 85-92 Piero Scaruffi: Consciousness as a feature of matter, p 92-94 Danko Georgiev: the Quantum neuron, p 94-98 Andrei Khrennikov: Quantum like brain and other metaphoric QM models, p 98-102 Hu and Wu/ Persinger: Spin mediated consciousness, p 103- 106 Chris Clarke: Qualia and free will, p 106- 109 Herms Romijn: Photon mediated consciousness and recent models, p, 110-114 Stuart Kauffman: Consciousness & the poised state p, 114-116 Post-Bohmian concepts of an of a universal quantum field, p 117-121 Dirk Meijer: Cyclic operating mental workspace, p 121-131 Amit Goswami: The vacuum as a universal information field, p 132-146 Simon Raggett: A final attempt to a theory on consciousness p, 146-157 Note on cited sources, p 158 References, p 158-175 Internetsites, p 175 1 Introduction in quantum aspects of brain function Since the development of QM and relativistic theories in the first part of the 20th century, attempts have been made to understand and describe the mind or mental states on the basis of QM concepts (see Meijer, 2014, Meijer and Korf, 2013,).
  • Relativistically Invariant Bohmian Trajectories of Photons

    Relativistically Invariant Bohmian Trajectories of Photons

    Relativistically invariant Bohmian trajectories of photons Joshua Foo ( [email protected] ) University of Queensland https://orcid.org/0000-0002-6435-0187 Estelle Asmodelle University of Queensland Austin Lund Centre for Quantum Computation and Communications Technology https://orcid.org/0000-0002-1983- 3059 Timothy Ralph University of Queensland Article Keywords: Bohmian mechanics, quantum theory, photons Posted Date: August 10th, 2021 DOI: https://doi.org/10.21203/rs.3.rs-783149/v1 License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License Relativistically invariant Bohmian trajectories of photons Joshua Foo,1, ∗ Estelle Asmodelle,1 Austin P. Lund,2, 1 and Timothy C. Ralph1, † 1Centre for Quantum Computation & Communication Technology, School of Mathematics & Physics, The University of Queensland, St. Lucia, Queensland, 4072, Australia 2Dahlem Center for Complex Quantum Systems, Freie Universitat¨ Berlin, 14195 Berlin, Germany (Dated: August 5, 2021) Bohmian mechanics is a nonlocal hidden-variable interpretation of quantum theory which predicts that par- ticles follow deterministic trajectories in spacetime. Historically, the study of Bohmian trajectories has been restricted to nonrelativistic regimes due to the widely held belief that the theory is incompatible with special relativity. Here we derive expressions for the relativistic velocity and spacetime trajectories of photons in a Michelson-Sagnac-type interferometer. The trajectories satisfy quantum-mechanical continuity,
  • Proposal to Observe the Nonlocality of Bohmian Trajectories with Entangled Photons

    Proposal to Observe the Nonlocality of Bohmian Trajectories with Entangled Photons

    Proposal to Observe the Nonlocality of Bohmian Trajectories with Entangled Photons The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Braverman, Boris, and Christoph Simon. “Proposal to Observe the Nonlocality of Bohmian Trajectories with Entangled Photons.” Physical Review Letters 110.6 (2013). © 2013 American Physical Society As Published http://dx.doi.org/10.1103/PhysRevLett.110.060406 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/78322 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. week ending PRL 110, 060406 (2013) PHYSICAL REVIEW LETTERS 8 FEBRUARY 2013 Proposal to Observe the Nonlocality of Bohmian Trajectories with Entangled Photons Boris Braverman1,2 and Christoph Simon2 1Department of Physics, MIT-Harvard Center for Ultracold Atoms and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2Institute for Quantum Information Science and Department of Physics and Astronomy, University of Calgary, Calgary T2N 1N4, Alberta, Canada (Received 11 July 2012; published 7 February 2013) Bohmian mechanics reproduces all statistical predictions of quantum mechanics, which ensures that entanglement cannot be used for superluminal signaling. However, individual Bohmian particles can experience superluminal influences. We propose to illustrate this point using a double double-slit setup with path-entangled photons. The Bohmian velocity field for one of the photons can be measured using a recently demonstrated weak-measurement technique. The found velocities strongly depend on the value of a phase shift that is applied to the other photon, potentially at spacelike separation.
  • Measuring Non-Hermitian Operators Via Weak Values

    Measuring Non-Hermitian Operators Via Weak Values

    Measuring Non-Hermitian Operators via Weak Values Arun Kumar Pati,1, ∗ Uttam Singh,1, y and Urbasi Sinha2, z 1Harish-Chandra Research Institute, Allahabad 211 019, India 2Raman Research Institute, Bangalore 560 080, India In quantum theory, a physical observable is represented by a Hermitian operator as it admits real eigenvalues. This stems from the fact that any measuring apparatus that is supposed to measure a physical observable will always yield a real number. However, reality of eigenvalue of some operator does not mean that it is necessarily Hermitian. There are examples of non-Hermitian operators which may admit real eigenvalues under some symmetry conditions. However, in general, given a non-Hermitian operator, its average value in a quantum state is a complex number and there are only very limited methods available to measure it. Following standard quantum mechanics, we provide an experimentally feasible protocol to measure the expectation value of any non-Hermitian operator via weak measurements. The average of a non-Hermitian operator in a pure state is a complex multiple of the weak value of the positive semi-definite part of the non-Hermitian operator. We also prove a new uncertainty relation for any two non-Hermitian operators and show that the fidelity of a quantum state under quantum channel can be measured using the average of the corresponding Kraus operators. The importance of our method is shown in testing the stronger uncertainty relation, verifying the Ramanujan formula and in measuring the product of non commuting projectors. I. INTRODUCTION lation [8–16]. In recent years, weak values have found nu- merous applications.