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. For example, the Panchratnam geomet- One of the basic postulates of quantum mechanics limits ric phase is nothing but the phase of a complex valued weak the possible quantum mechanical observables to be the Her- value that arises in the context of weak measurements [17]. mitian ones [1]. The Hermiticity of the quantum mechanical It has been shown that weak measurements can be used for observables seems to be a compelling and plausible postulate interrogating quantum systems in a coherent manner [18, 19]. as the eigenvalues of Hermitian operators are real. Moreover, In addition, it plays important role in understanding the uncer- a Hermitian Hamiltonian yields a unitary evolution leading to tainty principle in the double-slit experiment [20, 21], resolv- the conservation of probability. But the reality of the spec- ing Hardy’s paradox [22], analyzing tunneling time [23, 24], trum of the quantum mechanical observables does not imply protecting quantum entanglement from decoherence [25, 26], that the observables must be Hermitian. In fact, there are cer- modifying the decay law [27]. Remarkably, it is possible to tain class of operators that are not Hermitian yet their spec- express the wavefunction as a weak value of a projector and trum is real. The reason for the reality of such operators is this paved the way to measure the wavefunction of single pho- argued to be the underlying symmetry of the operators with ton directly [28, 29]. Similarly, in quantum metrology the certain other restrictions. This has resulted in the attempts to phase sensitivity of a quantum measurement is given by the lift the postulate of Hermiticity and allow for more general variance of the imaginary parts of the weak values of the gen- operators. It is known that there are non-Hermitian operators erators over the different measurement outcomes [30]. For which possess real eigenvalues if one imposes some symme- a very recent review on weak measurements one can look at try conditions, namely the PT −symmetry, which is unbro- Ref. [31]. ken. PT −symmetry is said to be not spontaneously broken Despite having complex spectrum, in general, the non- if the eigenfunctions of the non-Hermitian operator are itself Hermitian operators have found applications in theoretical PT −symmetric. Such kind of operators which respect the work as a mathematical model for studying open quantum sys- unbroken PT −symmetry are ingredients of PT −symmetric tems in nuclear physics [32] and quantum optics [33], among quantum mechanics [2–4]. others to name a few. In these fields, the non-Hermitian In quantum theory, the concept of weak measurement was Hamiltonian appears as an effective description for the sub- arXiv:1406.3007v2 [quant-ph] 9 Nov 2015 introduced by Aharonov-Albert-Vaidman [5–7] to study the system of the full system. The adiabatic measurements on sys- properties of a quantum system in pre and postselected en- tems evolving according to effective non-Hermitian Hamil- sembles. In this formalism the measurement of an observ- tonians are analyzed in Ref. [34] and it is established that able leads to a weak value of the observable with unexpect- the outcome of an adiabatic measurement of a Hermitian ob- edly strange properties. In fact, the weak value is shown to servable is the weak value associated with the two-state vec- be complex, in general, and can take values outside the eigen- tor comprising of forward and backward evolving eigenstates spectrum of the observable. The concept of weak measure- of the non-Hermitian Hamiltonian. Non-Hermitian operators ments has been generalized further beyond its original formu- that can be expressed as a product of two non commuting Hermitian operators do appear in the formalism that describes quantum states using quasiprobability distribution such as the Dirac distribution [35–38], the Moyal distribution [39, 40], ∗ [email protected] etc. Also, weak measurement of a Hermitian operator on a y [email protected] system having non-Hermitian Hamiltonian is considered in z [email protected] Ref. [41]. Apart from these limited expositions of the mea- 2 surement of non-Hermitian operators, not much is discussed II. NON-HERMITIAN OPERATORS towards experimental methods to measure the expectation val- ues of such operators. Here we propose an experimentally ver- For the sake of clarity and completeness, in this section we ifiable procedure to measure the complex expectation value of will review the non-Hermitian operators in quantum mechan- a general non-Hermitian operator. The key to measurement of ics. The abstract mathematical description of quantum me- non-Hermitian operators is the notion of the polar decomposi- chanics is facilitated by the introduction of a separable Hilbert tion of any operator and the process of weak measurement. In space H which by definition is complete and endowed with this paper, we show that the average of a non-Hermitian op- an inner product [42]. The states of the physical system and erator is a complex multiple of the weak value of the positive the physical observables are mapped one to one to the rays semi-definite part of the non-Hermitian operator. By experi- in the Hilbert space and Hermitian operators on the Hilbert mentally verifiable procedure to measure non-Hermitian oper- space, respectively. The physically measurable quantity asso- ators we mean that the expectation value of a non-Hermitian ciated to the operator O of a system in state j i 2 H is the operator in a quantum state is inferred from a direct measure- expectation value h jOj i. It is assumed here that the oper- ment of the weak value of its Hermitian positive definite part ator O is Hermitian so that its expectation value is real. It is employing the theoretically determined expectation value of worth pointing out that the notion that an operator is Hermi- the unitary part of the non-Hermitian operator. Significantly, tian or not depends on an inner product, e.g. O is Hermitian if our method can be used to measure the matrix elements of O = Oy where adjoint Oy of O is defined as any non-Hermitian operator. It is important to note that our proposed method to measure the expectation value require a h jOjφi = hOy jφi; (1) priori knowledge of both the operator to be measured and the for all j i 2 H and jφi 2 H in the domain of O. A closely state in which it is being measured. However, this situation related concept is of a self adjoint operator which is a Hermi- may arise naturally in many contexts, therefore it does not tian operator with the same domain for its adjoint [43]. With limit the applicability of our method severely. Besides the a given Hilbert space and inner product defined over it, the mentioned situations in quantum optics where non-Hermitian operators O for which O 6= Oy is called as a non-Hermitian operators appear, one noteworthy example is the quantum sys- operator. Since the notion of Hermiticity or non-Hermiticity tem described by evolution in presence of gain or loss [4]. is relative to some inner product, a non-Hermitian operator Also, we would like to mention here that our approach is dif- relative to some inner product can be turned into a Hermitian ferent from the one that is used in the direct measurement of operator relative to some other inner product. A very good ex- wavefunction [28, 29] and goes beyond it as the former is ap- position of this fact can be found in Refs. [44–46]. The non- plicable to any non-Hermitian operator. Moreover, we prove a Hermiticity arising from the representation of an operator by new uncertainty relation for any two non-Hermitian operators square integrable functions such that these functions are not and show that it can also be tested experimentally. As an ap- in the domain of the operator is not considered here. In this plication, we show that the uncertainty in the Kraus operators work we focus on the measurability of non-Hermitian opera- governs the fidelity of the output state for a quantum chan- tors keeping the notion of inner product fixed once and for all.