Measuring Non-Hermitian Operators Via Weak Values

Measuring Non-Hermitian Operators via Weak Values

Arun Kumar Pati,1, ∗ Uttam Singh,1, † and Urbasi Sinha2, ‡ 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 † [email protected] system having non-Hermitian Hamiltonian is considered in ‡ [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 values 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 |ψi ∈ H is the operator in a quantum state is inferred from a direct measure- expectation value hψ|O|ψi. 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 = O† where adjoint O† 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 hψ|O|φi = hO†ψ|φi, (1) priori knowledge of both the operator to be measured and the for all |ψi ∈ H and |φi ∈ 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= O† 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. nel. If the total uncertainties in the Kraus operators is less, The non-Hermitian operators naturally occur in effective de- then the fidelity will be more. We illustrate our main results scriptions of open quantum systems [32], systems in presence with several examples. Our method allows one to measure of loss and gain [4] and quantum optics [33], etc (see also the average of creation and annihilation operators in any state [45]). In general, the expectation value of a non-Hermitian and provides an experimental method to test the Ramanujan operator in some state is a complex number. Various physical formula for the sum of square roots of first s natural numbers. interpretations of the complex expectation of non-Hermitian operators in a quantum state are discussed in Ref. [45] for ex- The paper is organized as follows: In Sec.II we give a very ample in the scattering experiments. The adiabatic measure- brief review of non-Hermitian operators in quantum mechan- ments with an experimental proposal on systems evolving ac- ics. We discuss our method to measure expectation values cording to effective non-Hermitian Hamiltonians can be found of non-Hermitian operators in Sec. III. In Sec.IV, we find in Ref. [34] and the results on weak measurement of a Hermi- an uncertainty relation for non-Hermitian operators and using tian operator on a system having non-Hermitian Hamiltonian the results of the previous section, we make a connection of can be found in Ref. [41]. the uncertainty relation to experiments. To exemplify this, we In the next section, we provide our protocol to measure the provide the uncertainty relation for creation and annihilation complex expectation values of non-Hermitian operators em- operators. In Sec.V, we find a relation between fidelity of a ploying weak measurements. quantum channel and uncertainty of non-Hemitian Kraus operators of the channel and provide an example of amplitude damping channel as an illustration. In Sec.VI, we give a III. EXPECTATION VALUE OF A NON-HERMITIAN comprehensive account of the interesting applications of our OPERATOR results in the context of testing stronger uncertainty relation, measuring product of projection operators and in verifying Let us consider a non-Hermitian operator A. The expecta- Ramanujan’s formula. Finally, we conclude in Sec. VII to- tion value of such an operator in a quantum state |ψi, given gether with the discussion and implications of our results. by hψ|A|ψi, is in general a complex number. This makes 3 √ it unobservable in a laboratory experiment. But here we S = URU † = AA† is a positive semi-definite operator. In present a formalism to overcome this problem. To present this case the average of A in a pure state |ψi is given by the main idea, we need the polar decomposition of a matrix. Let A ∈ Cm×n, m ≥ n. Then, there exists a matrix hAi = hψ|A|ψi = hψ|SU|ψi m×n U ∈ C and a unique Hermitian positive semi-definite hψ|S|χi n×n † = hψ|χi = hSiwhψ|χi, (5) matrix R ∈ C such that A = UR, with U U√ = I. The hψ|χi ψ χ positive semi-definite matrix R is given by R = A†A, even if A is singular. If rank (A) = n then R is positive definite and w where |χi = U|ψi and ψhSi is the weak value of posi- U is uniquely determined [47, 48]. χ tive semi-definite operator S, given by hψ|S|χi . Following the Let us consider a quantum system initially in the state |ψi ∈ hψ|χi d same procedure as above one can measure the weak value of H = C . Suppose we are interested in measuring the average of a non-Hermitian operator A in the state |ψi. Consider the S with preselection in the state |χi = U|ψi and postselection d×d in the state |ψi. Now the weak value, hSiw, multiplied by polar decomposition of an operator A ∈ C , given by A = ψ χ UR, where R is a positive semi-definite operator and U is a hψ|χi yields hψ|A|ψi. Furthermore, we have unitary operator. The average of a non-Hermitian operator in w w the pure state |ψi, is given by hψ|A|ψi = ψhSiχ hψ|χi = φhRiψ hφ|ψi. (6)

∆A∆B ≥ | [S PS ]w| · |hφ|χi|, (10) φ A B χ 00 20 40 60 80 100 s w where φ[SAPSB]χ is the weak value of the non-Hermitian operator S PS and it can be determined using our experi- A B FIG. 1. (Color online) The plot of LHS and RHS of Eq. (14), given mentally viable method. For the case of Hermitian operators, respectively by Eq. (15) and Eq. (16), as a function of s. Both the we have the Robertson uncertainty relation [50]. axes are dimensionless. In the plot triangles show LHS and circles As an example of the generalized uncertainty relation, show RHS. This figure clearly shows that the uncertainty relation given by Eq. (10), we consider the creation and annihilation given for non-Hermitian operators is satisfied by the creation and operators, which are non-Hermitian, for a single mode elec- annihilation operators in phase states. tromagnetic field, in the phase state |θmi [51, 52]. The phase states are the eigenstates of the Hermitian phase operator. The s = hθ |aâˆ†|θ i, hθ |aˆ|θ i† = hθ |aˆ†|θ i = phase operator arises in the context of the polar decomposition 2 m m √ m m m m −1 −iθm Ps † 2 of creation and annihilation operators and it is known that the (s + 1) e n=0 n and hθm|(â ) |θmi = (s + −2 −2iθm Ps p polar decomposition of creation and annihilation operators for 1) e n=0 n(n − 1), we have the radiation field has difficulties related to the unitary part of s the decomposition [53–55]. This problem is addressed as the s X √ ∆â†∆â = (∆â†)2 = − (s + 1)−2 nm . (15) non-existence of Hermitian phase operator for the infinite di- 2 mensional Hilbert space [56, 57]. The problem is resolved m,n=0 by taking Hilbert space to be finite dimensional and taking the and limit at the end of all the calculations. In the finite dimensional † 2 † 2 Hilbert space there is a well defined Hermitian phase operator |h(â ) i − haˆ i | called as the Pegg-Barnett phase operator [51]. This leads to s s √ −1 X p −1 X the polar decompositions of the creation and the annihilation = (s + 1) n(n − 1) − (s + 1) nm . operators, which are given by n=0 m,n=0 (16) ˆ p iφθ ˆ aˆ = e N and One can see from Fig. (1) that the uncertainty relation given † p −iφˆ by Eq. (14) for creation and annihilation operators is indeed aˆ = Neˆ θ , (11) satisfied. ˆ where φθ is the Hermitian phase operator,

s V. UNCERTAINTY FOR KRAUS OPERATORS AND ˆ X FIDELITY OF QUANTUM STATES φθ = θm|θmihθm|, (12) m=0 Here we will show how our proposal of measuring the av- with θm = θ0 + 2mπ/(s + 1) and |θmi are the orthonormal erage of non-Hermitian operator could be of high value. This phase states, given by will also show that the uncertainty in the non-Hermitian op- s erator can have some real physical meaning. Suppose that −1/2 X inθm a quantum system, initially in the state |ψi, passes through |θmi = (s + 1) e |ni. (13) a quantum channel. The state of the system after passing n=0 through the quantum channel is given by ˆ The phase states satisfy e±iφθ |θ i = e±iθm |θ i. Now, the X m m |ψihψ| → ρ = E(|ψihψ|) = E |ψihψ|E†, (17) generalized uncertainty relation, given by Eq. (8), for the cre- k k ation and annihilation operators of a single mode electromag- k 0 netic field, in the phase state |θmi reads as where Eks are the Kraus elements of the channel. Now, the fidelity between the pure initial state and the mixed final state ∆â†∆â ≥ |h(â†)2i − haˆ†i2|, (14) is given by

p † † X 2 where ∆A = hθm|A A|θmi − hθm|A |θmihθm|A|θmi F = hψ|ρ|ψi = |hψ|Ek|ψi| . (18) † † with A = a, a . Using the expressions hθm|aˆ aˆ|θmi = k 5

Eq. (18) shows that, by measuring the average of the non- Lower bound ∆E . ∆E Hermitian operators Ek in the state |ψi, one can find the fi- 0.25 1 2 delity between the input and the output states. Note that, usu- Upper bound ally, to measure the fidelity of a channel, one has to do a quan- 0.2 tum state tomography of the final state and then calculate the quantity hψ|ρ|ψi. However, as stated above, by weakly mea- 0.15 suring the positive semi-definite part of the Kraus operators, 0.1 one can measure the average of the Karus operators and hence the channel fidelity. 0.05 Lower and upper bounds Now consider the variance of Ek in the state |ψi. This is 0 0 0.2 0.4 0.6 0.8 1 given by p

2 † † ∆Ek = hψ|EkEk|ψi − hψ|Ek|ψihψ|Ek|ψi. (19) FIG. 2. (Color online) The lower and upper bounds on ∆E1∆E2. Here both the axes are dimensionless. In the plot blue squares show P † ∆E ∆E , green triangles and red solid line show the lower bound Summing both the sides and using the relation E Ek = I, 1 2 k k (25) and upper bound 1−F , respectively on ∆E ∆E as a function we have 2 1 2 of p at fixed values of θ = π/2 and φ = π/4. X 2 F + ∆Ek = 1. (20) k √ Here, S1 = E1, U1 = I, S2 = p|0ih0| and U2 = σx. We have This relation gives us a physical meaning to the uncertainties ⊥ ⊥ w in the Kraus operators. This shows that if the total uncertainty |φ[S1|ψ ihψ |S2]χ | · |hφ|χi| in the Kraus operators is less, then the fidelity between the in- θ 4 θ √ p put and output states will be more. Thus, the fidelity and the = 2 cos φ cos2( ) sin ( ) p[1 − 1 − p], (25) 2 2 uncertainty play a complementary role in the quantum chan- θ θ √ √ 2 θ nel. Hence, to preserve a state more efficiently, one should ∆E1 = cos( 2 ) sin( 2 )[1 − 1 − p] and ∆E1 = p cos ( 2 ). have less uncertainties in the Kraus operators. For a quantum Fig. (2) shows the bounds on ∆E1∆E2 as a function of p at channel with two Kraus elements, the fidelity F and uncer- fixed value of θ = π/2 and φ = π/4, which validates Eq. (21) tainties of the Kraus operators satisfy for the amplitude damping channel.

1 − F ≥ ∆E ∆E ≥ | [S |ψ⊥ihψ⊥|S ]w| · |hφ|χi|, (21) 2 1 2 φ 1 2 χ VI. APPLICATIONS where E1 = S1U1, E2 = S2U2, |φi = U1|ψi, |χi = U2|ψi, ⊥ ⊥ In this section, we provide various interesting applications and |ψihψ| + |ψ ihψ | = I. of our results. In particular, we will show a way to test the We illustrate our uncertainty relation for Kraus operators stronger uncertainty relation [58], to measure the product of and its relation to ﬁdelity with the amplitude damping chan- projection operators. Interestingly, we show that our results nel. The Kraus operators for the amplitude damping channel can also be used to verify the Ramanujan’s sum formula [59]. are given by We, also, consider the application of our results in case of PT √ symmetric Hamiltonians. 1 0 0 p E = √ , and E = . (22) 1 0 1 − p 2 0 0 A. Testing stronger uncertainty relation θ iφ θ If we pass an arbitrary state |ψi = cos 2 |0i + e sin 2 |1i of the qubit through the amplitude damping channel, then the Uncertainty relation plays a fundamental role in quantum output state is given by mechanics and quantum information theory. Recently, the

2 stronger uncertainty relation [58] (compared to the Robertson X ρ = E |ψihψ|E† uncertainty relation [50]) is proved which shows that the sum k k of variances of two incompatible observables, A and B in a k=1 1 state |ψi, is given by = [e |0ih0| + e |1ih1| + e |0ih1| + e∗|1ih0|], (23) 2 1 2 3 3 ∆A2 + ∆B2 ≥ ±ihψ|[A, B]|ψi + |hψ|A ± iB|ψ¯i|2, (26) ¯ where e1 = 1 + p + (1 − p) cos θ, e2 = (1 − p)(1 − cos θ) and where |ψi is a state orthogonal to |ψi. For two canonically −iφ√ e3 = e 1 − p sin θ. The fidelity F = hψ|ρ|ψi is given by conjugate pair of observables such as position X and momentum P (~ = 1 and X,P are dimensionless), the stronger un- 1 certainty relation reads as F = [3 + p1 − p − p + 2p cos θ + (1 − p − p1 − p) cos 2θ]. 4 2 2 † ¯ 2 (24) ∆X + ∆P ≥ 1 + 2|hψ|a |ψi| , (27) 6 √ where a† = (X − iP )/ 2 is the creation operator and it is used to measure expectation value of any non-Hermitian oper- indeed a non-Hermitian operator. The Heisenberg-Robertson ator, the earlier method is applicable only to the cases of non- uncertainty relation [50] only implies that one has ∆X2 + Hermitian operators which are product of non-commuting ∆P 2 ≥ 1, while the new relation is stronger. We will show Hermitian operators. Next, we consider an example of non- that our protocol can be used to test this. The variances in the Hermitian operator which is not product of Hermitian opera- position and the momentum can be tested using the standard tors. method and the last term can be actually measured using our scheme. Specifically, we will show that it is modulus squared of the weak value of positive semi-definite part of the non- C. Average of the creation operator and the Ramanujan sum Hermitian operator multiplied by a real number. First note that † ¯ 2 ¯ 2 we can write |hψ|a |ψi| = |hψ|a|ψi| . Let a = UR, then we Let us consider the creation and the annihilation operators † ¯ 2 w 2 2 † ¯ have |hψ|a |ψi| = |φhRiψ | |hφ|ψi| , where |φi = U |ψi. for a single mode electromagnetic field. The polar decompo- Therefore, by measuring the non-Hermitian operator we can sitions of the creation and the annihilation operators are given test the stronger uncertainty relation. by Eq. (11). Now consider the expectation value of the creation operator in a general state from the (s + 1) dimensional Hilbert space spanned by phase states, given in Eq. (13). This B. Measurement of product of two non-commuting projectors is given by

w p ˆ p † ˆ −iφθ ˆ Consider measurement of Πi(B)Πj(C), with Πi(B) = hψ|aˆ |ψi = hψ| Ne |ψi = ψh Niχ · hψ|χi, (30) |ψiihψi| and Πj(C) = |φjihφj|, where |ψii and |φji (i, j = ˆ 1, 2.., d) are eigenstates of two non-commuting Hermitian op- where |χi = e−iφθ |ψi. Thus, by measuring the weak value erators B and C, respectively. The product Πi(B)Πj(C) is a of square root of number operator with preselection in the non-Hermitian operator. In fact, the average of this operator state |χi and postselection in the state |ψi leads to the ex- in a quantum state is nothing but the discrete version of the pectation value of aˆ† in the state |ψi. Consider a general Dirac distribution [35–38]. We will show that our method can Ps Ps 2 state |ψi = m=0 cm|mi with m=0 |cm| = 1. Here Π (B)Π (C) ˆ be applied to measure the expectation value of i j −iφθ Ps −i(s+1)θ0 |χi = e |ψi = m=1 cm−1|mi + cse |0i and using the polar decomposition and the weak measurement. w √ p Ps c c∗ m h Nˆi = m=1 m−1 m For the non-Hermitian operator A = hψi|φji|ψiihφj|, let ψ χ ∗ −i(s+1)θ0 Ps ∗ . Therefore, we csc0 e + m=1 cm−1cm the polar decomposition be denoted by A = UR, where † Ps ∗ √ have hψ|a |ψi = m=1 cm−1cm m. For equally super- R = |hψi|φji||φjihφj| and U is determined by relation iνm p posed number state, i.e., cm = e / (s + 1), we have iη iη hψi|φj i U|φji = e |ψii, where e = . Such a unitary oper- |hψi|φj i| s ator is given by e−iν X √ hψ|a†|ψi = m. (31) s + 1 d−1 m=1 X U = U(m) = eiη |ψ ihφ |, (28) k⊕m k Using the result that for any real number r with r ≥ 1 and k=0 positive integer n with j ⊕ m = i and ⊕ denotes the addition modulo d. Now s the expectation value of Π (B)Π (C) in a state |ψi is given X 1/r r r+1 1 1 i j m = (s + 1) r − (s + 1) r − Φ (r), (32) by r + 1 2 s m=1

w 0 hψ|Πi(B)Πj(C)|ψi = hψ|UR|ψi = ψ0 hRiψ · hψ |ψi, (29) where Φs(r) is a function of r with s as a parameter and is bounded between 0 and 1/2 [59√, 61]. Putting r = 2 in the 0 † Ps 2 3/2 1 where |ψ i = U |ψi. Thus, the expectation value of above formula, we get m=1 m = 3 (s + 1) − 2 (s + 1/2 Πi(B)Πj(C) in the state |ψi is given by the weak value 1) − Φs(2), where 0 ≤ Φs(2) ≤ 1/2. Therefore, we have 0 w hψ |R|ψi † e−iν 2 3/2 1 1/2 0 hRi = R ψ ψ hψ0|ψi of multiplied by a complex number hψ|a |ψi = s+1 [ 3 (s + 1) − 2 (s + 1) − Φ(1/2)]. In- hψ0|ψi. terestingly, one can invert Eq. (31) to get The weak average [29], without postselection, of a non- s iν Hermitian operator A in a state ρ is equal to, in general com- X √ e † w m = hψ|a |ψi. (33) plex, expectation value of A in the state ρ, i.e., hA iρ = s + 1 m=1 Tr[Aρ]. Following [38, 60], one can devise an experimental method to measure this complex expectation value of A. Therefore, one can use the expectation value of the creation The method is shown only for non-Hermitian operators which operator employing our method based on weak measurements are product of non commuting Hermitian operators. In this to estimate the sum of the square roots of the ﬁrst s natural method the interaction Hamiltonian for system and apparatus numbers and then this result can be compared to the Ramanu- PN is designed to be H = g Ai ⊗ Pi in order to measure jan formula [59] for the above series. This is another interest- Q i=1 expectation value h i Aii. Unlike our method, which can be ing application of our formalism. 7

D. Measurement of PT symmetric Hamiltonian where U11 = [(A+(s−t))B− +(A−(s−t))B+ −2t(B+ − iθ B−)]re , U12 = t[(A − (s − t))B− + (A + (s − t))B+ − 2 2r (B+ −B−), U21 = s[(A+(s−t))B− +(A−(s−t))B+ − There exists a class of Hamiltonians which are non- 2 2r (B+ − B−) and U22 = [(A − (s − t))B− + (A + (s − Hermitian and yet they possess real eigenvalues when they −iθ respect unbroken PT symmetry [2,3]. However, in general t))B+ − 2s(B+ − B−)]re . For the special case of s = t, they posses non-normalizable eigenstates and complex eigen- r 6= ±s and r > s, we have H = UR, where values, so one may think that we cannot measure their ex- r se−iθ eiθ 0 R = ,U = . (41) pectation values. But using our formalism which is based seiθ r 0 e−iθ on weak measurements one can in principle measure them. Therefore, given a non-Hermitian Hamiltonian, one can check For other special case of s = t, r 6= ±s and r < s, we have whether its expectation value indeed gives a complex number. H = UR, where The simplest example of a general PT symmetric Hamil- s re−iθ 0 1 tonian in 2 × 2, is given by [3] R = ,U = . (42) reiθ s 1 0 reiθ t H = , (34) Now the expectation value of A in a general single qubit state s re−iθ |ψi = cos(η/2)|0i + eiξ sin(η/2)|1i is given by where r, s, t, θ are the real parameters. The eigenvalues are hψ|H|ψi = r cos θ + s cos ξ sin η + ir sin θ cos η. (43) p given by = r cos θ ± st − r2 sin2 θ and corresponding ± A |ψi eigenstates of this Hamiltonian are given by The above expectation value of in the state can be real- ized if we experimentally measure the Hermitian operator R iα/2 −iα/2 with the preselection in the state |ψi and postselection in the 1 e 1 e † |+i = √ −iα/2 ; |−i = √ iα/2 , state U |ψi and then multiply the weak value thus obtained 2 cos α e 2 cos α −e with the complex number hψ|U|ψi. where α is deﬁned by the relation sin α = √r sin θ. Let the st √ polar decomposition of H be H = UR, where R = H†H and for r2 6= st, U = HR−1. Here, VII. CONCLUSION

r2 + s2 r(s + t)e−iθ R2 = . (35) In this paper, we have addressed the question of experimen- r(s + t)eiθ r2 + t2 tal feasibility of measuring the expectation value of any non- Hermitian operator in a pure quantum state. We show that the Therefore, the positive semi-definite operator R for the non- expectation value of a non-Hermitian operator in a quantum Hermitian operator H is given by state is equal to the weak value of the positive semi-definite part of the operator, modulo a complex number. Our method 1 R R to measure the expectation value require a priori knowledge R = √ 11 21 , (36) 2 2A R12 R22 of both the operator to be measured and the state in which it is being measured. However, this situation may arise naturally in where R11 = (A − (s − t))B− + (A + (s − t))B+, R12 = many contexts, therefore our method is widely applicable. We −iθ iθ 2r(B+ −B−)e , R21 = 2r(B+ −B−)e and R22 = (A+ have provided several examples to illustrate this technique. In (s − t))B− + (A − (s − t))B+ with particular, we have provided a relation between the average of the Kraus elements of a channel and the channel fidelity. p A = 4r2 + (s − t)2 (37) We have also applied our method to measure the expectation value of creation operator in a general state. This leads to an p 2 2 2 B± = 2r + s + t ± (s + t)A. (38) interesting link between the sum of the square roots of first s natural numbers and the expectation value of the creation Now, we have operator. Furthermore, we have proved an uncertainty relation for any two non-Hermitian operators. Our method also 1 S S helps to test the stronger uncertainty relation, experimentally. R−1 = √ 11 12 , (39) Our paper may open up the possibility of considering the non- 2AB+B− S21 S22 Hermitian operators not only as a mathematical tool but also an experimental arsenal such as in scattering experiments in where S11 = (A + (s − t))B− + (A − (s − t))B+, S12 = −iθ iθ quantum physics and quantum information theory, in general. −2r(B+ − B−)e , S21 = −2r(B+ − B−)e , and S22 = −1 (A − (s − t))B− + (A + (s − t))B+. Using U = HR , we have ACKNOWLEDGEMENTS Uttam Singh acknowledges the research fellowship of Depart- 1 U U U = √ 11 12 , (40) ment of Atomic Energy, Government of India. 2AB+B− U21 U22 8

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