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Introduction to Weak Measurements and Weak Values

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Introduction to Weak Measurements and Weak Values Introduction to Weak Measurements and Weak Values Boaz Tamir 1 & Eliahu Cohen 2 1 Faculty of Interdisciplinary Studies, S.T.S. program, Bar-Ilan University, Ramat-Gan, Israel. E-mail: [email protected] 2 School of Physics and Astronomy, Tel Aviv University, Tel Aviv, Israel. E-mail: [email protected] Editors: Avshalom C. Elitzur & Henry Krips Article history: Submitted on April 4, 2012; Accepted on May 2, 2013; Published on May 15, 2013. e present a short review of the theory of weak ing the state into eigenvectors. This is done by a weak measurement. This should serve as a map for coupling between the measurement device and the sys- Wthe theory and an easy way to get familiar tem. Weak measurements generalize ordinary quantum with the main results, problems and paradoxes raised (projective) measurements: following the weak coupling by the theory. the state vector is not collapsed but biased by a small Quanta 2013; 2: 7–17. angle, and the measurement device does not show a clear eigenvalue, but a superposition of several values. One question is immediately raised: what type of information 1 Introduction can weak measurement reveal? Weak measurement is increasingly acknowledged as one While trying to answer the above question we real- of the most promising research tools in quantum mechan- ize that weak measurement theory presents a plethora of ics. Tasks believed to be self contradictory by nature such quantum strange phenomena, not yet completely under- as ‘determining a particle’s state between two measure- stood. In the paper we shall present and discuss some of ments’ prove to be perfectly possible with the aid of this the paradoxes and peculiarities of the theory. technique. Similarly for the theory within which weak measurement has been conceived, namely the two-state 2 Weak measurements: the vector formalism. Within this framework several new questions can be raised, for which weak measurement one-state vector formalism turns out again to be the most suitable tool for seeking We start with a short review of ordinary strong measure- answers. In the following we very briefly present some ment, then we define the notion of weak measurement as a of the techniques, problems and paradoxes arising from generalization of strong measurement. Next we describe the theory. some properties of weak measurement. Weak measurements can reveal some information about the amplitudes of a quantum state without collaps- 2.1 Strong quantum measurements Let j i be a pure vector state. Consider the standard for- This is an open access article distributed under the terms of the Creative Commons Attribution License CC-BY-3.0, which malism of quantum measurement theory. We differentiate permits unrestricted use, distribution, and reproduction in any medium, between the outcome of the measurement m, the probabil- provided the original author and source are credited. ity to get that measurement outcome p(m), and the state Quanta j DOI: 10.12743/quanta.v2i1.14 May 2013 j Volume 2 j Issue 1 j Page 7 of the system after the measurement j mi [1–4]. 2.2 Weak measurements: definition We can describe the measurement process using the There are many motivations for generalizing quantum formalism of vector operators. Let Mˆ be a set of lin- m projective measurements into weak measurements: main- ear operators satisfying P Mˆ y Mˆ = Iˆ then we will say m m taining the initial state while gradually accumulating in- that the set defines a quantum measurement where the formation [5], determining the system’s state in-between probability is defined by two strong measurements [6] and revealing unusual weak y values [7] are just three examples, others will be described p(m) = h jMˆ Mˆ mj i (1) m below. and the vector state (non normalized) after the measure- Weak measurement should be treated as a generaliza- ment is tion of quantum strong measurement. In weak measure- ment theory both the system and the measurement device j mi = Mˆ mj i (2) are quantum systems [8]. Weak measurement consists of y If Mˆ m are also projective operators i.e. Mˆ m Mˆ m = Mˆ m two steps. In the first step we weakly couple the quantum then we say that the measurement is projective. measurement device to the quantum system. In the sec- ond step we strongly measure the measurement device. Example 1 The collapsed state of the measurement device is referred to as the outcome of the weak measurement process. For Consider the state a measurement to be weak, the standard deviation of the measurement outcome should be larger than the differ- 1 j i = p (j0i + j1i) (3) ence between the eigenvalues of the system. We will now 2 describe this process in details. The procedure resembles the strong measurement process described in [1], how- where j0i and j1i are the eigenvectors of the Pauli spin ever here we use a very weak entanglement between the matrix in the z-direction system and the measurement device (see [8]). ! 1 0 Let jφdi denote the wave function of the measurement Sˆ = (4) z 0 −1 device. When represented in the position basis it will be written as Z Measuring the state using the projectors Mˆ 0 = j0ih0j, jφi = jφdi = φ(x)jxidx (7) x Mˆ 1 = j1ih1j gives the result 0 or 1 respectively with probability 1 . Following the measurement, the state of where x is the position variable of the measuring needle. 2 ˆ ˆ the system is j0i or j1i respectively. Let Xd be the position operator such that Xdjxi = xjxi ˆ ˆ In some cases it is enough to know the probabilities of (here, we use X to distinguish the operator X from its ˆ the measurement outcome e.g. when the final state is not eigenvector jxi and eigenvalue x, the same for P, the so important. In such cases it is enough to demand the subscript d is used for measuring device). We will also assume that φ(x) behaves normally around 0 with some existence of a set of positive operators Eˆm such that variance ∆ = σ2: X ˆ ˆ Em = I (5) 2 − 1 −x2=4σ2 φ x πσ 4 e m ( ) = (2 ) (8) where the probability to get m is We will later (strongly) measure jφdi i.e. collapse the device’s needle to get a value which is the weak measure- p(m) = h jEˆmj i (6) ment’s outcome. Let S denote our system to be measured. Suppose Aˆ is This is known as positive operator-valued measurement an Hermitian operator on the system S . Suppose Aˆ has N (POVM). As a general scheme for implementing a POVM eigenvectors ja ji such that Aˆja ji = a jja ji. one can couple the system with an ancilla which repre- Consider the general state vector j i expressed in the sents the environment, evolve the whole system and an- eigenbasis of Aˆ: cilla using a unitary operator, and then measure the ancilla X space using a projective measurement. This produces a j i = α jja ji (9) series of positive operators on the original system. The j technique is frequently used in quantum operations and Consider the interaction Hamiltonian Hˆint [8, chapter 7] is also known as the operator-sum representation (see also [4, chapter 8]). Hˆ = Hˆint = g(t)Aˆ ⊗ Pˆd (10) Quanta j DOI: 10.12743/quanta.v2i1.14 May 2013 j Volume 2 j Issue 1 j Page 8 Here g(t) is a coupling impulse function satisfying We will now strongly measure the needle of the measuring device. Suppose the needle collapses to the vector jx0i, Z T g(t)dt = 1 (11) then our system is now in the state 0 2 2 (x0−~=2) (x0+~=2) − 2 − 2 [e 4σ αj0i + e 4σ βj1i] ⊗ jx0i (18) where T is the coupling time and Pˆd is the operator con- jugate to Xˆd such that [Xˆd; Pˆd] = {~. without the integration. The eigenvalue x0 could be any- We shall start the measuring process with the vector where around 0 or 1, or even further away especially if σ is big enough i.e. the measurement is very weak. j i ⊗ jφ(x)i (12) If the needle collapses to a value x0 around 1 this means that the amplitude to post-select j0i is a little higher than in the tensor space of the two systems. Then we apply the the amplitude to post-select j1i and vice versa. So the Hamiltonian collapse of the needle biases the system’s vector. However ˆ e−{Ht=~j i ⊗ jφ(x)i (13) if σ is very big with respect to the difference between the eigenvalues of Sˆ then the bias will be very small and It is easy to see that on each of the vectors ja i⊗jφ(x)i the z j the outcome system’s vector will be very similar to the Hamiltonian Hˆ takes Xˆ to Xˆ +a , (Heisenberg evolution) d d j original vector. Z T To sum up, the system’s vector is being biased a little in @Xˆd Xˆd(T) − Xˆd(0) = dt a direction that corresponds to the needle’s outcome value. @t 0 This is a generalization of strong measurements. On the Z T { = [Hˆ ; Xˆd]dt = a j (14) one hand the information we get, that is the value of the 0 ~ needle, is very vague, and on the other hand the system’s (see [9, section 8.4]). The corresponding transformation vector does not collapse but is being biased a little. If the of the coordinates of the wave function is measurement is getting stronger we will have a clearer value, very close to one of the eigenvalues of the system −{HTˆ =~ X e j i ⊗ jφi = α jja ji ⊗ jφ(x − a j)i (15) and an almost collapse of the system’s vector i.e.
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