Theory and Experiment for Resource-Efficient Joint Weak

Theory and experiment for resource-eﬃcient joint weak- measurement

Aldo C. Martinez-Becerril1, Gabriel Bussières1, Davor Curic2, Lambert Giner1,3, Raphael A. Abrahao1,4, and Jeﬀ S. Lundeen1,4

1Department of Physics and Centre for Research in Photonics, University of Ottawa, 25 Templeton Street, Ottawa, Ontario K1N 6N5, Canada 2 Complexity Science Group, Department of Physics and Astronomy, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 3 Département de Physique et d’Astronomie, Université de Moncton, 18 Ave. Antonine-Maillet, Moncton, New Brunswick E1A 3E9, Canada 4Joint Centre for Extreme Photonics, University of Ottawa - National Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario K1A 0R6, Canada

Incompatible observables underlie pil- 1 Introduction lars of quantum physics such as contextu- ality and entanglement. The Heisenberg Modern quantum measurement techniques have uncertainty principle is a fundamental lim- pushed forward our understanding and ability to itation on the measurement of the product manipulate quantum particles. Often, fundamen- of incompatible observables, a ‘joint’ mea- tal and practical measurements involve the prod- surement. However, recently a method us- uct of two or more observables of a quantum sys- ing weak measurement has experimentally tem. In particular, correlations of incompatible demonstrated joint measurement. This or non-commuting observables A and B, defined method [Lundeen, J. S., and Bamber, C. by [A, B] ≡ AB − BA =6 0, are central to Phys. Rev. Lett. 108, 070402, 2012] deliv- our understanding of entanglement [1,2] and the ers the standard expectation value of the Heisenberg uncertainty principle. The standard product of observables, even if they are in- procedure to measure such product observables compatible. A drawback of this method would be to measure observable A, then measure is that it requires coupling each observ- B. This fails since the first measurement col- able to a distinct degree of freedom (DOF), lapses the state of a particle into an eigenstate of i.e., a disjoint Hilbert space. Typically, A, erasing the information about B and random- this ‘read-out’ system is an unused inter- izing its value. nal DOF of the measured particle. Un- fortunately, one quickly runs out of inter- In contrast, weak measurement preserves the nal DOFs, which limits the number of ob- quantum state of the system and thereby allows servables and types of measurements one one to obtain correlations between any chosen can make. To address this limitation, set of general observables, including incompati- we propose and experimentally demon- ble ones [3–10]. Weak measurement is a type arXiv:2103.16389v1 [quant-ph] 30 Mar 2021 strate a technique to perform a joint weak- of non-destructive quantum measurement that measurement of two incompatible observ- minimizes disturbance of the measured system ables using only one DOF as a read-out [11]. To perform such a measurement, the ob- system. We apply our scheme to directly servable is weakly coupled to a separate read-out measure the density matrix of photon’s po- system (the ‘pointer’) that indicates the average larization. result of the measurement. Weak measurement has a broad range of applications from amplify- ing tiny signals [12–14] to fundamental studies on the meaning of a quantum state [15, 16]. Particu- Aldo C. Martinez-Becerril: [email protected] Jeff S. Lundeen: http://www.photonicquantum.info/ larly relevant to this paper are Refs. [4,5, 17, 18], which showed that if two observables are weakly

1 measured, the average measurement outcome is quires verification of the quality (i.e., ‘fidelity’) simply the expectation value of the product of of resource quantum states. The experimental those two observables, hBAi. Remarkably, this demonstration of the direct measurement of the holds even if A and B are incompatible, which wave function opened up new research lines in would make BA non-Hermitian and. nominally, quantum state estimation. The directness of the unobservable. method means that one can obtain the complex More recently the weak measurement formal- amplitudes of a quantum state, in any chosen ba- ism was expanded to deal with composite sys- sis [15]. This avoids the need to solve inverse tems and performing a measurement of the prod- problems, which is one of the key goals of cur- uct of two or more observables, known as a joint rent research in quantum state estimation [33– weak-measurement. Joint weak-measurement 35]. Further work demonstrated how to estimate has proven to be useful, for example, in exper- a general quantum state by directly measuring imental realizations of the Cheshire cat [19, 20], the density matrix [6, 18], or by directly mea- the Hardy’s paradox [21, 22], the study of quan- suring phase-space quasiprobability distributions tum dynamics, to give insight into the role of time of states, such as the Dirac distribution [36, 37]. ordering in the quantum domain [23, 24]. The Similarly, we apply our single-pointer joint weak- ability to jointly measure incompatible observ- measurement method to directly determine any ables has also shown to have many applications chosen element of the density matrix. Specifi- in the field of quantum metrology [25–30]. cally, we obtain the density matrix of a photon’s polarization using a single pointer for the two req- Known methods for the realization of a joint uisite observables. weak-measurement are resource intensive. Specif- The rest of the paper is organized as follows. ically, they require either interactions that involve We start by describing weak measurement in three or more particles or a separate read-out terms of raising and lowering operators. Then, system for each observable. With a few excep- we outline the theory of our technique to perform tions [21], due to the absence of two-particle in- a joint weak-measurement and introduce the im- teractions, even single-observable weak measure- portant ingredient, the fractional Fourier trans- ment resorts to a strategy of using internal de- form. Next, we present the experimental demon- grees of freedom (DOF) as the read-out systems stration of our technique and an application to [4,5, 17, 18]. For example, one can measure quantum state estimation. Finally, we summa- the polarization of a photon by using its posi- rize our work and point out some future possible tion DOF as a read-out [31]. For a joint mea- directions. surement, this strategy is particularly limiting given that quantum particles have a limited number of DOF. For instance, for a photon there 2 Weak measurement are just four DOF: polarization, and a three- dimensional wavevector (which, in turn, incor- In this section, we introduce a theoretical porates frequency-time and transverse position- model for quantum measurement, von Neu- momentum). Due to this limitation, joint weak- mann’s model, that is typically used to describe measurement experiments have never progressed weak measurement. The model involves a mea- beyond the product of two observables [7, 32]. To sured quantum system S and a pointer system P overcome this constraint, the present work theo- [38]. The latter indicates the measured value, the retically introduces and experimentally demon- read-out, on a meter. A key aspect of the model strates a technique to perform a joint weak- is that the pointer is also quantum mechanical. measurement of multiple observables using a sin- Before the measurement, S and P are in an ini- gle DOF as the read-out system. tial product state, |IiS ⊗ |φiP . Here, ⊗ indicates We implement our technique to directly mea- a tensor product between different Hilbert spaces sure quantum states. This is a type of quan- and the subscript is a label of the system. Both of tum state estimation where the state is fully de- these symbols will be omitted in the rest of the termined by the shift of the pointer. Quantum paper. As usual, we assume that the pointer’s state estimation has become an invaluable tool initial spatial wave function φ(x) is a Gaussian in the subject of quantum information, which re- centered at zero [38]:

2 the value of A (though it is not particularly ob- φ(x) ≡ hx|φi vious in this harmonic oscillator formulation).

x2 So far the model is general and independent 1 − 2 (1) = e 4σx , of the measurement strength. Now we consider 2 1 (2πσx) 4 the weak measurement regime. A weak measure- γ 1 where σx is the standard deviation of the position ment is characterized by , which allows one U |Ii | i probability-distribution. to approximate the evolved state as A 0 = |Ii | i γA |Ii | i γ The pointer’s initial state happens to be same 0 + 1 to first order in . In the weak as the ground state of a harmonic oscillator. regime, the entanglement between the pointer Thus, following Ref. [5], we define a lowering and measured system is reduced, and the ini- |Ii operator a as the operator that annihilates this tial state of the particle is largely preserved. pointer state, a |φi = 0. By this logic, from here Following the work on [11], a post-selection on a |F i on we label the pointer’s initial state as |0i = |φi. final sytem state is performed. Mathemati- hF | As a standard lowering operator, a can be written cally, this amounts to project onto and renor- malizing, after which the pointer’s final state is in terms of the position x and momentum p of the hF |A|Ii |φ0i = |0i + γ |1i pointer as follows a = x/(2σx) + ipσx/¯h. (Note, hF |Ii . This pointer’s final state we use the natural length-scale σx of the system is largely left unchanged. That is, it is mostly left | i γ in place of the mass m and angular frequency ω in 0 , but a small component proportional to , | i that usually appear in the harmonic oscillator: is transferred to 1 due to the interaction with q ¯h S. σx = .) Associated with a, there is a rais- 2mω Our goal is to identify in what manner the ing operator a† that fulfills [a, a†] = 1. Similarly † n pointer is shifted by the interaction. Expecta- |ni (√a ) | i we can define number states = n! 0 . For- tions of the position and momentum of the final mulating the model in terms of lowering, raising pointer, respectively, appear as the real and imag- operators and number states has proven fruitful hai ≡ hφ0|a|φ0i 1 hxi i σx hpi inary parts of = 2σx + ¯h . in the past [5, 17, 18] and will be important for Thus, using |φ0i from just above, one finds what follows. Suppose we want to measure observable A of hF |A|Ii S. Then, in the von Neumann model, one couples hai = γ hF |Ii S to P by the following Hamiltonian, (3) ≡ γ hAiw .

H ≡ gAp Consequently, the pointer is shifted from having g¯h † (2) hxi = hpi = 0 to indicating an average out- = i A(a − a), 1 σ 2σx hAi hxi i x hpi come w = 2σxγ + ¯hγ . This average here g is a real parameter that indicates the in- pointer shift was introduced by Aharonov, Al- teraction strength and we have used the usual de- bert, and Vaidman in Ref. [11] and is called the composition of p in terms of a and a†. We stress ‘weak value’. Unlike in the standard expectation that there is no harmonic potential in the system value, |F i= 6 |Ii and, thus, the weak value is a and thus no quantum harmonic oscillator. We are potentially complex quantity. In summary, the following Ref. [5] and simply using the formalism real and imaginary parts of the weak value are of raising and lowering operators to analyze the the average shifts of the position and momentum eﬀect of the interaction on the pointer state. of the pointer, which, in turn, are given by the We now consider the state of the total sys- expectation value of the lowering operator. tem after the unitary evolution induced by H: tH ∞ γn −i h¯ P n † UA |Ii |0i = e |Ii |0i = n=0 n! A (a − 3 Joint weak-measurement a n |Ii | i γ ≡ gt ) 0 . Here, 2σx is a unitless parameter that quantiﬁes the measurement strength. For composite systems, one is interested in the In general, the evolved system is in an entangled average value of the product of observables such state between S and P. For a strong interaction as hBAi. Universally, this involves correlations (γ 1), in each trial, a measurement of the po- between two measurement outcomes (e.g., as in sition of the pointer will unambiguously indicate Bell’s inequalities). In the von Neumann model,

3 this corresponds to correlations between pointer sured by coupling it to the same photon’s trans- distributions. This is true for both strong and verse spatial DOF. In absence of inter-particle in- weak measurements. In the latter case, the aver- teractions, this facilitates the use of weak mea- age outcome should be the joint weak-value, surement, but quickly uses up all available internal DOF. In turn, this limits the number of hF |BA|Ii observables in the product and limits the num- hBAiw ≡ . (4) hF |Ii ber of DOF that can be used in the measured A number of techniques have been proposed and system for other quantum information tasks. It demonstrated to observe the joint weak-value in is natural to ask: can we perform a joint weak- pointer correlations. We now briefly review these measurement with a single pointer? The main techniques. contribution of the present work is to introduce First, we review the case of compatible opera- and experimentally demonstrate such a tech- tors A and B. These could be two different ob- nique. Our technique uses a sequence of two servables of a single particle or observables acting standard von Neumann interactions, each given on two different particles. Ref. [4] proposed using by Eq.2. Unlike the previous techniques, the a separate von Neumann interaction (i.e., Eq.2) two interactions couple the system to the same and pointer for each observable (pointers 1 and pointer. As in section2, the total initial state 2). This was simplified in [5], which found that is |Ii |0i . The first interaction UA couples the 2 ha1a2i = hBAiw /γ . This strategy of perform- pointer to A, while the second UB couples the ing two separate weak measurements was exper- same pointer to B. The action of two von Neu- imentally demonstrated in [21]. mann unitaries with equal interaction strength γ γB(a†−a) γA(a†−a) A more challenging case, and the subject of is UBUA |Ii |0i = e e |Ii |0i = this work, is the one in which A and B act P∞ γn+m m n † n+m m,n=0 n!m! B A (a − a) |Ii |0i. This is on the same particle, but are incompatible e.g., the final state of the total system after the two two complementary observables such as position interactions. and momentum. Furthermore, the product BA Motivated by the techniques outlined above, is not Hermitian, thus it is not considered a which used correlations between two different valid observable in standard quantum mechan- lowering operators ha1a2i, we will aim to find the ics. For example, naively replacing A with BA expectation of the product of two identical low- in the von Neumann Hamiltonian, Eq.2, results ering operators, a2 . Thus, we must expand the in non-unitary time evolution. However, in the pointer state after the interaction to second or- weak regime, a measurement of A largely pre- der in the interaction strength γ. There are three serves the quantum state of the particle allowing second-order terms: m = n = 1; m = 0, n = 2; a subsequent measurement of B. The correla- and m = 2, n = 0. Along with the zero and first tions between the outcomes of the two measure- order terms, this gives ments gives hBAiw. A technique along these lines was proposed in [17]. As with the compatible ob- U U | i |Ii | i γ A B | i servable case above, it used a separate von Neu- B A 0 = 0 + ( + ) 1 + mann interaction and pointer for each observable 2 √ γ 2 2 3 (pointer 1 for B and pointer 2 for A). In [18], 2BA+A +B ( 2 |2i−|0i)+O γ |Ii . 2 the required correlation between the pointers was 2 (5) shown to be ha1a2i = hBAiw /γ and experimentally demonstrated in [6,7]. In summary, for Now we post-select the system on a final state both compatible and incompatible observables, |F i. To zero order in γ, the renormalized the same technique works. The drawback of the pointer’s final state is technique is that it requires one pointer for each observable. 0 1 φ = hF |Ii |0i + γ hF | A + B |Ii |1i + In particular, this requirement of one pointer hF |Ii per observable is resource intensive. In most im- ! γ2 √ plementations of weak measurement, pointers are hF | 2BA + A2 + B2 |Ii 2 |2i − |0i . internal DOF of the measured particle. For ex- 2 ample, in [31] a photon’s polarization is mea- (6)

4 As per our aim, we now calculate the expectation The reason we have introduced this new ob- value a2 for |φ0i: servable is that the square of d will contain the desired cross terms. Calculating d2 and solving D E D E D E a2 = 2γ2 hBAi +γ2 A2 + B2 . (7) for the cross terms we find w w w ! d2 x2 p2 xp px σ σ − − . This equation contains the weak value of the + = x p 2 2 2 2 (9) σd σx σp product observable hBAiw but also other nontriv- 2 2 ial weak values, A w and B w. However, if we Upon substituting Eq.9 in Eq.8, we obtain an limit the two observables to be projectors, then expression for the real and imaginary parts of A2 A B2 B = and = . This turns the nontrivial hBAiw: weak values into single-observable weak values, * 2 2 + which we can replace with hA + Biw = hai /γ. 1 x p gtx Re hBAiw = 2 2 − 2 − 2 (10) Using this and rearranging Eq.7 to solve for 8γ σx σp σx hBAiw we arrive at and 1 D 2E hBAi = a − γ hai . (8) * 2 2 2 + w 2γ2 1 d x p gt p Im hBAiw = 2 2 2 − 2 − 2 − . 8γ σd σx σp σx σp In this way, we have expressed the joint weak- (11) value solely in terms of expectation values on the Note that every term in Eqs. 10- 11 is a ratio of pointer’s final state. However, an additional step two variables with the same units, therefore each is still necessary. While the expectation value of term is unitless. For the same reason, experimen- a single lowering operator is easily measured in tal scaling factors e.g., a magnification in the x an experiment by measuring x and p in sepa- domain, cancel out. Hence, characterization of rate trials, powers of lowering operators cannot experimental scaling factors is not required for be measured as easily. To solve this, we express the use of our technique. a2 a x i p using = 2σx + 2σp , where we have used In summary, Eqs. 10- 11 express the full com- ¯h BA σxσp = 2 (which is valid since the pointer is in plex joint weak-value for product observable the minimum uncertainty state |0i). Doing so, in terms of Hermitian observables on the pointer’s leads to the appearance of cross terms such as final state. As expected, the joint weak-value ap- xp + px, which do not correspond to a straight- pears in second order powers of x and p and forwardly physical observable. our new observable d. This comprises our pro- To overcome this problem, we can use the Her- posed technique to weakly measure the product mitian observable d which is an equally weighted of incompatible observables using only a single x p d = √σd x + p . pointer. combination of and : 2 σx σp Here, σx, σp and σd are the standard deviations of the pointer in x, p and d spaces, respectively. The 4 Experiment d observable naturally appears in a variety of quantum systems. In the Heisenberg picture in In this section, we present the experimental quantum optics, the x field quadrature rotates to demonstration of our proposed technique using d after an eighth of a period of oscillation; this photons. Specifically we perform a joint weak- is equivalent to an x-p phase-space rotation of measurement of incompatible polarization pro- R(π/2), with R = 1/2 where R is the rotation jectors. The experimental setup is shown in order. Similarly, x rotates to p after a quarter Fig.1. The measured observable will be in the period (R = 1). Just as the Fourier Transform photon’s polarization DOF. The pointer is the links x and p, the fractional Fourier Transform photon’s transverse x position with probability- (FrFT) was introduced to calculate the effect of distribution given by the squared of the state in a rotation order R on a state in the Schrödinger Eq.1 with σx = 403 µm. The photon source is a picture [39]. In summary, there are established He:Ne laser at 633 nm with a power of 1.19 mW. practical methods to physically implement FrFTs The setup can be divided into state preparation, and measure d. weak measurements, strong measurement stages,

5 He:Ne laser PBS HWP at 휃/2 QWP BBO HWP BBO HWP PBS

State preparation Weak measurements Strong measurement Read-out system

Read-out system 4f system output 40 cm 18 cm

푓1 = 100 cm 푓2 = 120 cm 100 cm 4f lens-pair

220 cm 120 cm 29 cm 29 cm CMOS camera

Swappable lenses 푓푝= 100 cm 푓푑= 100 cm 푓푥= 12.5 cm

Figure 1: Experimental setup for performing a joint weak-measurement of a photon’s polarization state using a single pointer, the photon’s transverse x position. We work with three sets of pure polarization states |ψ1i = θ |Hi θ |V i, |ψ i θ |Hi i θ |V i and |ψ i √1 |Hi − ie2iθ |V i . State preparation To cos + sin 2 = cos + sin 3 = 2 : produce such states, we use a polarizing beamsplitter (PBS), a half wave plate (HWP) set at θ/2 and a quarter wave ◦ ◦ plate (QWP). The QWP is removed for preparing |ψ1i, and it is set at 0 and 45 for |ψ2i and |ψ3i, respectively. Weak measurements : A first walk-off crystal (BBO) implements a weak measurement of πj where j can be ◦ ◦ ◦ |Hi or |V i. A HWP at 22.5 and a second BBO effectively perform a weak measurement of π45◦ = |45 i h45 |, |Hi+|V i with | ◦i √ . Strong Measurement A final HWP and a PBS implement a strong measurement in the 45 = 2 : {|Hi , |V i} basis. Read − out system : A 4f lens-pair (f1 = 100 cm and f2 = 120 cm) is required to obtain the probability distributions involved in Eqs. 10- 11. For p, we use a Fourier transform lens of focal length fp = 100 cm, for d a FrFT lens of focal length fd = 100 cm and for x an imaging lens of focal length fx = 12.5 cm. Each lens is set at the specified distance from a fixed CMOS camera, the obtained images are used to calculate the required expectation values as described in the text.

and a read-out section. In order to test our tech- ization observable A = |Hi hH| to the photon’s nique, we prepare a range of polarization states transverse spatial position x that plays the role |Ii = α |Hi + β |V i, where |Hi (|V i) is the hor- of the pointer. The strong measurement regime izontal (vertical) polarization. For state prepa- is characterized by ∆x greater than σx, in which ration, we use a polarizing beam splitter (PBS) the eigenstates of A are fully separated. Our ex- followed by a half wave plate (HWP), set at an periment is performed in the weak measurement angle of θ/2 with respect to the |Hi polarization, regime where ∆x is less than σx. and a quarter wave plate (QWP) (see the caption In order to measure the product of two observ- in Fig.1 for setting details). ables with our technique, the setup performs two A von Neumann measurement of polarization weak measurements in a row. Each von Neumann can be performed with a birefringent crystal (e.g., interaction (i.e., Eq.2) is achieved with a sepa- a BBO crystal) acting as a beam displacer. This rate BBO crystal. Both crystals are aligned such optical component transversely shifts the photon that they shift the transverse proﬁle of horizon- by ∆x = gt = 150 µm if the photon is in the |Hi tally polarized photons in the horizontal direction polarization state and leaves it unshifted if it is x, leaving the transverse proﬁle in the y direc- in |V i. In this way, the crystal couples the polar- tion unchanged. Thus, they couple to the same

6 pointer, the x DOF. The first BBO implements values of fx, fp, and fd were chosen so that each 0 a measurement of A = πH = |Hi hH|. Before measured x distribution spans many pixels. By the second BBO, there is a HWP oriented at switching in one lens at a time, fx, fp, or fd, 22.5◦. This effectively rotates the second mea- the camera effectively measures the correspond- ◦ ◦ sured observable to B = π45◦ = |45 i h45 |, with ing observable. | ◦i |Hi√+|V i A key experimental simplification is that we do 45 = 2 . These two measurements and their read-out constitute an experimental appli- not need to experimentally or theoretically deter- 0 cation of our joint weak-measurement technique mine the proportionality constants between x at that uses a single pointer. Lastly, a strong mea- the camera and x, p, and d. The imaging mag- 0 surement of polarization observable πj (j = H or nification between x and x is an example of such V ) is performed. a proportionality constant. Since they depend on the focal lengths and lens-camera distances, In our experiment, we need the ability to these constants are difficult to experimentally de- measure three incompatible observables of the termine precisely. Instead, Eqs. 10- 11 show that pointer, x, p, and d. This is the read-out of the each observable is divided by the width of the result of the weak measurement. As we will ex- pointer’s initial distribution in that observable, plain, lens transformations will allow us to switch e.g. p/σp. Consequently, the units cancel and all between these spatial observables, transforming 0 calculations can be conducted directly in terms them to a final transverse position x on a cam- of x0, i.e. camera pixel index. era. We measure the probability distribution of the observables in Eqs. 10- 11 on a monochrome 8 bit CMOS camera with a pixel width in x0 of 2.2 µm. To make room for the optical lengths re-

quired for the lens transformations we add a 4f 푤

푉 흅 lens-pair to the imaging system (f1 = 100 cm °

and f2 = 120 cm). The 4f is positioned such that 45 흅 f1 is 100 cm after the crystals. This ensures that the spatial wave function at the exit surface of the second crystal is recreated 120 cm after the f2 lens. Our goal is to leave the camera fixed in Angle 휃 of polarization state [deg] place while different lenses are inserted in order to measure x, p, and d. Figure 2: Joint weak-value of the product of incom- To measure p, and d we use an optical FrFT patible observables π45◦ πV with a post-selection on the 1 2iθ state |Hi. The input state is |Ii = √ |Hi−ie |V i of the spatial DOF. The special case of rotation 2 order R = 1 (a standard Fourier Transform), is obtained by setting the preparation HWP at θ/2 and the QWP at ◦. The real and imaginary parts of the weak already widely used; the transverse position x0 at 45 value are displayed with markers, while solid lines corre- f one focal length after lens p = 100 cm is pro- spond to the joint weak-value, Eq.4. The error bars are portional to p at any distance before the lens. calculated solely from measurement statistics. Hence, lens fp can be placed at any distance after the 4f lens pair as long as it is fp distance from The data acquisition consisted of taking five the camera. Less common is the optical spatial camera images per pointer observable (i.e., per FrFT, which was introduced in [40–42]. At a dis- lens configuration). A background image, taken 0 tance z after lens fd = 100 cm, x will be pro- with the laser blocked, was subtracted from each. portional to the d observable a distance z before The resulting image was integrated along the ver- Rπ Rπ 0 the lens. Here, z = fd tan 4 sin 2 . For d, tical direction y , and normalized to the brightest the phase-space rotation parameter R equals 1/2 image obtained in that configuration. The re- making z = 29 cm. This d lens transformation sulting one dimensional probability distribution fixes the distance of the camera from the 4f lens P (x0), corresponds to the probability of detect- 0 pair. Lastly, a single lens (fx = 12.5 cm) placed ing a photon in position x with final polarization 160 cm after f2 relays the image from the 4f lens |Hi or |V i. With the fx lens in place, this is ef- 0 pair. This ensures that 18 cm after fx, x on the fectively an x read-out. The expectation values camera is proportional to x at the crystals. The required for the joint weak-value (Eqs. 10- 11)

7 0 can be obtained as hxi /σx = hx i /σx0 , where measured observable hCi = tr [Cρ]. Thus, the hx0i = R P (x0)x0dx0. For p and d a similar pro- direct measurement procedure results in a joint cedure is followed. weak-average, ρ(i, j) = 2 tr [πiπ45◦ πjρ]. There- As our first demonstration of the technique, we fore, by varying the first and last projectors, weakly measure the non-Hermitian product ob- the density matrix can be directly determined servable π45◦ πV for a range of input states, |Ii. element-by-element. Specifically, we set the state preparation HWP To measure the density matrix experimentally, at an angle of θ/2 and the QWP at 45◦ in order we changed the HWPs settings to scan over the |Ii √1 |Hi − ie2iθ |V i π π to produce the state: = 2 . projectors i and j. As shown in [18], the fi- We increment θ from 0◦ to 180◦ in steps of 4◦. nal observable in the product can be measured For each input state and each image, the expec- either weakly or strongly. The last PBS imple- tation values for the joint weak-value (Eqs. 10- ments a strong measurement. For each pair of 11) were evaluated. The uncertainties were es- projectors, we measure the required expectation timated by the standard deviation in the joint values in Eqs. 10- 11. weak-value across the five recorded images. We test our direct measurement with three sets Curves for the real and imaginary parts of the of pure polarization states. A general polariza- joint weak-value are shown in Fig.2. The exper- tion state |ψi = α |Hi + β |V i has density ma- imental values closely follow the expected curves trix ρ ≡ |ψi hψ| = |α|2 |Hi hH| + αβ∗ |Hi hV | + calculated from the nominal input state |Ii. How- βα∗ |V i hH| + |β|2 |V i hV | . The states sets are ever, they do not agree within error. These devia- |ψ1i = cos θ |Hi + sin θ |V i, |ψ2i = cos θ |Hi + i θ |V i |ψ i √1 |Hi − ie2iθ |V i tions are likely due to imperfections in the wave- sin and 3 = 2 . These plates. These imperfections will also propagate are visualized in the Poincaré sphere in Fig.3a. to the alignment of the displacement axes of the For all cases, the first HWP varies the parameter BBO crystals since the waveplates are used in θ scanning the interval [0◦, 180◦]. The QWP is ◦ ◦ the alignment process. Such imperfections have removed for |ψ1i, and it is set at 0 and 45 for been shown to be the dominant source of sys- |ψ2i and |ψ3i, respectively. tematic error in similar past experiments [6, 43]. Our results are shown in Fig.3b - d. The solid Nonetheless, the results demonstrate the validity lines correspond to the real and imaginary parts of our proposed technique using a single pointer. of the elements of the theoretical density matrix and the points are the corresponding experimen- 5 Direct Measurement of the Quan- tal joint weak-values. The latter should be equal to the real and imaginary parts of the density tum State matrix elements and indeed follow the expected curve for each element. As before, deviations are We now move to a more sophisticated demonstra- thought to be the result of systematic errors in tion of our technique, the direct measurement of the polarization optics. This direct determination each element of the density matrix of polariza- of the density matrix demonstrates the utility of tion states. Such a direct measurement was intro- our technique for weak measurement applications duced in [6, 18]. In particular, the off-diagonal el- in quantum information. ements known as coherences can be retrieved and used to certify entanglement without the need to fully reconstruct a quantum state [44]. Unlike in 6 Discussion and Conclusion Refs. [6, 18], which used two pointers to measure BA, here we use only a single pointer for the Before summarizing, we discuss some special same task. cases and extensions of our technique. First, the A joint weak-measurement of the product special case where B = A and the observable πiπ45◦ πj, with i, j = H or V (with no post- is general i.e., not necessarily a projector. In selection) gives the element ρ(i, j) of the density this case, the unitary evolution (Eq.5) of the matrix. As shown in [18], the average outcome two von Neumann interactions (i.e., Eq.2) is of a weak measurement without post-selection is equivalent to a single unitary given by UAUA = † † † the ‘weak average’ (rather than the weak value), eγA(a +a)eγA(a +a) = e2γA(a +a). This corre- which is equal to the expectation value of the sponds to a unitary of a single von Neumann

8 ȁ퐻ۧ a 1 b 2 ห−45°ൿ 3

ȁ푅ۧ ȁ퐿ۧ 1 path along 휌 ห45°ൿ

Element of of Element Angle 휃 of polarization state [deg] ȁ푉ۧ

c d

along path 2 path along

along path 3 path along

휌

휌 Element of of Element Angle 휃 of polarization state [deg] of Element Angle 휃 of polarization state [deg]

Figure 3: The polarization states used to test the method can be visualized in frame a. These states are located on the three great circles (labeled 1, 2 and 3) in the Poincaré sphere passing through the states |Hi , |V i , |45◦i = |Hi+|V i |Hi−|V i |Hi−i|V i |Hi+i|V i √ , |− ◦i √ , |Ri √ and |Li √ . In frames b to d, we show experimental elements 2 45 = 2 = 2 = 2 of the density matrix ρ of the polarization states |ψ1i = cos θ |Hi + sin θ |V i, |ψ2i = cos θ |Hi + i sin θ |V i and |ψ i √1 |Hi − ie2iθ |V i , respectively. Solid lines correspond to the theory. States following path 1 corresponds 3 = 2 to linear polarization (no QWP in the setup), states along paths 2 and 3 were obtained by setting the QWP at 0◦ and 45◦, respectively. Error bars are calculated using the standard deviation in the joint weak-value across the ﬁve recorded images, as previously employed in Fig.2.

interaction of the A observable with a doubled of the products πβπα and adding the results, the interaction strength 2γ. By measuring x, p, and joint weak-value hBAiw can be obtained. In sum- d on the pointer and using Eqs. 10- 11, we can mary, our method can be used to weakly measure 2 then find the weak value A w. This behaviour the product of general incompatible observables can be generalized so that a single von Neumann A and B. D E interaction can be used to measure AN . To Our technique is also applicable to observables w do so, one will need to measure corresponding on separate quantum systems, e.g., A is mea- powers of observables on the pointer, e.g., xN as sured on a first particle and B is measured on well as the N -th power of hybrid observables such a second particle. The standard procedure for as d. weak measurement would couple BA to a single pointer using H = gBAp, which is Hermitian In the derivation of our technique, we focused now. This, however, requires a three-particle in- on the case that A and B are projectors. How- teraction which is challenging. Our method uses ever, for general A and B, measuring the product a two-particle interaction on each system while of projectors is enough to obtain the joint weak- still only using a single pointer. One would first value hBAiw. This can be seen if we express use the standard von Neumann interaction Eq.2, P A in its spectral decomposition A = α απα. to couple the pointer to A, then couple the same Here α is an eigenvalue corresponding to the pointer to B. eigenstate |αi, and πα = |αi hα| [45]. Analo- The technique introduced in this paper can also P gously, for B we have B = β βπβ. Therefore, be extended to general types of pointers such as P BA ≡ α,β αβπβπα. Thus, by measuring each spin pointers. Indeed, previous work showed that

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