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 num- ber 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.