Minimizing back-action through entangled measurements

Introduction—The enthusiasm for probing and describing microscopic systems in the quantum domain motivates the search for ways to measure quantum processes, akin to standard measurements of operators. This task is notoriously challenging due to the fragility of quantum superpositions and invasiveness of measurements, epitomized in the quantum back-action (QBA) effect Braginskii and Vorontsov (1975); Caves et al. (1980); Braginsky and Khalili (1996); Wiseman and Milburn (2010). Given two measurements separated by a certain time interval, there is QBA when the statistics of the second measurement are altered by the presence of the first measurement. It is mostly considered an unpleasant hindrance in experiments, and significant effort has been invested to design back-action-evading (a.k.a. quantum non-demolition) measurements Braginskii and Vorontsov (1975); Caves et al. (1980); Wiseman (1995); Braginsky and Khalili (1996); Kippenberg and Vahala (2008); Wiseman and Milburn (2010); Erhart et al. (2012); Hacohen-Gourgy et al. (2016); Møller et al. (2017); Kono et al. (2018); Pezzè et al. (2018); Wu et al. (2019); Boulebnane et al. . Besides its fundamental interest, QBA also has important consequences for the definition of heat and work fluctuations in quantum thermodynamics Allahverdyan (2014); Solinas and Gasparinetti (2015); Talkner and Hänggi (2016); Bäumer et al. (2018); Levy and Lostaglio and quantum transport Hovhannisyan and Imparato (2019).

In its essence, QBA is an instance of the uncertainty principle Peres (1993). Say, after measuring an observable $A$, the system’s state, $\rho$, undergoes a unitary evolution $U$, generated by its internal Hamiltonian. Then, an observable $A^{\prime}$ is measured. Viewing in the Heisenberg picture, we thus have two operators, $A$ and $A^{\prime}_{H}=U^{\dagger}A^{\prime}U$, simultaneously measured on the state $\rho$, and when $[A,A^{\prime}_{H}]\neq 0$, these measurements are incompatible, both in the sense of the uncertainty relations à la Heisenberg Heisenberg (1927); Białynicki-Birula and Mycielski (1975); Peres (1993); Maccone and Pati (2014) and error-disturbance relations à la Arthurs-Kelly Arthurs and Kelly (1965); Ishikawa (1991); Peres (1993); Ozawa (2003); Allahverdyan et al. (2013); Branciard (2013); Erhart et al. (2012); Busch et al. (2014); Buscemi et al. (2014); Perarnau-Llobet and Nieuwenhuizen (2017). Measurement incompatibility can also be described by constructing joint (quasi)probability distributions, with the condition that the marginal distributions be given by the Born rule Wigner (1932); Terletsky (1937); Margenau and Hill (1961); Mückenheim et al. (1986); Nazarov and Kindermann (2003); Hofer (2017). Quasiprobability distributions, generically arising in (sequential) weak measurement scenarios Mitchison et al. (2007); Avella et al. (2017); Kim et al. (2018); Shojaee et al. (2018); Hovhannisyan and Imparato (2019); Pfender et al. (2019); Cujia et al. (2019); Monroe et al. , necessarily feature negative probabilities Margenau and Hill (1961); Cahill and Glauber (1969); Busch (1985); Uffink (1994); Hartle (2004); Allahverdyan (2014); Potts (2019) (which is related to contextuality Spekkens (2008); Pusey (2014); Lostaglio (2018)); joint distributions that do not feature negative probabilities require going beyond the axiomatics of standard “precise” probability theory Allahverdyan and Danageozian (2018).

In this work, we will take a novel approach towards reducing QBA, by using collective measurements on multiple copies of the system of interest. The logic of our approach is as follows. When $[\rho,A]=0$, the first measurement does not disturb the state, therefore, the second measurement is not affected by it. That is, there is no QBA when the state is, in a sense, classical, and the joint probability is $p^{\mathrm{cl}}_{ij}=\operatorname{Tr}(\rho P_{i}P^{\prime}_{j})\geq 0$, where $P_{i}$ and $P^{\prime}_{j}$ are the eigenprojectors of $A$ and $A_{H}^{\prime}$, respectively ($A=\sum_{i}a_{i}P_{i}$, $A_{H}^{\prime}=\sum_{i}a^{\prime}_{i}P^{\prime}_{i}$). This statistics coincides with that of the measure-evolve-measure protocol, widely used and discussed in quantum transport Shelankov and Rammer (2003); Nazarov and Kindermann (2003); Esposito et al. (2009); Hovhannisyan and Imparato (2019) and thermodynamics Jarzynski and Wójcik (2004); Talkner et al. (2007); Esposito et al. (2009); Perarnau-Llobet et al. (2017); Levy and Lostaglio .

When $[\rho,A]\neq 0$, the standard measure-evolve-measure protocol has QBA, and we wish to find new measurement schemes to reduce it (while agreeing with it for $[\rho,A]=0$). However, it was recently shown that there cannot exist a state-independent generalized quantum measurement—a positive operator-valued measure (POVM)—that describes the joint statistics of $A$ and $A^{\prime}_{H}$ and coincides with $p^{\mathrm{cl}}_{ij}$ when $\rho$ commutes with $A$ Perarnau-Llobet et al. (2017). Hence, our idea is to consider schemes that are allowed to depend on the state, but do so as weakly as possible. To that end, we use measurements that can be realized as state-independent POVMs on several copies of the system ¹¹1Only when viewed from the perspective of a single system, these depend on the state. This is motivated by a QBA-reduction technique proposed and implemented in Refs. Perarnau-Llobet et al. (2017); Wu et al. (2019), for the problem of measuring quantum-mechanical work. There, it was also proven that the QBA cannot be removed no matter how many copies one takes; however, significant reduction can be observed even with two copies. Interestingly, the QBA reduction in Refs. Perarnau-Llobet et al. (2017); Wu et al. (2019) was achieved by using factorized measurements (i.e., the POVM elements are in the form of a Kronecker product).

Here, focusing on the two-copy scenario, we go beyond these results and take this scheme to its extreme. By using entangled POVMs, we show that one can reduce the QBA even further, eliminating it completely in an extended parameter range. These enhancements are only possible due to the entangled nature of the measurements, as we also prove in Ref. ²²2See Supplemental Material at URL, which contains the mathematical details of the theoretical results and technical details of the experimental setup, and further references Virmani and Plenio (2003); Sanpera et al. (1998); Verstraete et al. (2001) that the proposal of Refs. Perarnau-Llobet et al. (2017); Wu et al. (2019) is the optimal not only among factorized POVMs, but among all separable, i.e., possibly correlated but non-entangled, POVMs. We measure the QBA by comparing the statistics of $A^{\prime}_{H}$ when $A$ is performed with that when $A$ is not performed. We determine the ultimate QBA-evading measurement on two copies, which turns out to be entangled. Somewhat counterintuitively, zero QBA can be achieved in a class of continuously parameterized highly coherent processes. We experimentally implement a class of such protocols in the discrete photonics regime, for observing minimal QBA in different quantum processes. Our results show the capability of achieving minimal QBA in our photonic setup with considerably low experimental imperfections, and demonstrate the significant advantage gained over non-entangled measurements.

Theoretical framework and results—Let us formalize the discussion above. The statistics of $A$’s measurement is given by probabilities $p^{(1)}_{i}=\operatorname{Tr}(\rho P_{i})$ and, if $A$ is not performed, the statistics of $A_{H}^{\prime}$ is given by $p^{(2)}_{j}=\operatorname{Tr}(\rho P^{\prime}_{j})$. The desired goal is to find a $p^{(12)}_{ij}=\operatorname{Tr}(M^{ij}\rho)$, where $M^{(ij)}$ constitute a POVM ³³3Namely, $M^{(ij)}$ are non-negative Hermitian operators such that $\sum_{ij}M^{(ij)}=\mathbbm{1}$ Wiseman and Milburn (2010), such that $\sum_{j}p^{(12)}_{ij}=p^{(1)}_{i}$ and $\sum_{i}p^{(12)}_{ij}=p^{(2)}_{j}$, for any state $\rho$. Had such state-independent POVM exist, we would say that $A$ and $A^{\prime}_{H}$ are jointly measurable Busch (1985); Uffink (1994). However, it is well-known that no such state-independent POVM exists that satisfies this goal for general $\rho$’s when $[A,A^{\prime}_{H}]\neq 0$ Busch (1985); Uffink (1994); Ozawa (2003); Branciard (2013). We deal with this impossibility in the following manner. First, we notice that, when $[\rho,H]=0$,

where $M_{\mathrm{TPM}}^{(ij)}=P_{i}P^{\prime}_{j}P_{i}$, describes a valid joint statistics and represents two projective measurements (hence the label “TPM”) of $A$ and $A^{\prime}_{H}$, performed one after another. Then, we recall that state-independent POVMs, that produce $p^{\mathrm{cl}}_{ij}$ whenever $[\rho,A]=0$, essentially coincide with $M_{\mathrm{TPM}}^{(ij)}$ Perarnau-Llobet et al. (2017), meaning that there is no room for tweaking the QBA.

Therefore, inspired by an approach developed in the context of quantum thermodynamics Perarnau-Llobet et al. (2017); Wu et al. (2019), we extend the class of POVMs by considering state-independent measurements on two copies of the system: we take a general POVM, $\mathcal{M}=\{M^{(ij)}\}$ ($M^{(ij)}\geq 0$ and $\sum_{ij}M^{(ij)}=\mathbbm{1}$),

and require it to satisfy the condition

in other words, to reproduce $p^{\mathrm{cl}}_{ij}$ for any incoherent $\rho$. Furthermore, since this measurement is intended to describe the QBA effect, we also require the statistics produced by $\mathcal{M}$ for the first event (measurement of $A$) to coincide with $p^{(1)}_{i}$, for any $\rho$; i.e., $\mathcal{M}$ also satisfies

Lastly, noticing that the set $C_{1}\cap C_{2}$ has more than one element (as opposed to single-copy state-independent measurements, for which the above two requirements necessitate the measurement to coincide with $\mathcal{M}_{\mathrm{TPM}}$), we fulfill our main goal by finding $\mathcal{M}^{\mathrm{opt}}=\arg\min\limits_{\mathcal{M}\in C_{1}\cap C_{2}}\mathcal{B}[\mathcal{M}]$, with $\mathcal{B}$ being the magnitude of the QBA itself:

where $p_{j}[\mathcal{M}]:=\sum_{i}p_{ij}[\mathcal{M}]$ is the actual statistics of the second measurement. Put differently, we are looking for the POVM that delivers the correct statistics for classical states, always yields the Born rule for the first measurement, and minimizes the QBA for the second measurement (see also the discussion in Ref. Note (2)). Here, we quantified the QBA by the Euclidean distance between the probability distribution of the second measurement as produced by the joint POVM, $p_{j}$, and the probability distribution of the second measurement had the first measurement not taken place, $p^{(2)}_{j}$. In principle, one could choose other distance measures in (5), but for our problem of interest—a two-level system—all of them will depend only on the ground-state probability and hence yield the same $\mathcal{M}^{\mathrm{opt}}$.

For simplicity, we will concentrate on qubits and consider the case of $A=A^{\prime}$. Moreover, in the single-qubit Hilbert spaces, we will work in the eigenbasis of $A$, $\{|0\rangle,|1\rangle\}$, and, since constant shifts, and multiplication by constants, of $A$ do not affect the phenomena at hand, w.l.o.g, we can choose $A=|1\rangle\langle 1|$. Below, we will sketch how $\mathcal{M}^{\mathrm{opt}}$ is calculated in this case, and refer the reader to Ref. Note (2) for full details.

We start by considering requirement $C_{1}$. With our choice of operators, the TPM POVM elements will be $M_{\mathrm{TPM}}^{(ij)}=|i\rangle\langle i|\cdot|\langle j|U|i\rangle|^{2}$, resulting in

where $i,j$ are $0$ or $1$ and $\rho_{ii}:=\langle i|\rho|i\rangle$ describes the probability to start in the state $|i\rangle$ and $|U_{ji}|^{2}:=\left|\langle j|U|i\rangle\right|^{2}$ the transition probability from $|i\rangle$ to $|j\rangle$. Now, taking into account that all four matrices $M^{(ij)}$ do not depend on the state, Eq. (8) fixes all diagonal elements of these matrices up to some constants $a$ and $\widetilde{a}$ (see Note (2) for the explicit construction). In particular, one diagonal element will always be zero, so the matrices are at most of rank three.

In order to minimize the QBA, we have to divide the construction and analysis into two cases. First, highly coherent unitary operators, that is, evolutions that produce a lot of coherence given a classical state: $|U_{10}|^{2}\in\left[\frac{1}{3},\frac{2}{3}\right]$. Second, slightly coherent unitary operations: $|U_{10}|^{2}\in\left(0,\frac{1}{3}\right)\cup\left(\frac{2}{3},1\right)$. Interestingly, for the first case one can construct a POVM with zero QBA, while it cannot be reduced to zero for the second case.

In the case of highly coherent unitaries, we can write the optimal POVM as four rank-two matrices that are each a convex combination of $|\psi^{-}\rangle=(|01\rangle-|10\rangle)/\sqrt{2}$ and a non-trivial pure entangled state. Note that this POVM also satisfies requirement $C_{2}$, i.e., $\operatorname{Tr}[(M^{(00)}+M^{(01)})\rho^{\otimes 2}]=\rho_{00}$. In the case of slightly coherent unitaries, the optimal POVM contains two rank-three matrices and two rank-one matrices. In this case, there is always some back-action proportional to the initial coherence $\rho_{01}$ (see Ref. Note (2)).

In both cases, the optimal POVM are entangled, and one may wonder to which extent such quantum correlations are needed. We also show that the separable collective measurement previously proposed in Refs. Perarnau-Llobet et al. (2017); Wu et al. (2019) is in fact the optimal non-entangled POVM. Hence, any advantage reported in this work is possible due to the entangled nature of the measurement (see Ref. Note (2) for details).

Experiment—We now experimentally implement the minimally invasive entangled POVMs derived above.

Fig. 1 details both the conceptual design of our quantum walk and the corresponding experimental setup. For state preparation, we make use of multiple degrees of freedom of a down-converted single photon Takeuchi (2001) (spatial mode and polarization). Specifically, we encode the first copy in the spatial degree of freedom, and the second copy in the polarization degree of freedom Wu et al. (2019); Hou et al. (2018), preparing the state $|\Phi\rangle\otimes|\Phi\rangle$, where $|\Phi\rangle=\cos\alpha|0\rangle+\sin\alpha|1\rangle$ (see Ref. Note (2) for details).

Experimentally demonstrating the highly non-trivial entangled collective measurements is much more challenging than the separable measurements previously reported in Ref. Wu et al. (2019). The realization of the POVMs can be embedded in the dynamics of a one-dimensional discrete quantum walk Xiao et al. (2017); Hou et al. (2018); Xue et al. (2015); Zhao et al. (2015) containing two degrees of freedom: the walker position, $q$, and a coin state, $c$. The coin qubit and the walker in positions, respectively, $1$ and $-1$ are taken as the two-qubit system of interest, while the other positions of the walker are regarded as an auxiliary system. By engineering the coin operators followed by measuring the walker position after certain steps, a POVM can be deterministically realized with desired outcome at each position. In Figs. 1(a) and (b), the diagram of quantum walks for two cases are shown. The perfect QBA elimination in the case of highly coherent processes can be realized via a $7$ step quantum walk, while the slightly coherent case needs $8$ steps.

Our all-optical setup is capable of realizing the POVMs for both cases described above. Fig. 1(c) shows the experimental setup for realizing these POVMs. The whole setup is based on a single-photon interferometer. Robust interference between photons in different modes is required to ensure the high-fidelity-implementation of the measurement operators. Details are given in Ref. Note (2).

A big entanglement-induced advantage over previous schemes (TPM and the optimal separable collective measurement Perarnau-Llobet et al. (2017)) emerges when $|U_{01}|^{2}=1/3$. In particular, Fig. 2(a) shows an initial state (red) and final state (blue) in Bloch representation: a pure state $|+\rangle$ is initialized and evolves into $\sqrt{1/3}|+\rangle+\sqrt{2/3}|-\rangle$, where $|\pm\rangle=(|0\rangle\pm|1\rangle)/\sqrt{2}$. The experimentally obtained $\widetilde{p}_{ij}$ (where the $\sim$ above a quantity means that is evaluated experimentally) from entangled measurement are presented in Fig. 2(a) as white cubes, while the ideal values given by Eq. (2) are plotted as blue cubes. The expected results obtained from previous schemes are also shown as green (for separable collective measurement) and orange (for TPM) cubes. Fig. 2(b) shows the experimentally reconstructed POVMs, with an average fidelity up to $0.985$. The fidelity between two POVMs can be defined as $\mathcal{F}(M,M^{\prime})=\left(\sum_{i}w_{i}\sqrt{\mathcal{F}_{i}}\right)^{2}$, where $w_{i}=\sqrt{\operatorname{Tr}(M_{i})\operatorname{Tr}(M^{\prime}_{i})}/d$, and $\mathcal{F}_{i}$ is the fidelity between $\mu_{i}:=\frac{M_{i}}{\operatorname{Tr}(M_{i})}$ and $\mu^{\prime}_{i}:=\frac{M^{\prime}_{i}}{\operatorname{Tr}(M^{\prime}_{i})}$ Hou et al. (2018).

Fig. 3 shows the experimental reduction of QBA for a family of unitaries parameterized as $U_{00}=-U_{11}=\cos x$ and $U_{01}=U_{10}=\sin x$, where $x$ determines the “degree of coherence” of $U$. Fig. 4 shows a similar analysis for different input states and a fixed process. In both figures, the QBA is quantified through Eq. (5).

To observe the divergence between these methods, we first fix the initial state to be the maximally coherent state $|+\rangle$ and implement $67$ entangled POVMs so that both highly coherent and slightly coherent processes are included. The results are shown in Fig. 3: (a) shows the probabilities $p_{0}$ of ending states under various protocols, the experimental values for new entangled POVMs are shown as red disks; (b) shows the experimentally observed QBA for the new POVMs and the simulated values for QBA of previous schemes. The new entangled POVMs (red points) can, within tolerable experimental imperfection, produce probabilities for final states that are closest to the ideal case (black lines), resulting in a significant decrease in QBA compared to the separable measurement. Fig. 3(c) illustrates the fact that any advantage reported in this work is possible due to the entangled nature of the measurement, by showing a comparison between the maximal gain in QBA reduction entangled measurements offer over separable ones and the value of entanglement, as measured by the negativity Horodecki et al. (2009), of the measurement delivering that gain.

We also experimentally test the performance of the new scheme on different inputs $|\Phi\rangle$ (parameterized by $\alpha$) with fixed $x$. Specifically, we choose $U_{01}=\sqrt{1/3},\sqrt{1/2},1/2$, and the coherence of inputs, quantified as $\sin(2\alpha)$, varies from $0$ to $1$. The experimentally observed QBA for the entangled POVMs, and the simulated values for all schemes, are shown in Fig. 4, where we can see that the experimentally observed QBA is almost zero for two highly coherent processes and is significantly advantageous over the separable ones for slightly coherent process. The error bars in all figures are estimated by Monte Carlo simulations Note (2).

Conclusions—In this work, we demonstrated that harnessing non-trivial entangled measurements can significantly minimize the QBA. We devised the optimal measurement for reducing the QBA given two copies of a qubit, which turns out to be an entangled measurement. We also proved that the optimal non-entangled measurement coincides with the previous proposal of Perarnau-Llobet et al. (2017); Wu et al. (2019). More specifically, we show that for highly coherent unitary operations the QBA can, in principle, be eliminated. While such an elimination is impossible for slightly coherent unitaries, our new scheme offers a considerable advantage over those using only separable measurements Perarnau-Llobet et al. (2017); Wu et al. (2019).

Our experimental implementation of the optimal measurement demonstrates substantial entanglement-based advantages over previous schemes  Perarnau-Llobet et al. (2017); Wu et al. (2019). Our design can be realized with high average fidelity reaching up to $0.985$, which results in a good agreement between theoretical predictions and experimental observations.

It is worth emphasizing that the measurement step of our experiment is completely independent of the preparation step. In fact, other preparation procedures can be used. For example, by using quantum state joining Vitelli et al. (2013), one can convert the joint state of two photons (which could be generated, e.g., in an atom-photon interface Blinov et al. (2004); Wilk et al. (2007); Stute et al. (2012)) into a hybrid encoded single-photon state. Our experimental setup thus demonstrates the feasibility of in-depth investigation of fundamental principles in quantum mechanics within a variety of current technologies.