Non-projective Measurements Research Articles

Randomness-free detection of non-projective measurements: qubits & beyond

Abstract Non-projective measurements play a crucial role in various information-processing protocols. In this work, we propose an operational task to identify measurements that are neither projective nor classical post-processing of data obtained from projective measurements. Our setup involves space-like separated parties with access to a shared state with bounded local dimensions. Specifically, in the case of qubits, we focus on a bipartite scenario with different sets of target correlations. While some of these correlations can be obtained through non-projective measurements on a shared two-qubit state, it is impossible to generate these correlations using projective simulable measurements on bipartite qubit states, or equivalently, by using one bit of shared randomness and local post-processing. For certain target correlations, we show that detecting qubit non-projective measurements is robust under arbitrary depolarizing noise, except in the limiting case. We extend this task for qutrits and demonstrate that some correlations achievable via local non-projective measurements cannot be reproduced by both parties performing the same qutrit projective simulable measurements on their pre-shared state. We provide numerical evidence for the robustness of this scheme under arbitrary depolarizing noise. For a more generic consideration (bipartite and tripartite scenario), we provide numerical evidence for a projective-simulable bound on the reward function for our task. We also show a violation of this bound by using qutrit positive operator valued measures. From a foundational perspective, we extend the notion of non-projective measurements to general probabilistic theories (GPTs) and use a randomness-free test to demonstrate that a class of GPTs, called square-bits or box-world are unphysical.

Nonprojective Bell-state measurements

The Bell-state measurement (BSM) is the projection of two qubits onto four orthogonal maximally entangled states. Here we first propose how to appropriately define more general BSMs, which have more than four possible outcomes, and then study whether they exist in quantum theory. We observe that nonprojective BSMs can be defined in a systematic way in terms of equiangular tight frames of maximally entangled states, i.e., a set of maximally entangled states, where every pair is equally, and in a sense maximally, distinguishable. We show that there exists a five-outcome BSM through an explicit construction and find that it admits a simple geometric representation. Then we prove that there exists no larger BSM on two qubits by showing that no six-outcome BSM is possible. We also determine the most distinguishable set of six equiangular maximally entangled states and show that it falls only somewhat short of forming a valid quantum measurement. Finally, we study the nonprojective BSM in the contexts of both local state discrimination and entanglement-assisted quantum communication. Our results put forward natural forms of nonprojective joint measurements and provide insight into the geometry of entangled quantum states. Published by the American Physical Society 2024

Extracting Bayesian networks from multiple copies of a quantum system

Despite their theoretical importance, dynamic Bayesian networks associated with quantum processes are currently not accessible experimentally. We here describe a general scheme to determine the multi-time path probability of a Bayesian network based on local measurements on independent copies of a composite quantum system combined with postselection. We further show that this protocol corresponds to a nonprojective measurement. It thus allows the investigation of the multi-time properties of a given local observable while fully preserving all its quantum features.

Randomness-Free Test of Nonclassicality: A Proof of Concept.

Quantum correlations and nonprojective measurements underlie a plethora of information-theoretic tasks, otherwise impossible in the classical world. Existing schemes to certify such nonclassical resources in a device-independent manner require seed randomness-which is often costly and vulnerable to loopholes-for choosing the local measurements performed on different parts of a multipartite quantum system. In this Letter, we propose and experimentally implement a semi-device-independent certification technique for both quantum correlations and nonprojective measurements without seed randomness. Our test is semi-device independent in the sense that it requires only prior knowledge of the dimension of the parts. We experimentally show a novel quantum advantage in correlated coin tossing by producing specific correlated coins from pairs of photons entangled in their transverse spatial modes. We establish the advantage by showing that the correlated coin obtained from the entangled photons cannot be obtained from two two-level classical correlated coins. The quantum advantage requires performing qubit trine positive operator-valued measures (POVMs) on each part of the entangled pair, thus also certifying such POVMs in a semi-device-independent manner. This proof of concept firmly establishes a new cost-effective certification technique for both generating nonclassical shared randomness and implementing nonclassical measurements, which will be important for future multiparty quantum communications.

Single-energy-measurement integral fluctuation theorem and nonprojective measurements.

We study a Jarzysnki-type equality for work in systems that are monitored using nonprojective unsharp measurements. The information acquired by the observer from the outcome f of an energy measurement and the subsequent conditioned normalized state ρ[over ̂](t,f) evolved up to a final time t are used to define work, as the difference between the final expectation value of the energy and the result f of the measurement. The Jarzynski equality obtained depends on the coherences that the state develops during the process, the characteristics of the meter used to measure the energy, and the noise it induces into the system. We analyze those contributions in some detail to unveil their role. We show that in very particular cases, but not in general, the effect of such noise gives a factor multiplying the result that would be obtained if projective measurements were used instead of nonprojective ones. The unsharp character of the measurements used to monitor the energy of the system, which defines the resolution of the meter, leads to different scenarios of interest. In particular, if the distance between neighboring elements in the energy spectrum is much larger than the resolution of the meter, then a similar result to the projective measurement case is obtained, up to a multiplicative factor that depends on the meter. A more subtle situation arises in the opposite case in which measurements may be noninformative, i.e., they may not contribute to update the information about the system. In this case a correction to the relation obtained in the nonoverlapping case appears. We analyze the conditions in which such a correction becomes negligible. We also study the coherences, in terms of the relative entropy of coherence developed by the evolved post-measurement state. We illustrate the results by analyzing a two-level system monitored by a simple meter.

Contextuality without Incompatibility.

The existence of incompatible measurements is often believed to be a feature of quantum theory which signals its inconsistency with any classical worldview. To prove the failure of classicality in the sense of Kochen-Specker noncontextuality, one does indeed require sets of incompatible measurements. However, a more broadly applicable notion of classicality is the existence of a generalized-noncontextual ontological model. In particular, this notion can imply constraints on the representation of outcomes even within a single nonprojective measurement. We leverage this fact to demonstrate that measurement incompatibility is neither necessary nor sufficient for proofs of the failure of generalized noncontextuality. Furthermore, we show that every proof of the failure of generalized noncontextuality in a quantum prepare-measure scenario can be converted into a proof of the failure of generalized noncontextuality in a corresponding scenario with no incompatible measurements.

Measurement-induced population switching

Quantum information processing is a key technology in the ongoing second quantum revolution, with a wide variety of hardware platforms competing toward its realization. An indispensable component of such hardware is a measurement device, i.e., a quantum detector that is used to determine the outcome of a computation. The act of measurement in quantum mechanics, however, is naturally invasive as the measurement apparatus becomes entangled with the system that it observes. This always leads to a disturbance in the observed system, a phenomenon called quantum measurement backaction, which should solely lead to the collapse of the quantum wave function and the physical realization of the measurement postulate of quantum mechanics. Here we demonstrate that backaction can fundamentally change the quantum system through the detection process. For quantum information processing, this means that the readout alters the system in such a way that a faulty measurement outcome is obtained. Specifically, we report a backaction-induced population switching, where the bare presence of weak, nonprojective measurements by an adjacent charge sensor inverts the electronic charge configuration of a semiconductor double quantum dot system. The transition region grows with measurement strength and is suppressed by temperature, in excellent agreement with our coherent quantum backaction model. Our result exposes backaction channels that appear at the interplay between the detector and the system environments, and opens new avenues for controlling and mitigating backaction effects in future quantum technologies.

Certification of the maximally entangled state using nonprojective measurements

In recent times, device-independent certification of quantum states has been one of the intensively studied areas in quantum information. However, all such schemes utilize projective measurements which are practically difficult to generate. In this paper, we consider the one-sided device-independent scenario and propose a self-testing scheme for the two-qubit maximally entangled state using nonprojective measurements, in particular, three three-outcome extremal positive operator-valued measures. We also analyze the robustness of our scheme against white noise.

Certification of a non-projective qudit measurement using multiport beamsplitters

The most common form of measurement in quantum mechanics projects a wavefunction onto orthogonal states that correspond to definite outcomes. However, generalized quantum measurements that do not fully project quantum states are possible and have an important role in quantum information tasks. Unfortunately, it is difficult to certify that an experiment harvests the advantages made possible by generalized measurements, especially beyond the simplest two-level qubit system. Here we show that multiport beamsplitters allow for the robust realization of high-quality generalized measurements in higher-dimensional systems with more than two levels. Using multicore optical fibre technology, we implement a seven-outcome generalized measurement in a four-dimensional Hilbert space with a fidelity of 99.7%. We present a practical quantum communication task and demonstrate a success rate that cannot be simulated in any conceivable quantum protocol based on standard projective measurements on quantum messages of the same dimension. Our approach, which is compatible with modern photonic platforms, showcases an avenue for faithful and high-quality implementation of genuinely non-projective quantum measurements beyond qubit systems. Generalized measurements that do not correspond to conventional basis projections of the quantum wavefunction are a part of several important protocols in quantum information. These measurements can be certifiably performed on higher-dimensional systems using optical fibre technology.

Mermin and Svetlichny inequalities for non-projective measurement observables

The necessary and sufficient criteria for violating the Mermin and Svetlichny inequalities by arbitrary three-qubit states are presented. Several attempts have been made, earlier, to find such criteria, however, those extant criteria are neither tight for most of the instances, nor fully general. We generalize the existing criteria for Mermin and Svetlichny inequalities which are valid for the local projective measurement observables as well as for the arbitrary ones. We obtain the maximal achievable bounds of the Mermin and Svetlichny operators with unbiased measurement observables for arbitrary three-qubit states and with arbitrary observables for three-qubit states having maximally mixed marginals. We find that for certain ranges of measurement strengths, it is possible to violate Mermin and Svetlichny inequalities only by biased measurement observables. The necessary and sufficient criteria of violating any one of the six possible Mermin and Svetlichny inequalities are also derived.

Experimental quantum state discrimination using the optimal fixed rate of inconclusive outcomes strategy

The problem of non-orthogonal state discrimination underlies crucial quantum information tasks, such as cryptography and computing protocols. Therefore, it is decisive to find optimal scenarios for discrimination among quantum states. We experimentally investigate the strategy for the optimal discrimination of two non-orthogonal states considering a fixed rate of inconclusive outcomes (FRIO). The main advantage of the FRIO strategy is to interpolate between unambiguous and minimum error discrimination by solely adjusting the rate of inconclusive outcomes. We present a versatile experimental scheme that performs the optimal FRIO measurement for any pair of generated non-orthogonal states with arbitrary a priori probabilities and any fixed rate of inconclusive outcomes. Considering different values of the free parameters in the FRIO protocol, we implement it upon qubit states encoded in the polarization mode of single photons generated in the spontaneous parametric down-conversion process. Moreover, we resort to a newfangled double-path Sagnac interferometer to perform a three-outcome non-projective measurement required for the discrimination task, showing excellent agreement with the theoretical prediction. This experiment provides a practical toolbox for a wide range of quantum state discrimination strategies using the FRIO scheme, which can significantly benefit quantum information applications and fundamental studies in quantum theory.

Experimental certification of nonprojective quantum measurements under a minimum overlap assumption.

Certifying quantum measurements is increasingly important for foundational insights in quantum information science. Here, we report an experimental certification of unknown quantum measurements in a semi-device-independent setting. For the first time, we experimentally demonstrate that genuine three-outcome positive operator-valued measures (POVMs) can be certified under the assumption of a limited overlap between the prepared quantum states. The generalized quantum measurements are realized through discrete-time quantum walk and our experimental results clearly show that three-outcome POVMs can be certified even in the presence of noise. Finally, we experimentally investigate that optimal POVMs for performing unambiguous state discrimination can be self-tested. Our work opens new avenues for robust certification of quantum systems in the prepare-and-measure scenario.

Demonstration of optimal non-projective measurement of binary coherent states with photon counting

Quantum state discrimination is a central problem in quantum measurement theory, with applications spanning from quantum communication to computation. Typical measurement paradigms for state discrimination involve a minimum probability of error or unambiguous discrimination with a minimum probability of inconclusive results. Alternatively, an optimal inconclusive measurement, a non-projective measurement, achieves minimal error for a given inconclusive probability. This more general measurement encompasses the standard measurement paradigms for state discrimination and provides a much more powerful tool for quantum information and communication. Here, we experimentally demonstrate the optimal inconclusive measurement for the discrimination of binary coherent states using linear optics and single-photon detection. Our demonstration uses coherent displacement operations based on interference, single-photon detection, and fast feedback to prepare the optimal feedback policy for the optimal non-projective quantum measurement with high fidelity. This generalized measurement allows us to transition among standard measurement paradigms in an optimal way from minimum error to unambiguous measurements for binary coherent states. As a particular case, we use this general measurement to implement the optimal minimum error measurement for phase-coherent states, which is the optimal modulation for communications under the average power constraint. Moreover, we propose a hybrid measurement that leverages the binary optimal inconclusive measurement in conjunction with sequential, unambiguous state elimination to realize higher dimensional inconclusive measurements of coherent states.

Device-Independent Certification of Maximal Randomness from Pure Entangled Two-Qutrit States Using Non-Projective Measurements.

While it has recently been demonstrated how to certify the maximal amount of randomness from any pure two-qubit entangled state in a device-independent way, the problem of optimal randomness certification from entangled states of higher local dimension remains open. Here we introduce a method for device-independent certification of the maximal possible amount of random bits using pure bipartite entangled two-qutrit states and extremal nine-outcome general non-projective measurements. To this aim, we exploit a device-independent method for certification of the full Weyl–Heisenberg basis in three-dimensional Hilbert spaces together with a one-sided device-independent method for certification of two-qutrit partially entangled states.

Oblivious communication game, self-testing of projective and nonprojective measurements, and certification of randomness

We provide an interesting two-party parity oblivious communication game whose success probability is solely determined by the Bell expression. The parity-oblivious condition in an operational quantum theory implies the preparation non-contextuality in an ontological model of it. We find that the aforementioned Bell expression has two upper bounds in an ontological model; the usual local bound and a non-trivial preparation non-contextual bound arising from the non-trivial parity-oblivious condition, which is smaller that the local bound. We first demonstrate the communication game when both Alice and Bob perform three measurements of dichotomic observables in their respective sites. The optimal quantum value of the Bell expression in this scenario enables us to device-independently self-test the maximally entangled state and trine-set of observables, three-outcome qubit positive-operator-valued-measures (POVMs) and 1.58 bit of local randomness. Further, we generalize the above communication game in that both Alice and Bob perform the same but arbitrary (odd) number ($n> 3$) of measurements. Based on the optimal quantum value of the relevant Bell expression for any arbitrary $n$, we have also demonstrated device-independent self-testing of state and measurements.

Genuine Einstein-Podolsky-Rosen steering of three-qubit states by multiple sequential observers

We investigate the possibility of multiple uses of a single copy of a three-qubit state for detecting genuine tripartite Einstein-Podolsky-Rosen (EPR) steering. A pure three-qubit state of either the Greenberger-Horne-Zeilinger (GHZ) type or $W$ type is shared between two fixed observers in two wings and a sequence of multiple observers in the third wing who perform unsharp or nonprojective measurements. The measurement settings of each of the multiple observers in the third wing are independent and uncorrelated with the measurement settings and outcomes of the previous observers. In such setup, we investigate all possible types of ($2\ensuremath{\rightarrow}1$) and ($1\ensuremath{\rightarrow}2$) genuine tripartite EPR steering. For each case, we obtain an upper limit on the number of observers on the third wing who can demonstrate genuine EPR steering through the quantum violation of an appropriate tripartite steering inequality. We show that the GHZ state allows for a higher number of observers compared to the $W$ state. Additionally, ($1\ensuremath{\rightarrow}2$) genuine steering is possible for a larger range of sharpness parameters compared to ($2\ensuremath{\rightarrow}1$) genuine steering cases.

Maximal randomness from partially entangled states

We investigate how much randomness can be extracted from a generic partially entangled pure state of two qubits in a device-independent setting, where a Bell test is used to certify the correct functioning of the apparatus. For any such state, we first show that two bits of randomness are always attainable both if projective measurements are used to generate the randomness globally or if a nonprojective measurement is used to generate the randomness locally. We then prove that the maximum amount of randomness that can be generated using nonprojective measurements globally is restricted to between approximately 3.58 and 3.96 bits. The upper limit rules out that a bound of four bits potentially obtainable with extremal qubit measurements can be attained. We point out this is a consequence of the fact that nonprojective qubit measurements with four outcomes can only be self-tested to a limited degree in a Bell experiment.

Simulation of the complex dynamics of mean-field p -spin models using measurement-based quantum feedback control

We study the application of a new method for simulating nonlinear dynamics of many-body spin systems using quantum measurement and feedback [Mu\~noz-Arias et al., Phys. Rev. Lett. 124, 110503 (2020)] to a broad class of many-body models known as $p$-spin Hamiltonians, which describe Ising-like models on a completely connected graph with $p$-body interactions. The method simulates the desired mean-field dynamics in the thermodynamic limit by combining nonprojective measurements of a component of the collective spin with a global rotation conditioned on the measurement outcome. We apply this protocol to simulate the dynamics of the $p$-spin Hamiltonians and demonstrate how different aspects of criticality in the mean-field regime are readily accessible with our protocol. We study applications including properties of dynamical phase transitions and the emergence of spontaneous symmetry breaking in the adiabatic dynamics of the collective spin for different values of the parameter $p$. We also demonstrate how this method can be employed to study the quantum-to-classical transition in the dynamics continuously as a function of system size.

Mitigation of readout noise in near-term quantum devices by classical post-processing based on detector tomography

We propose a simple scheme to reduce readout errors in experiments on quantum systems with finite number of measurement outcomes. Our method relies on performing classical post-processing which is preceded by Quantum Detector Tomography, i.e., the reconstruction of a Positive-Operator Valued Measure (POVM) describing the given quantum measurement device. If the measurement device is affected only by an invertible classical noise, it is possible to correct the outcome statistics of future experiments performed on the same device. To support the practical applicability of this scheme for near-term quantum devices, we characterize measurements implemented in IBM's and Rigetti's quantum processors. We find that for these devices, based on superconducting transmon qubits, classical noise is indeed the dominant source of readout errors. Moreover, we analyze the influence of the presence of coherent errors and finite statistics on the performance of our error-mitigation procedure. Applying our scheme on the IBM's 5-qubit device, we observe a significant improvement of the results of a number of single- and two-qubit tasks including Quantum State Tomography (QST), Quantum Process Tomography (QPT), the implementation of non-projective measurements, and certain quantum algorithms (Grover's search and the Bernstein-Vazirani algorithm). Finally, we present results showing improvement for the implementation of certain probability distributions in the case of five qubits.

Self-testing nonprojective quantum measurements in prepare-and-measure experiments.

Self-testing represents the strongest form of certification of a quantum system. Here, we theoretically and experimentally investigate self-testing of nonprojective quantum measurements. That is, how can one certify, from observed data only, that an uncharacterized measurement device implements a desired nonprojective positive-operator valued measure (POVM). We consider a prepare-and-measure scenario with a bound on the Hilbert space dimension and develop methods for (i) robustly self-testing extremal qubit POVMs and (ii) certifying that an uncharacterized qubit measurement is nonprojective. Our methods are robust to noise and thus applicable in practice, as we demonstrate in a photonic experiment. Specifically, we show that our experimental data imply that the implemented measurements are very close to certain ideal three- and four-outcome qubit POVMs and hence non-projective. In the latter case, the data certify a genuine four-outcome qubit POVM. Our results open interesting perspective for semi-device-independent certification of quantum devices.