Collective measurements on identical and independent quantum systems can offer advantages in information extraction compared with individual measurements. However, little is known about the distinction between restricted collective measurements and genuine collective measurements in the multipartite setting. In this work we establish a rigorous performance gap based on a simple and old estimation problem, the estimation of a random spin state given three parallel spins. Notably, we derive an analytical formula for the maximum estimation fidelity of biseparable measurements and clarify its fidelity gap from genuine collective measurements. Moreover, we clarify the structure of optimal biseparable measurements. It turns out the maximum estimation fidelity can be achieved by two- and one-copy measurements assisted by one-way communication in one direction, but not the other way. Our work reveals a rich landscape of multipartite nonclassicality in quantum measurements instead of quantum states and is expected to trigger further studies.
Collective measurements on identical and independent quantum systems can offer advantages in information extraction compared with individual measurements. However, little is known about the distinction between restricted collective measurements and genuine collective measurements in the multipartite setting. In this work we establish a rigorous performance gap based on a simple and old estimation problem, the estimation of a random spin state given three parallel spins. Notably, we derive an analytical formula for the maximum estimation fidelity of biseparable measurements and clarify its fidelity gap from genuine collective measurements. Moreover, we clarify the structure of optimal biseparable measurements. It turns out the maximum estimation fidelity can be achieved by two- and one-copy measurements assisted by one-way communication in one direction, but not the other way. Our work reveals a rich landscape of multipartite nonclassicality in quantum measurements instead of quantum states and is expected to trigger further studies.
Nonclassical phenomena tied to entangled states are the focus of foundational studies and powerful resources in many applications. By contrast, the counterparts on quantum measurements are still poorly understood. Notably, genuine multipartite nonclassicality is barely discussed, let alone its experimental realization. Here we experimentally demonstrate the power of genuine tripartite nonclassicality in quantum measurements based on a simple estimation problem. To this end we realize an optimal genuine three-copy collective measurement via a nine-step two-dimensional photonic quantum walk with 30 elaborately designed coin operators. Then we realize an optimal estimation protocol and achieve an unprecedented high estimation fidelity, which can beat all strategies based on restricted collective measurements by more than 11 standard deviations. These results clearly demonstrate that genuine collective measurements can extract more information than local measurements and restricted collective measurements. Our Letter opens the door for exploring genuine multipartite nonclassical measurements and their power in quantum information processing.
We analyse the reconstruction of an unknown pure qubit state. We derive the optimal guess that can be inferred from any set of measurements on N identical copies of the system with the fidelity as a figure of merit. We study in detail the estimation process with individual von Neumann measurements and demonstrate that they are very competitive as compared to (complicated) collective measurements. We compute the expressions of the fidelity for large N and show that individual measurement schemes can perform optimally in the asymptotic regime.
Abstract Quantum estimation involving multiple parameters remains an important problem of both theoretical and practical interest. In this work, we study the problem of simultaneous estimation of two parameters that are respectively associate with the length and direction of the Bloch vector for identically prepared qubit states that is confined to a plane, where in order to obtain the optimal estimation precision for both parameters, collective measurements on multiple qubits are necessary. Upon treating N qubits as an ensemble of spin-1/2 systems, we show that simultaneous optimal estimation for both parameters can be attained asymptotically with a simple collective measurement strategy—first, we estimate the length parameter by measuring the populations in spaces corresponding to different total angular momentum values j , then we estimate the direction parameter by performing a spin projection onto an optimal basis. Furthermore, we show that when the state is nearly pure, for sufficiently but not arbitrarily large N , most information will be captured in the largest three j -subspaces. Then, we study how the total angular-momentum measurement can be realized by observing output signatures from a Bell multiport setup, either exactly for N = 2, 3, or approximately when the qubits are nearly pure for other N values. We also obtain numerical results that suggest that using a Bell multiport setup, one can distinguish between projection onto the j = N /2 and j = N /2 − 1 subspaces from their respective interference signatures at the output.
Quantum estimation involving multiple parameters remains an important problem of both theoretical and practical interest. In this work, we study the problem of simultaneous estimation of the length and direction of the Bloch vector for identically prepared qubit states that is confined to a plane, where in order to obtain the optimal estimation precision for both parameters, collective measurements on multiple qubits are necessary. Upon treating N qubits as an ensemble of spin-1/2 systems, we show that simultaneous optimal estimation for both parameters can be attained asymptotically with a simple collective measurement strategy -- First, we estimate the length parameter by measuring the populations in spaces corresponding to different total angular momentum values j, then we estimate the direction parameter by performing a spin projection onto an optimal basis. Furthermore, we show how the total angular-momentum measurement can be realized by observing output signatures from a Bell multiport setup, either exactly for N=2,3, or approximately when the qubits are nearly pure for other N values. Our results point to the conjecture that using a Bell multiport setup, one can distinguish between projection onto the j=N/2 and j=N/2−1 subspaces from their respective interference signatures at the output.
Quantum measurements can sometimes extract more information from multiple copies of a state than from each copy separately. One of the simplest settings is the discrimination of three qubit states when two identical copies are available and one is allowed to perform an arbitrary joint measurement. Recent numerical work identified the double-trine ensemble as the apparent optimal benchmark in this setting, but an analytic proof of its optimality remained open. In this paper, we prove that the double-trine ensemble is indeed globally optimal. More precisely, among all triples of equally likely qubit states, no choice of states can give a larger optimal two-copy discrimination success probability. The result holds in full generality, whether the signal states are pure or mixed, real or complex. This exactly solved case serves as a foundation for a broader analytic understanding of optimal collective measurements.
In multi-parameter quantum metrology, the resource of entanglement can lead to an increase in efficiency of the estimation process. Entanglement can be used in the state preparation stage, or the measurement stage, or both, to harness this advantage; here we focus on the role of entangling measurements. Specifically, entangling or collective measurements over multiple identical copies of a probe state are known to be superior to measuring each probe individually, but the extent of this improvement is an open problem. It is also known that such entangling measurements, though resource-intensive, are required to attain the ultimate limits in multi-parameter quantum metrology and quantum information processing tasks. In this work we investigate the maximum precision improvement that collective quantum measurements can offer over individual measurements for estimating parameters of qudit states, calling this the 'collective quantum enhancement'. We show that, whereas the maximum enhancement can, in principle, be a factor of n for estimating n parameters, this bound is not tight for large n . Instead, our results prove an enhancement linear in dimension of the qudit is possible using collective measurements and lead us to conjecture that this is the maximum collective quantum enhancement in any local estimation scenario.