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arXiv:quant-ph/0410169 · Physical Review Letters

Efficient Universal Programmable Quantum Measurements

Giacomo Mauro D’Ariano, Paolo Perinotti

Published Mar 8, 2005• 42 citations
arXivHTMLPDF

Abstract

A universal programmable detector is a device that can be tuned to perform any desired measurement on a given quantum system, by changing the state of an ancilla. With a finite dimension d for the ancilla only approximate universal programmability is possible, with size d=f(epsilon(-1)) increasing the function of the "accuracy" epsilon(-1). In this Letter we show that, much better than the exponential size known in the literature, one can achieve polynomial size. An explicit example with linear size is also presented. Finally, we show that for covariant measurements exact programmability is feasible.

DOI
10.1103/physrevlett.94.090401

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A universal programmable detector is a device that can be tuned to perform any desired measurement on a given quantum system, by changing the state of an ancilla. With a finite dimension d for the ancilla only approximate universal programmability is possible, with size d=f(epsilon(-1)) increasing the function of the "accuracy" epsilon(-1). In this Letter we show that, much better than the exponential size known in the literature, one can achieve polynomial size. An explicit example with linear size is also presented. Finally, we show that for covariant measurements exact programmability is feasible.
We review some partial results for two strictly related problems. The first problem consists in finding the optimal joint unitary transformation on system and ancilla which is the most efficient in programming any desired channel on the system by changing the state of the ancilla. In this respect we present a solution for dimension two for both system and ancilla. The second problem consists in finding the optimal universal programmable detector, namely a device that can be tuned to perform any desired measurement on a given quantum system, by changing the state of an ancilla. With a finite dimension for the ancilla only approximate universal programmability is possible, with minimal dimension increasing function versus the precision of the approximation. We show that one can achieve a dimension growing polynomially versus the precision, and even linearly in specific cases.
The expectation value of an arbitrary operator O can be obtained via a universal measuring apparatus that is independent of O, by changing only the data-processing of the outcomes. Such a ``universal detector'' performs a joint measurement on the system and on a suitable ancilla prepared in a fixed state, and is equivalent to a positive operator valued measure (POVM) for the system that is ``informationally complete''. The data processing functions generally are not unique, and we pose the problem of their optimization, providing some examples for covariant POVM's, in particular for SU(d) covariance group.
PACS. 03.65.Ta – Foundations of quantum mechanics; measurement theory. PACS. 03.65.Wj – State reconstruction, quantum tomography. Abstract. – We address the problem of achieving a universal measuring apparatus that allows to estimate the expectation value of any arbitrary operator by changing only the data processing of the outcomes. The “universal detector ” performs a joint measurement on the system and on a suitably prepared ancilla. We characterize such universal detectors, and show how they can be obtained either via Bell measurements or via local measurements and classical communication between system and ancilla. Quantum technology is nowadays turning from the stage of experimental setup design to that of quantum system engineering. The aim is to produce tools for communication, infor-mation processing, and storage, which rely on the principles of quantum mechanics, with the chance of achieving much higher speeds and capacities than those of classical devices. In this scenario, a new kind of quantum lab can be devised, in which universality and programmability are crucial features, with different tasks achieved by a basic set of devices. A “universal detector ” would allow the estimation of ensemble averages of arbitrary op-
A programmable quantum processor uses the states of a program register to specify one element of a set of quantum channels which is applied to an input register. It is well-known that such a device is impossible with a finite-dimensional program register for any set that contains infinitely many unitary quantum channels (Nielsen and Chuang's No-Programming Theorem ), meaning that a universal programmable quantum processor does not exist. The situation changes if the system has symmetries. Indeed, here we consider group-covariant channels. If the group acts irreducibly on the channel input, these channels can be implemented exactly by a programmable quantum processor with finite program dimension (via teleportation simulation , which uses the Choi-Jamiolkowski state of the channel as a program). Moreover, by leveraging the representation theory of the symmetry group action, we show how to remove redundancy in the program and prove that the resulting program register has minimum Hilbert space dimension. Furthermore, we provide upper and lower bounds on the program register dimension of a processor implementing all group-covariant channels approximately.
Programmable photonic quantum processors face a critical challenge: despite significant advances in quantum state preparation and manipulation, measurements remain limited to projective techniques. Here, we demonstrate a programmable measurement processor that overcomes this limitation by enabling arbitrary quantum measurements within a scalable circuit framework. Our large-scale integrated photonic architecture achieves precise coherent control of ancillary quantum systems, realizing a universal four-dimensional quantum measurement device. We benchmark the processor by performing measurement tomography on 100 randomly selected measurements, achieving an average fidelity of 97.7%. The processor's performance exceeds the theoretical limits of projective measurements in three key quantum information tasks: state discrimination (with 23 times lower error), state estimation (with 10.6% higher fidelity), and randomness generation (with 37% more randomness yield), demonstrating its high operational quality. This work establishes a fully programmable quantum measurement processor, advancing the development of universal quantum operations for photonic quantum information processing by providing the key missing component.
We prove that for a programmable measurement device that approximates every POVM with an error $\ensuremath{\leqslant}\ensuremath{\delta}$, the dimension of the program space has to grow at least polynomially with $\frac{1}{\ensuremath{\delta}}$. In the case of qubits we can improve the general result by showing a linear growth. This proves the optimality of the programmable measurement devices recently designed in G. M. D'Ariano and P. Perinotti, Phys. Rev. Lett. 94, 090401 (2005).