Ideal Projective Measurements Have Infinite Resource Costs

Institute for Quantum Optics and Quantum Information - IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria

Abstract

We show that it is impossible to perform ideal projective measurements on quantum systems using finite resources. We identify three fundamental features of ideal projective measurements and show that when limited by finite resources only one of these features can be salvaged. Our framework is general enough to accommodate any system and measuring device (pointer) models, but for illustration we use an explicit model of an $N$-particle pointer. For a pointer that perfectly reproduces the statistics of the system, we provide tight analytic expressions for the energy cost of performing the measurement. This cost may be broken down into two parts. First, the cost of preparing the pointer in a suitable state, and second, the cost of a global interaction between the system and pointer in order to correlate them. Our results show that, even under the assumption that the interaction can be controlled perfectly, achieving perfect correlation is infinitely expensive. We provide protocols for achieving optimal correlation given finite resources for the most general system and pointer Hamiltonians, phrasing our results as fundamental bounds in terms of the dimensions of these systems.

The notion of measurement forms an integral (and often heatedly debated) part of quantum mechanical reasoning. An ideal measurement provides information about the measured system, but also disturbs the latter. Nonetheless, according to the so-called projection postulate of quantum mechanics, the measurement outcome allows one to make precise statements about the system after its interaction with the measuring device. Understanding the process of measurement as an interaction of a system with a pointer elucidates the thermodynamic nature of the process. While previous considerations of the first and second law only implied a modest cost in energy and compensation of entropy, we show that the third law of thermodynamics imposes the most drastic of restrictions. Indeed, ideal measurements are as impossible as cooling a system to the ground state. Impossible because they would require infinite amounts of time or energy, or exact control over infinitely complex measuring devices. Consequently, all practical measurements are non-ideal and to approximate good measurements large amounts of work need to be expended.

Here, we investigate the structure and resource costs of non-ideal measurements and identify three fundamental properties that all ideal measurements possess. We argue that when limited by finite resources the three properties cannot hold simultaneously – one must choose which one of the properties to keep. Under the assumption that at least one property holds, we then derive expressions for the energy cost of performing such an imperfect measurement.

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