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arXiv:0902.0935 · New Journal of Physics

Entanglement detection with bounded reference frames

Fabio Costa, Nicholas Harrigan, Terry Rudolph, Časlav Brukner

Published Dec 4, 2009• 14 citations
arXivHTMLPDF

Abstract

Quantum experiments usually assume the existence of perfect, classical reference frames (RFs), which allow for the specification of measurement settings (e.g. orientation of the Stern-Gerlach magnet in spin measurements) with arbitrary precision. If the RFs are 'bounded' (i.e. quantum systems themselves, having a finite number of degrees of freedom), only limited precision can be attained. Using spin coherent states as bounded RFs, we have found the minimum size needed for them to violate local realism for entangled spin systems. For composite systems of spin particles, RFs of very small size are sufficient for the violation; however, to see this violation for macroscopic entangled spins, the size of the RF must be at least quadratically larger than that of the spins. The unavailability of such RFs gives a possible explanation for the non-observance of violation of local realism in everyday experience.

DOI
10.1088/1367-2630/11/12/123007

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Quantum experiments usually assume the existence of perfect, classical reference frames (RFs), which allow for the specification of measurement settings (e.g. orientation of the Stern-Gerlach magnet in spin measurements) with arbitrary precision. If the RFs are 'bounded' (i.e. quantum systems themselves, having a finite number of degrees of freedom), only limited precision can be attained. Using spin coherent states as bounded RFs, we have found the minimum size needed for them to violate local realism for entangled spin systems. For composite systems of spin particles, RFs of very small size are sufficient for the violation; however, to see this violation for macroscopic entangled spins, the size of the RF must be at least quadratically larger than that of the spins. The unavailability of such RFs gives a possible explanation for the non-observance of violation of local realism in everyday experience.
Does there exist a limit for the applicability of quantum theory for objects of large mass or size, or objects whose states are of large complexity or dimension of the Hilbert space? The possible answers range from practical limitations due to decoherence within quantum theory to fundamental limits due to collapse models that modify quantum theory. Here, we suggest the viewpoint that there might be also fundamental limits without altering the quantum laws. We first demonstrate that for two quantum spins systems of a given spin length, no violation of local realism can be observed, if the measurements are sufficiently coarse-grained. Then we show that there exists a fundamental limit for the precision of measurements due to (i) the Heisenberg uncertainty relation which has to be applied to the measuring apparatus, (ii) relativistic causality, and (iii) the finiteness of resources in any laboratory including the whole universe. This suggests that there might exist a limit for the size of the systems (dimension of the Hilbert space) above which no violation of local realism can be seen anymore.
We show that the expectation value of squared correlations measured along random local directions is an identifier of quantum entanglement in pure states, which can be directly experimentally assessed if two copies of the state are available. Entanglement can therefore be detected by parties who do not share a common reference frame and whose local reference frames, such as polarizers or Stern-Gerlach magnets, remain unknown. Furthermore, we also show that in every experimental run, access to only one qubit from the macroscopic reference is sufficient to identify entanglement, violate a Bell inequality, and, in fact, observe all phenomena observable with macroscopic references. Finally, we provide a state-independent entanglement witness solely in terms of random correlations and emphasize how data gathered for a single random measurement setting per party reliably detects entanglement. This is only possible due to utilized randomness and should find practical applications in experimental confirmation of multiphoton entanglement or space experiments.
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Quantum physics violates the basic assumptions of local realism. Particles can be entangled with each other in ways that allow correlations stronger than possible in classical physics. This was first realized by Einstein, Podolsky and Rosen (EPR) in 1935 and led them to conclude that quantum theory was incomplete. It took about 30 years before J. S. Bell found observable consequences of this phenomenon by showing how entangled states violate inequalities for two-particle correlations derived purely from the assumption that the world obeys local realism. Violations of Bell inequalities were first observed experimentally in the Aspect experiment, which will be discussed below. Such experiments are technically challenging and care is needed to avoid any loopholes in the experimental setup that would still permit the description of the experiments by a local realistic theory. Bell inequalities describe violations of local realism on average, meaning that each individual experimental outcome could occur in a local realistic theory. It is the statistics of these outcomes that violates local realism. In contrast, another class of quantum states, so-called Greenberger–Horne–Zeilinger (GHZ) states, which contain three particles, can be shown to violate local realism in each measurement outcome. We will also discuss this class of states and experiments by Zeilinger's group, which demonstrated this type of violation of local realism for the first time. Our aim here is not to discuss the exact relations between quantum entanglement, non-locality and realism, and their physical and philosophical implications. While this is a highly interesting topic which attracts a lot of attention from researchers, such an exploration would go beyond the scope of this book.