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arXiv:1011.4252 · Physical Review A

Optimal primitive reference frames

David H. Jennings

Published Jul 5, 2011• 1 citation
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Abstract

We consider the smallest possible directional reference frames allowed and determine the best one can ever do in preserving quantum information in various scenarios. We find that for the preservation of a single spin state, two orthogonal spins are optimal primitive reference frames; and in a product state, they do approximately $22%$ as well as an infinite-sized classical frame. By adding a small amount of entanglement to the reference frame, this can be raised to $2(2/3){}^{5}=26%$. Under the different criterion of entanglement preservation, a very similar optimal reference frame is found; however, this time it is for spins aligned at an optimal angle of ${87}^{\ifmmode^\circ\else\textdegree\fi{}}$. In this case $24%$ of the negativity is preserved. The classical limit is considered numerically, and indicates under the criterion of entanglement preservation, that ${90}^{\ifmmode^\circ\else\textdegree\fi{}}$ is selected out nonmonotonically, with a peak optimal angle of ${96.5}^{\ifmmode^\circ\else\textdegree\fi{}}$ for $L=3$ spins.

DOI
10.1103/physreva.84.012306

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•Oct 5, 2026•0 citations
We consider the smallest possible directional reference frames allowed and determine the best one can ever do in preserving quantum information in various scenarios. We find that for the preservation of a single spin state, two orthogonal spins are optimal primitive reference frames; and in a product state, they do approximately 2222%22 as well as an infinite-sized classical frame. By adding a small amount of entanglement to the reference frame, this can be raised to 2(2/3)5=262(2/3){}^{5}=26%2(2/3)5=26. Under the different criterion of entanglement preservation, a very similar optimal reference frame is found; however, this time it is for spins aligned at an optimal angle of 87\ifmmode∘\else\textdegree\fi{87}^{\ifmmode^\circ\else\textdegree\fi{}}87\ifmmode∘\else\textdegree\fi. In this case 2424%24 of the negativity is preserved. The classical limit is considered numerically, and indicates under the criterion of entanglement preservation, that 90\ifmmode∘\else\textdegree\fi{90}^{\ifmmode^\circ\else\textdegree\fi{}}90\ifmmode∘\else\textdegree\fi is selected out nonmonotonically, with a peak optimal angle of 96.5\ifmmode∘\else\textdegree\fi{96.5}^{\ifmmode^\circ\else\textdegree\fi{}}96.5\ifmmode∘\else\textdegree\fi for L=3L=3L=3 spins.
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.
The optimal NNN qubit states featuring highest sensitivity to small misalignment of Cartesian reference frames are found using the quantum Cram\'er-Rao bound. It is shown that the optimal states are supported on the symmetric subspace and hence are mathematically equivalent to a single spin J=N∕2J=N∕2J=N∕2. Majorana representation of spin states is used to reveal a beautiful connection between the states optimal for aligning reference frames and the platonic solids.
Abstract Symmetry principles are fundamental in physics, and while they are well understood within Lagrangian mechanics, their impact on quantum channels has a range of open questions. The theory of asymmetry grew out of information-theoretic work on entanglement and quantum reference frames, and allows us to quantify the degree to which a quantum system encodes coordinates of a symmetry group. Recently, a complete set of entropic conditions was found for asymmetry in terms of correlations relative to infinitely many quantum reference frames. However, these conditions are difficult to use in practice and their physical implications unclear. In the present theoretical work, we show that this set of conditions has extensive redundancy, and one can restrict to reference frames forming any closed surface in the state space that has the maximally mixed state in its interior. This in turn implies that asymmetry can be reduced to just a single entropic condition evaluated at the maximally mixed state. Contrary to intuition, this shows that we do not need macroscopic, classical reference frames to determine the asymmetry properties of a quantum system, but instead infinitesimally small frames suffice. Building on this analysis, we provide simple, closed conditions to estimate the minimal depolarization needed to make a given quantum state accessible under channels covariant with any given symmetry group.
In the quantization of gauge theories and quantum gravity, it is crucial to treat reference frames such as rods or clocks not as idealized external classical relata, but as internal quantum subsystems. In the Page-Wootters formalism, for example, evolution of a quantum system S is described by a stationary joint state of S and a quantum clock, where time dependence of S arises from conditioning on the value of the clock. Here, we consider (possibly imperfect) internal quantum reference frames R for arbitrary compact symmetry groups, and show that there is an exact quantitative correspondence between the amount of entanglement in the invariant state on RS and the amount of asymmetry in the corresponding conditional state on S. Surprisingly, this duality holds exactly regardless of the choice of coherent state system used to condition on the reference frame. Averaging asymmetry over all conditional states, we obtain a simple representation-theoretic expression that admits the study of the quality of imperfect quantum reference frames, quantum speed limits for imperfect clocks, and typicality of asymmetry in a unified way. Our results shed light on the role of entanglement for establishing asymmetry in a fully symmetric quantum world.
At eleven parameters a frame is a body with a momentum. The quantum step of the theory makes that momentum an operator, and a frame body in a superposition of momenta is a quantum reference frame. This letter asks what using such a frame does to it, in a model with one assumption. An ideal meter comoving with the frame body reads the canonical spins of test bodies, and nothing else. Relative to the frame each spin is turned by the Wigner rotation of the composition law, by an angle that depends on the frame's own rapidity. So every spin the frame serves is a weak measurement of the frame's velocity. The quantum Fisher information per spin is the square of the derivative of the Wigner angle, in closed form. For a frame at rest it is tanh 2 (θt/2), θt the test body's rapidity, and it is bounded by 1/ cosh 2 θF . With polarised spins and the records kept, n spins sharpen the frame's rapidity, to the van Trees bound 1/(σ -2 + n F ) within one percent. With the records discarded, or with unpolarised spins, the frame's momentum coherence decays by the overlap of the two rotated spin states per spin. For a Gaussian frame its purity is 1/ 1 + N F σ 2 , equal to the purity of the spins relative to the frame. The origin of the frame is the conjugate reference. Each discarded spin kicks the frame's position by ±ℏφ ′ /2M , a random walk, and the kept records widen it by the Heisenberg bound. The velocity reference is kept or sharpened while the origin degrades, the two being the conjugate blocks of the eleven. The frame of an entangled pair with a sharp total momentum has nothing to learn and no origin to lose. The sixteen of a body being the operators of two qubits, a boost is a coupling that is not a gate, and its cost in a simulation is stated. The statements are checked numerically.