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Rotational covariance restricts available quantum states

Fynn Otto, Konrad Szymański

Abstract

Quantum states of angular momentum and spin generally are not invariant under rotations of the reference frame. Therefore, they can be used as a resource of relative orientation, which is encoded in the asymmetry of the state under consideration. In this paper we introduce the analytical characterization of the rotational information by parameterizing the group characteristic function by polynomial functions. By doing so, we show that the set of states achievable through transformations lacking a reference frame (rotationally covariant ones) admits an analytical characterization and can be studied through the use of semidefinite optimization techniques. We demonstrate the developed methods via examples, and provide a physical scenario in which a reference-independent operation performs a metrologically useful operation: the preparation of a state of light improving interferometer sensitivity, which equivalently can be realized as a postprocessing step.

Rotational covariance restricts available quantum states

Abstract

Quantum states of angular momentum and spin generally are not invariant under rotations of the reference frame. Therefore, they can be used as a resource of relative orientation, which is encoded in the asymmetry of the state under consideration. In this paper we introduce the analytical characterization of the rotational information by parameterizing the group characteristic function by polynomial functions. By doing so, we show that the set of states achievable through transformations lacking a reference frame (rotationally covariant ones) admits an analytical characterization and can be studied through the use of semidefinite optimization techniques. We demonstrate the developed methods via examples, and provide a physical scenario in which a reference-independent operation performs a metrologically useful operation: the preparation of a state of light improving interferometer sensitivity, which equivalently can be realized as a postprocessing step.
Paper Structure (19 sections, 17 theorems, 117 equations, 4 figures)

This paper contains 19 sections, 17 theorems, 117 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Theoretical background
  3. Phase reference and $\operatorname{U}(1)$ group
  4. General results
  5. Three-dimensional rotations: $\operatorname{SU}(2)$
  6. Relevant mathematical structures
  7. Polynomial $\operatorname{SU}(2)$ representation
  8. Semidefinite programs
  9. Application to $\operatorname{SU}(2)$-covariant transformations
  10. Pure state transformations
  11. Transformations of mixed states
  12. Phase estimation experiment
  13. Conclusions
  14. Majorana stars and polynomial $\operatorname{SU}(2)$ representation
  15. Strict treatment of the characteristic function in the polynomial form
  16. ...and 4 more sections

Key Result

Proposition 1

The $\operatorname{U}(1)$-covariant quantum channel $\mathcal{E}$ such that $\sum \sqrt{p_n} \ket{n} \xmapsto{\mathcal{E}} \sum_{n\in\mathbb{N}_0} \sqrt{q_n} \ket{n}$ exists if and only if there exists $\delta\in\mathbb{N}$ and a sequence $w_n$ such that $\sum_{n\in\mathbb{N}} w_n=1$, $w_n\ge0$ a

Figures (4)

  • Figure 1: The constraint of a quantum channel being symmetric with respect to a certain symmetry group is called covariance and can be interpreted in the two pictured ways. Top: a state $\rho$ is transferred between two laboratories, but as a result of lack of common reference the laboratory B receives a state modified by a unitary $U$ describing a particular element of a symmetry group. The channel $\mathcal{E}$ is applied to the modified state $\mathcal{U}(\rho)=U\rho U^\dagger$ and it is sent back; the result from perspective of A is equal to $\mathcal{E}(\rho)$ if the channel $\mathcal{E}$ is covariant. Bottom: equivalently, the channel $\mathcal{E}$ is covariant if the result of state transformation by sequential application of $\mathcal{E}$ and $\mathcal{U}$ does not depend on the order of operations (for all $\mathcal{U}$ and $\rho$).
  • Figure 2: Example of probability distributions compatible with \ref{['prop:uocov']}: deterministic transformation of $\ket\psi=\sum_n \sqrt{p_n} \ket{n}$ to $\ket\phi=\sum_n \sqrt{q_n} \ket{n}$ is possible with a $\operatorname{U}(1)$-covariant channel, because $p_n$ is a convolution of $q_n$ with an auxiliary $w_n$.
  • Figure 3: An unknown phase shift $\theta$ can be determined by measuring difference of the photon numbers at the output of the interferometer. The accuracy of phase estimation can be increased by the action of a device $D$ transforming the initial state $\ket\gamma\otimes\ket0$ into a more sensitive one $\ket\gamma\otimes\ket\tau$. Since the device action commutes with the interferometer, the preparation stage can be replaced by postprocessing by the same device, with the same metrological improvement.
  • Figure 4: For a pure spin state $\ket\psi$ with definite total angular momentum $j$, the Majorana stars is a collection of $2j$ points on an unit sphere fully defining the state in question. The points correspond to directions $\vec{n}$ along which the state has zero overlap with a coherent state, $\langle -\vec{n} \vert \psi \rangle=0$.

Theorems & Definitions (34)

  • Definition 1
  • Proposition 1
  • Definition 2
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Corollary 1
  • Proposition 5
  • Proposition 6
  • Definition 3
  • ...and 24 more