Quantum Reference Frames

A quantum reference frame is a physical system whose position, orientation, or clock reading is used to describe other systems while retaining its quantum degrees of freedom. Changing such a frame can correlate variables that were independent in another description. It does not let an observer choose measured outcomes by adopting a viewpoint.

Relative coordinates are observables

For two one-dimensional particles, define center-of-mass and relative variables

$$X=\frac{m_Ax_A+m_Bx_B}{M},\qquad r=x_B-x_A, \qquad M=m_A+m_B,$$ $$P=p_A+p_B,\qquad p_r=\frac{m_Ap_B-m_Bp_A}{M}.$$

They obey $[X,P]=[r,p_r]=i\hbar$ with vanishing cross commutators. For a translation-invariant interaction $V(r)$,

$$H=\frac{P^2}{2M}+\frac{p_r^2}{2\mu}+V(r), \qquad \mu=\frac{m_Am_B}{M}.$$

This familiar separation already shows that the kinetic energy of relative motion involves the reduced mass. Replacing a finite quantum frame by an infinitely heavy fixed origin is an approximation with dynamical consequences.

An uncertain frame creates relational correlations

As a simplified kinematic coordinate change relative to a laboratory origin $C$, let $A$ have position $x$ and $B$ position $y$. In an $A$-centered description the old origin has coordinate $u=-x$, while $B$ has coordinate $r=y-x$. The Jacobian has absolute value one, so

$$\psi'(u,r)=\psi(-u,r-u)$$

preserves normalization. A product laboratory wavefunction $f(x)g(y)$ becomes $f(-u)g(r-u)$, generally not a product in $(u,r)$.

For narrow packets, a laboratory state with $A$ in an equal superposition near $x=\pm a$ and $B$ localized near $0$ becomes a correlation in which both $C$ and $B$ lie near $u=r=\mp a$ relative to $A$. What had been assigned to frame-position uncertainty now appears in relational correlations. Ideal position eigenkets are only shorthand; normalizable wave packets are required for a physical state.

This coordinate example omits boosts, dynamical clock changes, and constraints. A complete frame transformation also transforms conjugate momenta, Hamiltonians, and measurement operations. Giacomini, Castro-Ruiz, and Brukner develop such transformations explicitly.

Invariant predictions require transformed observables

For a unitary description change $S$, use $\rho'=S\rho S^\dagger$ and $E'=SES^\dagger$. Then

$$\operatorname{tr}(\rho'E')=\operatorname{tr}(\rho E).$$

Changing the state while leaving the measurement fixed generally asks a different physical question. Statements that entanglement is frame dependent refer to a changed relational subsystem decomposition or accessible degrees of freedom, not to an arbitrary disagreement about one fixed detector probability.

An uncertain reference may also limit what operations are available. Lack of a shared phase reference, for example, can make some coherences operationally inaccessible. This differs from an objective destruction of the global state.

Exercise

For independent laboratory positions with variances $\sigma_A^2,\sigma_B^2$, calculate $\operatorname{Var}(u)$, $\operatorname{Var}(r)$, and $\operatorname{Cov}(u,r)$ under the coordinate change above.

Solution

They are $\sigma_A^2$, $\sigma_A^2+\sigma_B^2$, and $\sigma_A^2$, respectively. The shared uncertainty in $x_A$ produces correlation between $u=-x_A$ and $r=x_B-x_A$. As the frame becomes sharply localized, this kinematic covariance tends to zero. For quantum wavefunctions, the same change of variables can produce nonfactorization relative to the new subsystem split.

References

• Giacomini, F., Castro-Ruiz, E., and Brukner, Č., “Quantum mechanics and the covariance of physical laws in quantum reference frames,” Nature Communications 10, 494 (2019), arXiv:1712.07207 — states, operations, and dynamics under quantum frame changes.