Perspective on: Switching Quantum Reference Frames for Quantum Measurement

This is a Perspective on "Switching Quantum Reference Frames for Quantum Measurement" by Jianhao M. Yang, published in Quantum 4, 283 (2020).

By Pierre Martin-Dussaud (Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France and Basic Research Community for Physics e.V.).

Quantum reference frames

In the last few years, the communities of quantum information and quantum gravity have been working together on the notion of $\textit{quantum reference frames}$. The notion itself is not new (similar ideas go back to 1967 at least [1,2]) but recent results shed new light on it. Let me recall a few important facts.

In its original version, quantum mechanics divides the world in two: the quantum system and the classical observer. However, we have good reasons to believe that the observer is nothing but a quantum system itself. In other words, the delimitation between the quantum and the classical realms is not fundamental but can be set between any two quantum systems. Thus, physics is about describing systems from the perspective of other systems, as advocated in the $\textit{relational interpretation}$ of quantum mechanics [3].

Such a view is very familiar in general relativity. To extract observational predictions from the theory, one has to specify an observer with respect to which time, positions, velocities and accelerations are measured. Although general relativity is given abstractly as a $\textit{perspective-neutral theory}$, any of its operational implementations requires to adopt the view of some reference frame. Conversely, the various perspectives on a system relate to one another by changes of reference frame, which form a symmetry group. The absolute core of the theory is given by some abstract mathematical objects, invariant under the action of the symmetries.

Keeping in mind the horizon of quantum gravity, reference frames should be considered themselves as quantum systems. From this simple fact, hard questions arise, like: what are the relevant transformations to switch from one quantum perspective to another? or how does the world look like for a reference frame in a state of superposition?

Switching between perspectives

In [4], a concrete example is worked out and enables to draw few lessons. The model consists of three systems $A$, $B$, $C$ of which one considers the relative positions. The density matrix $\rho_{AB}^{(C)}$, describing the state of $A$ and $B$ from $C$-perspective, is transformed into the state of $B$ and $C$ from $A$-perspective, as
\begin{equation}
\rho_{BC}^{(A)} = \hat{S} \, \rho_{AB}^{(C)} \, \hat{S}^\dagger,
\end{equation}
with the unitary operator
\begin{equation}
\hat{S} = \hat{\mathcal{P}}_{AC} \, e^{\frac{i}{\hbar} \hat{x}_A \hat{p}_B}.
\end{equation}
To be precise, we should say that there is not a single notion of a jump from one perspective to another, as its definition relies upon an initial choice of prefered variables (here the positions). Among other conceptual takes, the formalism shows that entanglement is a reference-frame dependent notion.

Also, a change between two reference frames induces different stories for the evolution of states. As stressed by von Neumann, there are two ways for a quantum state to evolve: the unitary time evolution, given by the Schrödinger equation, and the non-unitary measurement projection, aka collapse of the wave-function. The former amounts to finding the good transformation for the hamiltonian. The latter is maybe the most intriguing: if $C$ performs a measurement on $A$ and $B$ via an apparatus $E$, how does the same process look like from the perspective of $A$? The answer is both simple and striking: for $A$, it is $B$ and $C$ which are measured by $E$ (and the observable is different).

A first-principle approach

In the original paper [4], the results mentioned above are derived from some operational considerations, more suited for concrete usage in quantum information. This approach has the drawback of remaining a bit opaque to further generalisations. Hopefully, another approach has been proposed soon after to derive the same transformations from first principles [5]. The method gets inspiration from the theory of hamiltonian constrained systems, familiar to the quantum gravity community.

At the classical level, the equivalence of the many point-of-views entails a symmetry principle, which imposes constraints on the phase space of a theory. Then, choosing a reference frame is tantamount to fixing a gauge. The quantum picture can be recovered following two paths of quantisation: the Dirac quantisation (quantise then impose the constraints) and the reduced quantisation (the other way around). Although both procedures are equivalent in simple cases, they differ in spirit.

On the one hand, the reduced quantisation can be understood as the quantisation from an internal perspective. For instance, it results in a Hilbert space $\mathcal{H}_{BC|A}$ from $A$-perspective.

On the other hand, Dirac quantisation is perspective-neutral. It results in some “agnostic” Hilbert space $\mathcal{H}_{phys}$. The latter recovers the usual description of quantum mechanics, from the point of view of an ideal classical observer, a ‘point of view from nowhere’. It also carries some redundancies that contain non physically meaningful information. Then, choosing a reference frame can be achieved as a $\textit{quantum symmetry reduction}$, which consists in a mapping between $\mathcal{H}_{phys}$ and $\mathcal{H}_{BC|A}$. Switching from one reference frame to another can now be better understood through the intermediate step of the perspective-neutral Hilbert space $\mathcal{H}_{phys}$.

A perspective-neutral measurement?

Despite the success of the first-principle approach of [5], not all of the results of [4] had been explained in a perspective-neutral context. In particular, the measurement process remained to bit fit into it. It is this stone that has been brought by the recent work of Yang [6].

It is argued that the unitary time-evolution can be implemented at the perspective-neutral level of $\mathcal{H}_{phys}$, contrary to the measurement projection that can only be formulated after the quantum symmetry reduction to $\mathcal{H}_{BC|A}$ has been performed. There is one noticeable exception to that rule: when the measured variable is independent of the variables involved in the change of reference frames.

The results of [4], that shows how the measurement process looks like from different perspectives, are recovered. Moreover, it is shown how the projection operator should be transformed when the measurement apparatus is taken as the quantum reference frame itself.

In my opinion, future work should consist in a generalisation of the ideas to more general systems than the toy-model here considered. This may also facilitate to take the conceptual lessons out of the formalism. Finally, the hamiltonian phase space approach may have to be overtaken to express general changes of reference frames, related by Lorentz transformations or diffeomorphisms. This would bring us closer to quantum gravity.

► BibTeX data

► References

[1] Y. Aharonov and L. Susskind, ``Charge Superselection Rule,'' Phys. Rev. 155 no. 5, (Mar., 1967) 1428–1431.
https:/​/​doi.org/​10.1103/​PhysRev.155.1428

[2] B. S. DeWitt, ``Quantum Theory of Gravity. I. The Canonical Theory,'' Phys. Rev. 160 no. 5, (Aug., 1967) 1113–1148.
https:/​/​doi.org/​10.1103/​PhysRev.160.1113

[3] C. Rovelli, ``Relational Quantum Mechanics,'' International Journal of Theoretical Physics 35 no. 8, (1996) 1637–1678, arXiv:quant-ph/​9609002.
https:/​/​doi.org/​10.1007/​BF02302261
arXiv:quant-ph/9609002

[4] F. Giacomini, E. Castro-Ruiz, and Č. Brukner, ``Quantum Mechanics and the Covariance of Physical Laws in Quantum Reference Frames,'' Nature Communications 10 no. 1, (Jan., 2019) 494.
https:/​/​doi.org/​10.1038/​s41467-018-08155-0

[5] A. Vanrietvelde, P. A. Hoehn, F. Giacomini, and E. Castro-Ruiz, ``A Change of Perspective: Switching Quantum Reference Frames via a Perspective-Neutral Framework,'' Quantum 4 (Jan., 2020) 225, arXiv:1809.00556.
https:/​/​doi.org/​10.22331/​q-2020-01-27-225
arXiv:1809.00556

[6] J. M. Yang, ``Switching Quantum Reference Frames for Quantum Measurement,'' arXiv:1911.04903 [quant-ph] (Mar., 2020) , arXiv:1911.04903 [quant-ph].
https:/​/​doi.org/​10.22331/​q-2020-06-18-283
arXiv:1911.04903

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