Quantum Frame Relativity of Subsystems, Correlations and Thermodynamics

TL;DR

This work develops a perspective-neutral framework for quantum reference frames (QRFs) and proves that the partitioning of a system into subsystems is inherently QRF-dependent, generalizing the relativity of simultaneity from special relativity. By restricting to ideal QRFs associated with finite Abelian groups, it introduces TPS-invariant subalgebras that identify observables invariant up to local unitaries under QRF changes, and analyzes how reduced subsystem states, correlations, and dynamics transform across QRFs. It then applies these algebraic tools to quantum thermodynamics, showing that equilibrium states, temperatures, and heat/work exchange are generally QRF-relative, with conditions for cross-frame invariance and notable phenomena such as frame-induced sign changes in temperature. The results illuminate how correlations and dynamical locality are frame-dependent, while preserving a core covariance structure that mirrors special relativity, and suggest extensions to gauge theories and gravity via relational subsystems and edge-mode perspectives. Overall, the paper reveals a rich, internally consistent picture of how physics at the subsystem level reorganizes under shifts of internal quantum reference frames, with meaningful consequences for thermodynamics and open-system dynamics.

Abstract

It was recently noted that different internal quantum reference frames (QRFs) partition a system in different ways into subsystems, much like different inertial observers in special relativity decompose spacetime in different ways into space and time. Here we expand on this QRF relativity of subsystems and elucidate that it is the source of all novel QRF dependent effects, just like the relativity of simultaneity is the origin of all characteristic special relativistic phenomena. We show that subsystem relativity, in fact, also arises in special relativity with internal frames and, by implying the relativity of simultaneity, constitutes a generalisation of it. Physical consequences of the QRF relativity of subsystems, which we explore here systematically, and the relativity of simultaneity may thus be seen in similar light. We focus on investigating when and how subsystem correlations and entropies, interactions and types of dynamics (open vs. closed), as well as quantum thermodynamical processes change under QRF transformations. We show that thermal equilibrium is generically QRF relative and find that, remarkably, QRF transformations not only can change a subsystem temperature, but even map positive into negative temperature states. We further examine how non-equilibrium notions of heat and work exchange, as well as entropy production and flow depend on the QRF. Along the way, we develop the first study of how reduced subsystem states transform under QRF changes. Focusing on physical insights, we restrict to ideal QRFs associated with finite abelian groups. Besides being conducive to rigour, the ensuing finite-dimensional setting is where quantum information-theoretic quantities and quantum thermodynamics are best developed. We anticipate, however, that our results extend qualitatively to more general groups and frames, and even to subsystems in gauge theory and gravity. [abridged]

Figures (12)

• Figure 1: Just like different inertial observers in special relativity decompose spacetime in different ways into space and time (relativity of simultaneity, left), different internal quantum reference frames (QRFs) partition the total system in different ways into subsystems (quantum relativity of subsystems, right). More precisely, two QRFs $R_1$ and $R_2$ decompose the total algebra $\mathcal{A}_{\text{phys}}$ of gauge-invariant (external-frame-independent) observables in different ways into subalgebras corresponding to a subsystem $S$ (their joint complement), and "the other frame". In fact, we demonstrate that the relativity of subsystems also arises for special relativity with internal frames (tetrads) and implies the relativity of simultaneity. The relativity of subsystems is thus a generalisation of the relativity of simultaneity. Just as the latter is the source of all the characteristic special relativistic effects and thereby can be viewed as the essence of special covariance, subsystem relativity is the source of all the QRF dependence of physical properties and thereby can similarly be regarded as the essence of QRF covariance. Given their intimate relation, their physical consequences should be seen in a similar light. While $R_1$ and $R_2$ describe $S$ by distinct invariant subalgebras (blue and red planes on the right), they will agree on the description of internal relational observables of $S$, which comprise the overlap of these subalgebras (green line on the right). This is analogous to how different Lorentz observers agree on scalars (and may agree on some spatial directions).
• Figure 2: Internal frame changes in special relativity and in the perspective-neutral approach to QRF covariance: gauge-fixing the frame-dressed observable $v_a$ to tetrads being in a certain orientation (e.g. aligning the background coordinates with them) is the analogue of reducing the quantum relational observable $O_{\mathds1_2\otimes f_S|R_1}^{g_1}$ into an internal QRF perspective (down pointing arrows in (a) and (b), respectively). Here, $f_S$ plays the analogue role of the bare $v_{\mu}$ in the coordinates adapted to the first tetrad and, in analogy with the gauge-fixing of the tetrads in (a), we set $g_1=g_2=e$ in (b). The QRF counterpart of the Lorentz coordinate change $\Lambda_{\mu'}{}^{\mu}$ is then the quantum coordinate transformation $\hat{V}_{1\to2}^{e,e}$ mapping the observables in $R_1$-perspective to the corresponding observables in $R_2$-perspective.
• Figure 3: Gauge-induced QRF transformation of Hilbert spaces (quantum coordinate change): $V_{1\to2}^{g_1,g_2}$ given in \ref{['eq:Vitoj']} maps from $\mathcal{H}_{2}\otimes\mathcal{H}_S$ in the perspective of QRF $R_1$ in orientation $g_1$ via the perspective-neutral Hilbert space $\mathcal{H}_{\text{phys}}$ to $\mathcal{H}_{1}\otimes\mathcal{H}_S$ in the perspective of $R_2$ in orientation $g_2$. For comparison with Fig. \ref{['Fig:jumpSR']}, \ref{['Fig:jumpQRF']}, we set $g_1=g_2=e$.
• Figure 4: Pictorial illustration of different notions of subsystems and why external (kinematical) and internal notions of correlations and thermal properties are distinct. (a) A composite subsystem $S$ (green balls) and two internal QRFs $R_1$ (blue ball) and $R_2$ (red ball) as seen from the perspective of an external (possibly fictitious) reference frame. (b) The kinematical notion of subsystem, i.e. the one relative to the external frame, is distinct from a relational one, i.e. one defined relative to an internal QRF. For example, the description of $S$ is left invariant under a reorientation of $R_1$ relative to the external frame, but changes relative to $R_1$ because the relations between $S$ and $R_1$ change. For this reason, the TPS on the kinematical Hilbert space $\mathcal{H}_{\rm kin}$ between $R_1,R_2$ and $S$ does not carry over to the external-frame-independent physical Hilbert space $\mathcal{H}_{\rm phys}\subsetneq\mathcal{H}_{\rm kin}$.
• Figure 5: Pictorial illustration of why internal notions of subsystem generally depend on the choice of internal QRF. (a) The description of $S$ relative to $R_1$ is invariant under reorientations of $R_2$, but relative to $R_2$ it changes since the relations between $S$ and $R_2$ change. For this reason, the internal perspectives of $R_1$ and $R_2$ induce inequivalent TPSs on the internal-frame-neutral $\mathcal{H}_{\rm phys}$ between $R_2,S$ and $R_1,S$, respectively. (b) Relational observables describing $S$ relative to $R_1$ are invariant under reorientations of $R_2$ and vice versa. Hence, relational observables describing $S$ relative to both$R_1$ and $R_2$, i.e. residing in $\mathcal{A}^{\rm phys}_{S|R_1}\cap\mathcal{A}^{\rm phys}_{S|R_2}$, must be invariant under reorientations of both frames. These describe $\mathcal{G}$-invariant properties of $S$ that are independent of the relations between $S$ and both internal frames and correspond to all the relational observables encoding internal relations within $S$ (e.g., we could have chosen another internal QRF $R_3$ inside $S$). If $S$ is sufficiently complex, this subalgebra may be large, but is still a proper subalgebra of both $\mathcal{A}^{\rm phys}_{S|R_i}$, $i=1,2$, which also contain observables that depend non-trivially on the relation between $S$ and $R_i$. This is the content of Theorem \ref{['lem_TPSineqalg']}.

Theorems & Definitions (90)

• Definition 1: TPS
• Lemma 1
• Theorem 1
• Corollary 1.1
• Lemma 2
• Lemma 3
• Lemma 4
• Example 1
• Theorem 2: $\mathbf{S}$-operators
• Theorem 3: Frame-operators