Thermodynamic Constraints on the Emergence of Intersubjectivity in Quantum Systems

Abstract

Ideal quantum measurement requires divergent thermodynamic resources. This is a consequence of the third law of thermodynamics, which prohibits the preparation of the measurement pointer in a fully erased, pure state required for the acquisition of perfect, noiseless measurement information. In this work, we investigate the consequences of finite resources in the emergence of intersubjectivity as a model for measurement processes with multiple observers. Here, intersubjectivity refers to a condition in which observers agree on the observed outcome (agreement), and their local random variables exactly reproduce the original random variable for the system observable (probability reproducibility). While agreement and reproducibility are mutually implied in the case of ideal measurement, finite thermodynamic resources constrain each of them. Starting from the third law of thermodynamics, we derive how the achievability of ideal intersubjectivity is affected by restricted thermodynamic resources. Specifically, we establish a no-go theorem concerning perfect intersubjectivity and present a deviation metric to account for the influence of limited resources. We further present attainable bounds for the agreement and bias that are exclusively dependent on the initial state of the environment. In addition, we show that either by cooling or coarse-graining, we can approximate ideal intersubjectivity even with finite resources. This work bridges quantum thermodynamics and the emergence of classicality in the form of intersubjectivity.

Figures (5)

• Figure 1: Multiple macroscopic observers $\mathcal{P}_j$ monitor a central quantum system $S$ in order to extract information about a specific central observable. The presence of a thermal environment at temperature $T$ hinders the emergence of ideal intersubjectivity.
• Figure 2: Representation of coarse-graining: The environments are grouped into larger and larger macrofractions of size ${l^{(x)}_{\text{cg}}}$, from which information is extracted.
• Figure 3: In this figure we plot the exponential decay of $1-a(0\vert{l_{\text{cg}}})$ to extend the result of Theorem \ref{['th:idealintcg']}. Top panel: Logarithmic plot of $1-a(0\vert{l_{\text{cg}}})$ as a function of ${l_{\text{cg}}}$. Stars correspond to values evaluated with the definition in App. \ref{['app:coarse-grain-proof-dslarger']}; Dashed lines correspond to fit with $c_0 e^{c_1 {l_{\text{cg}}}}$. Table: Results of the fit. We assess the efficiency of the fit with the coefficient of determination $R^2$, see App. \ref{['app:coarse-grain-proof-dslarger']} for details. We considered initial probability $\bm{a}_{d_{\space\raisebox{0pt}{\tiny{$S$}}}} = \{a(0) = 1/d_{\space\raisebox{0pt}{\tiny{$S$}}} + 0.1, a(1)=..=a(d_{\space\raisebox{0pt}{\tiny{$S$}}}-1) = 1/d_{\space\raisebox{0pt}{\tiny{$S$}}}-0.1/(d_{\space\raisebox{0pt}{\tiny{$S$}}}-1) \}$, which is close to a maximally mixed initial configuration. The fit is performed for ${l_{\text{cg}}} > {l_{\text{cg}}}^{\text{min}}$ to demonstrate the asymptotic behaviour. In practice, we excluded the first point ${l_{\text{cg}}}=1$.
• Figure 4: We report minimum disagreement $\textup{Dis}(U_t)$ and bias $\textup{Bias}(U_t)$ on the left and right canvas, respectively, for a pure dephasing dynamics generated by a star shape spin interaction, see Eq. \ref{['eq:Hspin']}. The quantity are evaluated for initial state $\vert \psi_{0}\rangle = \sqrt{p_{0}} \vert 0 \rangle + \sqrt{1-p_{0}} \vert 1 \rangle$ with $p_{0}=0.2$, while the environments are in a thermal state with inverse temperature $\beta=1$ and the total number of environmental qubits is $N_{\space\raisebox{0pt}{\tiny{$P$}}} = 1024$. We evaluate the minimum of the two considered quantities over the time interval given as $t \in[0,6]$, and compare them with the corresponding coarse-grained version of the bound, namely $\delta^{(N_{\space\raisebox{0pt}{\tiny{$P$}}})}_{\bm{a},{l_{\text{cg}}}}$ and $\beta_{\bm{a},{l_{\text{cg}}}}^{(N_{\space\raisebox{0pt}{\tiny{$P$}}})}$. We observe a similar behaviour in both the bound and pure dephasing models, with the only distinction being that the decrease is slower in the pure dephasing case. Nevertheless, it can still be characterized as an exponential decay. Note that the axis uses an exponential scaling in the ${l_{\text{cg}}}$. For ${l_{\text{cg}}} = 64$, the bound for $\delta^{(N_{\space\raisebox{0pt}{\tiny{$P$}}})}_{\bm{a},{l_{\text{cg}}}}$ was too small to be represented in the plot.
• Figure 5: Plot of agreement and bias as a function of rescaled time $tg$, for an initial state of the system $\vert \psi_{0}\rangle = \sqrt{p_{0}} \vert 0 \rangle \langle + \sqrt{1-p_{0}} \vert 1 \rangle$ with $p_{0}=0.2$, while the environments are in a thermal state with inverse temperature $\beta=1$. Different colours correspond to different coarse-grainings ${l_{\text{cg}}}$. Here, $N_{\space\raisebox{0pt}{\tiny{$P$}}}=1024$ spins. The time scale $[0, \pi/(2g)]$, where the relevant dynamics occur, corresponds to half of the recurrence time $t_{\textup{r}} = \pi/g$.

Theorems & Definitions (5)

• Definition 1: Biased joint information broadcasting (BJIB)
• Theorem 1: Maximum agreement
• Theorem 2: Bias for optimal BJIB
• Theorem 3: Approaching ideal intersubjectivity with coarse-graining
• Lemma 1