Quantitative bounds to propagation of quantum correlations in many-body systems

Abstract

We investigate how much information about a quantum system can be simultaneously communicated to independent observers, by establishing quantitative limits to bipartite quantum correlations in many-body systems. As recently reported in Phys. Rev. Lett. 129, 010401 (2022), bounds on quantum discord and entanglement of formation between a single quantum system and its environment, e.g., a large number of photons, dictate that independent observers which monitor environment fragments inevitably acquire only classical information about the system. Here, we corroborate and generalize those findings. First, we calculate continuity bounds of quantum discord, which establish how much states with a small amount of quantum correlations deviate from being embeddings of classical probability distributions. Also, we demonstrate a universally valid upper bound to the bipartite entanglement of formation between an arbitrary pair of components of a many-body quantum system. The results confirm that proliferation of classical information in the Universe suppresses quantum correlations.

Figures (3)

• Figure 1: We consider a quantum Universe in which a system ${\cal S}$ interacts with an $N$-particle environment ${\cal E}$. We investigate fundamental bounds to bipartite quantum correlations, as quantified by quantum discord and the entanglement of formation, between ${\cal S}$ and an environment fragment ${\cal F}_k$.
• Figure 2: There exist upper limits to bipartite quantum correlations between a system ${\cal S}$ and the environment subsystems $\varepsilon_i$s of ${\cal E}$. Proliferation of classical information, as quantified by the amount of consensus about ${\cal S}$ that can be reached by observers eavesdropping on $\varepsilon_i$s, inhibits quantum discord and entanglement.
• Figure 3: We show the bound to the average entanglement of formation (Eq. (\ref{['entbound']})) in action. The following quantities are computed in the final state ${\bf U}_{{\cal SE}}(a)|+\rangle_{\cal S}|0\rangle_{\cal E}^{\otimes N}$, for different values of $N$ (see the main text for full details): the average entanglement of formation, $\bar{E}\left(\rho_{{\cal S}\varepsilon_{i}}\right)$ (blue line ---); the upper bound in Eq. (\ref{['entbound']}) (dashed blue line $\mathbin{\vcenter{\mathbin{\vcenter{\mathbin{\vcenter{$); the known entropic upper bound $\min\left\{H(\rho_{{\cal S}}),H(\rho_{\varepsilon_{i}})\right\}$ (dotdashed bordeaux line $\mathbin{\vcenter{\cdot\mathbin{\vcenter{\cdot\mathbin{\vcenter{$); the average classical correlations $\bar{J}\left(\rho_{{\cal S}\check{\varepsilon}_{i}}\right)$ (black line ---). The average values of classical and quantum correlations are the ones computed for an arbitrary pair ${\cal S}\varepsilon_i$, because of the symmetry under environment subsystem permutations of the final state of the Universe. The newfound bound is much more informative than the known entropic bound in the limit $a\rightarrow 0$, when the global state of the Universe is highly entangled. The study confirms that proliferation of a larger amount of classical correlations suppresses quantum correlations. Comparing these results with Fig. 3 of Ref.red, we note that the entanglement of formation declines by increasing $N$ much faster than quantum discord. We briefly discuss why the index $\delta_i$ is a good measure of the (lack of) consensus between two observers monitoring $\varepsilon_i$ and ${\cal E}_{/i}$, respectively. Assume $J\left(\rho_{{\cal S}\check{{\cal E}}_{/i}}\right)\geq J\left(\rho_{{\cal S}\check{\varepsilon}_{i}}\right)$. If $\delta_i=0$, then $J\left(\rho_{{\cal S}\check{{\cal E}} }\right)=J\left(\rho_{{\cal S}\check{{\cal E}}_{/i}}\right)=J\left(\rho_{{\cal S}\check{\varepsilon}_{i}}\right)$. The reverse implication is also true. Hence, the parameter $\delta_i$ is zero if and only if the same classical information about ${\cal S}$ is simultaneously available into $\varepsilon_i$ and ${\cal E}_{/i}$. That is, if and only if observers measuring on the two environment fragments are in perfect agreement. Further, if $\delta_i=1$, then $J\left(\rho_{{\cal S}\check{\varepsilon}_i}\right)=0$, and there is maximal disagreement between the observers. The reverse statement holds too. Introducing a measure of (lack of) objectivity about classical information is instrumental in proving a bound to bipartite quantum discord in many-body systems for any pure state of the universe $|\psi\rangle_{{\cal SE}}$. The result can be demonstrated as follows. First, one observes that I(\rho_{{\cal S}\varepsilon_{i}})+ I(\rho_{{\cal S}{\cal E}_{/i}})= 2\, H(\rho_{{\cal S}}),\,\forall\,i\RightarrowI(\rho_{{\cal S}\varepsilon_{i}})+ I(\rho_{{\cal S}{\cal F}_{k}})\leq 2\,H(\rho_{{\cal S}}),\,\forall\, i,\,\forall\, {\cal F}_k\subseteq {\cal E}_{/i}\Rightarrow\bar{I}(\rho_{{\cal S}{\cal E}_{/i}}):=\frac{1}{N} \sum_{i=1}^N I(\rho_{{\cal S}{\cal E}_{/i}})\geq H(\rho_{{\cal S}}).This preliminary result implies an upper bound to quantum discord when the state of the Universe is pure $\left(H(\rho_{{\cal S}})=J\left(\rho_{{\cal S}\check{\varepsilon}_i}\right)\right)$: I(\rho_{{\cal S}\varepsilon_{i}})= 2\, H(\rho_{{\cal S}})- I(\rho_{{\cal S}{\cal E}_{/i}})\RightarrowD(\rho_{{\cal S}\check{\varepsilon}_i})= 2\, H(\rho_{{\cal S}})- J(\rho_{{\cal S}\check{\varepsilon}_i})-I(\rho_{{\cal S}{\cal E}_{/i}})D(\rho_{{\cal S}\check{\varepsilon}_i})\leq 2\, H(\rho_{{\cal S}})+(\delta_i-1)\, H(\rho_{{\cal S}})-I(\rho_{{\cal S}{\cal E}_{/i}})\leq (\delta_i+1)\,H(\rho_{{\cal S}})-I(\rho_{{\cal S}{\cal E}_{/i}}).Therefore, there exists a related bound to the average quantum discord between central system and environment subsystem: \bar{D}(\rho_{{\cal S}\check{\varepsilon}_i}):= \frac{1}{N}\sum_{i=1}^N D(\rho_{{\cal S}\check{\varepsilon}_i}),\bar{D}(\rho_{{\cal S}\check{\varepsilon}_i})\leq \frac{1}{N}\sum_{i=1}^N\left\{(\delta_i +1)\,H(\rho_{{\cal S}})-I(\rho_{{\cal S}{\cal E}_{/i}})\right\}\leq \, (\delta +1)\,H(\rho_{{\cal S}})-\bar{I}(\rho_{{\cal S}{\cal E}_{/i}})\Rightarrow\bar{D}\left(\rho_{{\cal S}\check\varepsilon_{i}}\right)\leq\, \delta\,H(\rho_{{\cal S}}).Hence, consensus about classical information, i.e., the emergence of classical objectivity about properties of ${\cal S}$ by indirect observation (intercepting fragments of the environment), suppresses quantum correlations. An equivalent bound holds for the entanglement of formation. By employing the Koashi-Winter inequality in Eq. (\ref{['koa']}), and reminding the definition in Eq. (\ref{['delta']}), a few algebra steps show that, for pure states of the universe, one has E(\rho_{{\cal S}\varepsilon_i})=\, H(\rho_{{\cal S}})-J\left(\rho_{{\cal S}\check{\cal E}_{/i}}\right)\leq\, H(\rho_{{\cal S}}) -\min\left\{J\left(\rho_{{\cal S}\check{\varepsilon}_i}\right),J\left(\rho_{{\cal S}\check{\cal E}_{/i}}\right)\right\}\leq\, \delta_i\,H(\rho_{{\cal S}})\Rightarrow\bar{E}\left(\rho_{{\cal S}\varepsilon_{i}}\right):=\frac{1}{N}\sum_{i=1}^N E\left(\rho_{{\cal S}\varepsilon_{i}}\right)\leq\, \delta\,H(\rho_{{\cal S}}).We elucidate the meaning of the bound with a numerical study. Consider the quantum Universe to be in the initial uncorrelated state $|+\rangle_{{\cal S}} |0\rangle^{\otimes N}_{{\cal E}}$. Then, one applies the unitary ${\bf U}_{{\cal SE}}(a)\equiv\Pi_{i=1}^N {\bf U}_{{\cal S}\varepsilon_i}(a)$, where the two-site transformation ${\bf U}_{{\cal S}\varepsilon_i}(a)$ is the "c-maybe" gate on ${\cal S}\varepsilon_i$akram: $\left(1000010000a\sqrt{1-a^2}00\sqrt{1-a^2}-a\right), \,\,a\in[0,1].$This algorithm models the interaction of a quantum system ${\cal S}$ with a large photonic environment photon2darwin2. We calculate the local average bipartite classical correlations $\bar{J}\left(\rho_{{\cal S}\check{\varepsilon}_{i}}\right)$ and the average entanglement of formation $\bar{E}\left(\rho_{{\cal S}\varepsilon_{i}}\right)$ in the final state ${\bf U}_{{\cal SE}}(a)|+\rangle_{\cal S}|0\rangle_{\cal E}^{\otimes N}$. Their values can be computed analytically ranktwoakram. Also, we calculate the newfound bound Eq. (\ref{['entbound']}) and compare it against the known upper limit to the entanglement of formation, given by the smallest between the marginal entropies $H(\rho_{{\cal S}}), H(\rho_{\varepsilon_i})$. The results are plotted in FIG. \ref{['fig2']}. They show how the entanglement of formation obeys a "weak monogamy relation" dictated by the abundance of classical information about ${\cal S}$ simultaneously available throughout the environment. For $a\rightarrow 0$, the universe comes close to be in a (generalized) GHZ state and such a behaviour is magnified: quantum correlations in partitions ${\cal S}\varepsilon_i$ vanish, while classical information proliferation is maximized. This simple yet instructive model also shows the usefulness of the newfound bound of Eq. (\ref{['entbound']}). Indeed, it captures how local quantum correlations $E(\rho_{{\cal S}\varepsilon_i})$ are monotonically suppressed by increasing the strength of the interaction between system and environment ($a\rightarrow 0$), while the known entropic limit monotonically increases. Also, it signals that quantum correlations (in fact, both quantum discord and entanglement) cannot exist without classical ones, as we will demonstrate in the next Section. Here, we derive new results that show how bipartite quantum correlations are restricted in many-body systems. We observe that the findings of the previous section imply that, if $\delta=0$, and therefore $J\left(\rho_{{\cal S}\check{\varepsilon}_{i}}\right)=H(\rho_{\cal S}),\,\forall\,i$, then there is no environment fragment that can share quantum discord with ${\cal S}$. We prove a statement about the degenerate case of this scenario: all the subsystems store the very same amount of classical information about ${\cal S}$, but its value is zero, i.e., no classical correlations exist. Remark:There are not quantum correlations without classical correlations:J\left(\rho_{{\cal S}\check{\varepsilon}_{i}}\right)=0\Rightarrow D\left(\rho_{{\cal S}\check{\varepsilon}_{i}}\right) =0.Proof -- This claim can be proved in several ways. For example, from the Koashi-Winter relation, it follows that $J\left(\rho_{{\cal S}\check{\varepsilon}_{i}}\right)=0\Rightarrow E(\rho_{{\cal SE}_{/i}})=D\left(\rho_{{\cal S}\check{\cal E}_{/i}}\right)= H({\cal S})$. Since $E(\rho_{{\cal SE}_{/i}})+E\left(\rho_{{\cal S}{\varepsilon}_{i}}\right)= D\left(\rho_{{\cal S}\check{\varepsilon}_{i}}\right) + D(\rho_{{\cal S}\check{\cal E}_{/i}})$fanchini, one has $D\left(\rho_{{\cal S}\check{\varepsilon}_{i}}\right) =0$. Next, we explore more nuanced aspects of the transition from quantum to classical regimes. We ask whether quantum discord is "continuous", in the sense of taking small values for states that are geometrically close (and physically similar) to classically correlated density matrices. The bound in Eq. (\ref{['main']}) establishes that simultaneous maximal classical correlations between ${\cal S}$ and each $\varepsilon_i$ destroy quantum discord throughout the Universe. Hence, quantum information about ${\cal S}$ is not accessible to independent observers that monitor different $\varepsilon_i$. Proving that quantum discord is subject to sharp continuity bounds at the frontier with classical states would mean that, whenever a classical description of the correlation pattern is sufficiently precise, quantum correlations are inevitably negligible. That is, classical objectivity and a significant amount of quantum correlations cannot co-exist. It is known that $D\left(\rho_{{\cal S}\check{\cal F}_k}\right)=0$ if and only if there exists a measurement $\mathbf{M}_k$ such that $\rho_{{\cal SF}_{k}}=\rho_{{\cal SF}_{k,\mathbf{M}_k}}$. We here prove continuity bounds to quantum discord about the zero value. First we show that if a state of a partition ${\cal SF}_k$ (which we assume to be a full rank density matrix) is close to the set of post-measurement states $\rho_{{\cal SF}_{k, \mathbf{M}_k}}$, then its discord is small. Given the subset of the projective measurements $\{\mathbf{P}_k\}\subset \{\mathbf{M}_k\}$ which can be performed on ${\cal F}_k$, recalling the definition of relative entropy $H\left(\rho_{\cal X}||\rho_{{\cal Y}}\right):=\text{Tr}\{\rho_{\cal X}\log_2\rho_{\cal X}\}-\text{Tr}\{\rho_{\cal X}\log_2\rho_{\cal Y}\}$, one has D\left(\rho_{{\cal S}\check{\cal F}_k}\right)\leq \min_{\mathbf{P}_k}\left\{ I\left(\rho_{{\cal S}{\cal F}_k}\right)-J\left(\rho_{{\cal S}\check{\cal F}_k}\right)\right\}= \min_{\mathbf{P}_k}\left\{H\left(\rho_{{\cal S}{\cal F}_k}||\rho_{{\cal S}}\otimes \rho_{{\cal F}_k}\right) -H\left(\rho_{{\cal S}{\cal F}_{k,\mathbf{P}_k}}||\rho_{{\cal S}}\otimes \rho_{{\cal F}_{k,\mathbf{P}_k}}\right) \right\}= \min_{\mathbf{P}_k}\left\{ H\left(\rho_{{\cal S}{\cal F}_k}||\rho_{{\cal S}{\cal F}_{k,\mathbf{P}_k}}\right) -H\left(\rho_{{\cal F}_k}||\rho_{{\cal F}_{k,\mathbf{P}_k}}\right) \right\}\leq \min_{\mathbf{P}_k}\left\{ H\left(\rho_{{\cal S}{\cal F}_k}||\rho_{{\cal SF}_{k,\mathbf{P}_k}}\right) \right\}.Finally, we obtain $\min_{\mathbf{P}_k}\left\{ H\left(\rho_{{\cal S}{\cal F}_k}||\rho_{{\cal SF}_{k,\mathbf{P}_k}}\right) \right\} \leq \epsilon \Rightarrow D\left(\rho_{{\cal S}\check{\cal F}_k}\right) \leq \epsilon, \forall\,\epsilon.$Therefore, states that are geometrically close ($\epsilon\rightarrow 0$) to be embeddings of classical probability distributions (classical-quantum states) display small values of quantum discord. For the sake of completeness, we calculate the maximal relative entropy between a state and the closest classically correlated state when an upper bound to quantum discord, which we obtain by maximizing in Eq. (\ref{['class']}) over projective measurements rather than POVMs, takes arbitrary small values. As a preliminary step, we recall an upper limit to the relative entropy between two arbitrary states jens: H\left(\rho_{{\cal X}}||\rho_{{\cal Y}} \right)\leq (\lambda_{min}(\rho_{{\cal Y}})+d_{{\cal X},{\cal Y}})\, \log\left(1+\frac{d_{{\cal X},{\cal Y}}}{\lambda_{min}(\rho_{{\cal Y}})}\right)-\lambda_{min}(\rho_{{\cal X}})\,\log\left(1+\frac{d_{{\cal X},{\cal Y}}}{\lambda_{min}(\rho_{{\cal X}})}\right),d_{{\cal X},{\cal Y}}\equiv||\rho_{{\cal X}}-\rho_{{\cal Y}}||_1/2,in which $\lambda_{min}(\rho_{{\cal X}})$ is the smallest eigenvalue of $\rho_{{\cal X}}$. Then, calling $\tilde{\mathbf{P}}_k$ the projective measurement performed on ${\cal F}_k$ that maximizes the post-measurement mutual information (see Eq. (\ref{['class']})), we obtain H\left(\rho_{{\cal S}{\cal F}_k}||\rho_{{\cal S}{\cal F}_{k,\tilde{\mathbf{P}}_k}}\right) -H\left(\rho_{{\cal F}_k}||\rho_{{\cal F}_{k,\tilde{\mathbf{P}}_k}}\right)\leq \epsilon\RightarrowH\left(\rho_{{\cal S}{\cal F}_k}||\rho_{{\cal S}{\cal F}_{k,\tilde{\mathbf{P}}_k}}\right)\leq \epsilon +H\left(\rho_{{\cal F}_k}||\rho_{{\cal F}_{k,\tilde{\mathbf{P}}_k}}\right)H\left(\rho_{{\cal S}{\cal F}_k}||\rho_{{\cal S}{\cal F}_{k,\tilde{\mathbf{P}}_k}}\right)\leq \epsilon + f\left(\rho_{{\cal F}_k}, \tilde{\mathbf{P}}_k\right),where f\left(\rho_{{\cal F}_k}, \tilde{\mathbf{P}}_k\right) =\left\{\lambda_{min}\left(\rho_{{\cal F}_{k,\tilde{\mathbf{P}}_k}}\right)+d_{{\cal F}_k,{\cal F}_{k,\tilde{\mathbf{P}}_k}}\right\}\, \log\left\{1+\frac{d_{{\cal F}_k,{\cal F}_{k,\tilde{\mathbf{P}}_k}}}{\lambda_{min}\left(\rho_{{\cal F}_{k,\tilde{\mathbf{P}}_k}}\right)}\right\}-\lambda_{min}\left(\rho_{{\cal F}_k}\right)\,\log\left(1+\frac{d_{{\cal F}_k,{\cal F}_{k,\tilde{\mathbf{P}}_k}}}{\lambda_{min}(\rho_{{\cal F}_k})}\right).This constraint is certainly less neat than Eq. (\ref{['cont1']}) for generic mixed states. We leave to future studies to shape this claim, as we conjecture that a cleaner continuity bound may exist.We now generalize the bound to the bipartite entanglement in a star-like configuration (Eq. (\ref{['entbound']})). We focus on the correlation structure of the environment ${\cal E}$, which is a generic $N$-partite quantum system. We show that there is an upper bound to the bipartite entanglement of formation between two components of the environment, in terms of how much classical information is shared by the environment parts. We define a new disagreement quantifier: \delta^\varepsilon_i:=1- \frac{\min\limits_{\varepsilon_j} J\left(\rho_{\varepsilon_i\check\varepsilon_j}\right)}{H(\rho_{\varepsilon_i})}\, \in\,[0,1].The quantity manifestly enjoys the same properties of the parameter introduced in Eq. (\ref{['delta']}). Then, the entanglement of formation between an environment subsystem $\varepsilon_i$ and any other subsystem is limited by the (lack of) consensus about measurement outcomes (classical information) on $\varepsilon_i$ across the environment. By employing again the Koashi-Winter relation, one has J\left(\rho_{\varepsilon_i\check\varepsilon_j}\right)\geq\, (1-\delta^\varepsilon_i)\,H(\rho_{\varepsilon_i}),\,\forall\,i,j\RightarrowE\left(\rho_{\varepsilon_{i}{\cal E}_{/ij}}\right)\leq\, \delta^\varepsilon_i\,H(\rho_{\varepsilon_i}),\,\forall\,i,j \RightarrowE\left(\rho_{\varepsilon_i\varepsilon_j}\right)\leq\, \delta^\epsilon_i\,H(\rho_{\varepsilon_i}),\,\forall\, i,j.The bound is clearly saturated, for example, for the GHZ state.We have investigated quantitative limits to the propagation of quantum information in many-body systems. Specifically, we have extended the results of red, calculating a continuity bound to quantum discord nearby classical states (Eq. (\ref{['cont1']})), and proving an upper bound to the entanglement of formation (Eq. (\ref{['entbound2']})) between two arbitrary components of a multipartite system. Classical correlations are not subject to any limitations. Consequently, classical information can be freely broadcast from a source to an arbitrary number of receivers. Yet, the very same possibility that observers reach consensus on such classical information of target physical systems dictates bounds to quantum information, which are here formulated in terms of limits to quantum discord and the entanglement of formation. The results further corroborate the key ideas of Quantum Darwinism, a theoretical framework that explain the emergence of classical reality within quantum mechanics. We hope these findings will propel further studies on the subtleties of the transition between the quantum and classical regimes, which may lead to derive stronger bounds than the ones here demonstrated. 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