Collective purification of interacting quantum networks beyond symmetry constraints

Abstract

Following any quantum information processing protocol, it is essential to reset a mixed state of a many-body interacting spin-network to the computational-zero pure state. This task is challenging, both theoretically and experimentally, because of the quantum correlations. There is currently no effective cooling strategy for both high and low temperatures in such networks. Here we put forth a universal cooling strategy for multi-spin interacting networks. The strategy is based on the collective coupling of the system to an ancilla spin that intermittently dumps part of its entropy into an ultracold bath. Yet this strategy should overcome the symmetry-imposed correlations that impede the cooling. To avoid the prohibitive complexity of computing the dynamics, we resort to graph analysis of the network. %To approach the desired state, We show that a unique choice of alternating, non-commuting system-ancilla interaction Hamiltonians exists that breaks the symmetry constraints and allows the network to approach the desired pure state. We illustrate this universal purification strategy in diverse experimental settings.

Figures (10)

• Figure 1: Purification Protocol: Schematic diagram of the purification protocol: Interacting spin network cooling/purification via collective swapping of the network ($S$) entropy with an ancilla qubit ($A$) in recurring cycles. The ancilla is intermittently reset/purified by an ultracold (ideally, zero-temperature) bath $(B)$.
• Figure 2: Angular-momentum constraints on purification: Angular-momentum symmetry constraints on purification in the star or the equivalent isotropic Heisenberg-chain model: (a) A Scheme that depicts a system $S$ of either isolated or identically-coupled spins that are purified via the ancilla spin $A$ which is repeatedly reset by the cold bath $B$. (b) Block purification scheme of $S$ in angular momentum $j-m$ basis. In the initial fully mixed state, all $j-m$ states in $S$ have equal probabilities, thus all blocks have the same color. The goal is to concentrate the probabilities at the FGS state via $S-A$ resonant transfer (RT). The RT excitation exchange proceeds vertically, from $m$ to $m+1$ in each $j$ block, and is thus confined to conserved-$j$ subspaces (creating symmetry bottlenecks) for purification. (c) Polarizability as a function of the number of spins $N$ under angular-momentum symmetry constraints according to Eq. \ref{['eqnnP']}.
• Figure 3: Automorphism constraints on purification: Polarizable and unpolarizable networks under graph automorphism constraints: (a) Polarizability (✔) or non-polarizability (✗) of some representative networks (i)-(iv) via collective entropy swapping with a probe (ancilla) spin $A$ that is intermittently coupled to a cold bath. Network polarizability is obtained by graph-theoretic considerations regarding their automorphism orbits (AO). Nodes that belong to the same AO, are colored with the same color in the graph, whereas different colors divide the nodes into topologically equivalent sets. Visual inspection of network (i)-(iv) suffices to determine their polarizability bounds. (b) Numerically calculated network purity $\text{Tr~}\rho^2_S$ as a function of the number of ancilla-resets for the networks (i)-(iv) shown in (a). The calculations confirm our prediction that full purification (polarization) is only achievable for networks with non-degenerate automorphism orbits (AO). (c) left- A polarizable network $N$ coupled to an ancilla $A$ represented by an identity (open-chain) graph for which the rank is equal to the dimensionality ($\mathscr{R}(\mathbcal{M}) = \mathscr{D}(\mathbcal{M})$), right- an unpolarizable network for which $\mathscr{R}(\mathbcal{M}) < \mathscr{D}(\mathbcal{M})$. (d) Estimated purity versus spin number $N$ for open-chain graphs and complete graphs. Complete graphs (red dotted line) have maximal $\mathscr{N}(\mathbcal{M})$ (Eq. (\ref{['eq:nullity']})) and hence the lowest polarizability. (e) Same as (b) for network (i) with different anisotropy $\Delta$ parameters.
• Figure 4: Spectral symmetry constraints on purification: Networks where spectral symmetry (SPS) hinders full polarization. (a) The nodes where the support of the eigenvector(s) corresponding to the null subspace is zero, are marked with $0$. (b) SPS effects in diverse graph classes: (i) Path graph $P_5$ ($N=5$), (ii) complete bipartite graph $K_{3,3}$ ($N=6$), (iii) an identity graph ($N=5$), (iv) a graph with a non-identity nontrivial AO (mirror symmetry) ($N=6$). Nodes marked with 'O' denote null support of the kernel. (c) Numerically calculated time dependence of network purity $\text{Tr~}\rho^2_S$ as a function of the number of ancilla-resets for the networks. The calculations confirm that full purification (polarization) is only achievable for spin networks with non-degenerate automorphism orbits (AO) that also lack SPS.
• Figure 5: Purification using ADRT protocol: (a) Schematic representation of the ADRT purification protocol for a star model: a system $S$ of isolated spins via the ancilla spin $A$, showing its overwhelming ability to overcome symmetry constraints/bottlenecks compared to RT in Fig. \ref{['fig:DRT11']}. In the ADRT protocol, the excitation exchange takes place both horizontally and vertically (i.e., along $m$ and $j$), thus mixing all $j$-blocks. This allows us to achieve the desired final state. (b) The variation of the network purity with the number of cycles for the isolated spin model and the Heisenberg chain of $5$ spins with different anisotropy parameters $\Delta$. (c) The variation of the network purity with the number of cycles for the non-polarizable graph (i) shown in Fig. \ref{['fig:2']}(a) with different anisotropy parameters $\Delta$. Both plots (b) and (c) show that the desired state is attained using the ADRT protocol, unlike the RT protocol used in Fig. \ref{['fig:2']}(b).

Theorems & Definitions (1)

• Theorem 6