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We investigate a quantum dynamical phase transition induced by the competition between local unitary evolution and dissipation in a qubit chain with a strong, on-site $\mathbb{Z}_2$ symmetry. While the steady-state of this evolution is always maximally-mixed, we show that the dynamical behavior of certain non-local observables on the approach to this steady-state is dictated by a quantum Ising model with a $\textit{complex}$ transverse-field (cTFIM). We investigate these observables analytically, uncovering a dynamical phase transition as the relative rate of unitary evolution and dissipation is tuned. We show that the weak-dissipation limit corresponds to a cTFIM with a large magnitude of the imaginary transverse-field, for which the many-body "ground-state" (with smallest real eigenvalue) is gapless, exhibiting quasi-long-range correlations of the local magnetization with a continuously-varying exponent. Correspondingly, the dynamics of the non-local observables show oscillatory behavior with an amplitude decaying exponentially in time. The strong-dissipation limit corresponds to a gapped ferromagnetic phase of the cTFIM, and non-local observables show exponential decay on the approach to equilibrium. This transition in (1+1)-dimensions has a peculiar, "two-sided" nature appearing as either first- or second-order depending on the phase from which the transition is approached, an analytic result which is corroborated by numerical studies. In higher dimensions, we present a field-theoretic understanding of the first-order nature of this transition, when approaching from the ferromagnetic phase of the cTFIM, though the nature of the phase with large imaginary transverse-field remains to be understood., Comment: 15 + 14 pages, 8 figures

We study in detail the postselection problem in a specific model: bosons hopping on a lattice subjected to continuous local measurements of quadrature observables. We solve the model analytically and show that the postselection overhead can be reduced by postprocessing the entire measurement record into one or two numbers for each trajectory and then postselecting based only on these numbers. We then provide a step-by-step protocol designed to recover connected two-point functions of the quantum trajectories, which display an exponentially decaying profile that is not observable in the unconditional, trajectory averaged, state. With the analytical solution in hand, we analyse the features of this postprocessing stage with the intention of abstracting away the properties that make postselection feasible in this model and may help in mitigating postselection in more general settings. We also test the protocol numerically in a way that utilizes only experimentally accessible information, showing that various quantum trajectory observables can be recovered with a few repetitions of the numerical experiment, even after including inevitable coarse-graining procedures expected under realistic experimental conditions. Furthermore, all the information required to design the postprocessing stage is independently present both in the unconditional dynamics and also in the measurement record, thus bypassing the need to solve for the conditional evolution of the model. We finalize by providing experimental implementations of these models in cavity-QED and circuit-QED., Comment: 16 + 8 pages, 11 figures

We demonstrate that, starting with a simple fermion wave function, the steady mixed state of the evolution of a class of Lindbladians, and the ensemble created by strong local measurement of fermion density without post-selection can be mapped to the "Gutzwiller projected" wave functions in the doubled Hilbert space -- the representation of the density matrix through the Choi-Jamiolkowski isomorphism. A Gutzwiller projection is a broadly used approach of constructing spin liquid states. For example, if one starts with a gapless free Dirac fermion pure quantum state, the constructed mixed state corresponds to an algebraic spin liquid in the doubled Hilbert space. We also predict that for some initial fermion wave function, the mixed state created following the procedure described above is expected to have a spontaneous "strong-to-weak" U(1) symmetry breaking, which corresponds to the emergence of superconductivity in the doubled Hilbert space. We also design the experimental protocol to construct the desired physics of mixed states., Comment: 11 pages, 2 figures, more results and new authors added

Measurements can detect errors in a decohered quantum memory allowing active error correction to increase the memory time. Previous understanding of this mechanism has focused on evaluating the performance of error correction algorithms based on measurement results. In this work, we instead intrinsically characterize the information dynamics in a quantum memory under repeated measurements, using coherent information and relative entropy. We consider the dynamics of a $d$-dimensional stabilizer code subject to Pauli errors and noisy stabilizer measurements and develop a $(d+1)$-dimensional statistical mechanics model for the information-theoretic diagnostics. Our model is dual to the model previously obtained for the optimal decoding algorithm, and the potential decoding transition in the quantum memory again manifests as a thermal phase transition in the statistical mechanics model. We explicitly derive the model and study the phase transition in information encoding in three examples: surface codes, repetition codes, and the XZZX code., Comment: 27 pages, 9 figures

Quantum systems subject to random unitary evolution and measurements at random points in spacetime exhibit entanglement phase transitions which depend on the frequency of these measurements. Past work has experimentally observed entanglement phase transitions on near-term quantum computers, but the characterization approach using entanglement entropy is not scalable due to exponential overhead of quantum state tomography and post selection. Recently, an alternative protocol to detect entanglement phase transitions using linear cross-entropy was proposed which eliminates both bottlenecks. Here, we report the demonstration of this protocol in systems with one-dimensional and all-to-all connectivities on IBM's quantum hardware on up to 22 qubits, a regime which is presently inaccessible if post-selection is required. We demonstrate a collapse of the data into a scale-invariant form with critical exponents agreeing with theory within uncertainty. Our demonstration paves the way for studies of measurement-induced entanglement phase transitions and associated critical phenomena on larger near-term quantum systems., Comment: 23 pages, 15 figures

The linear cross-entropy (LXE) has been recently proposed as a scalable probe of the measurement-driven phase transition between volume- and area-law-entangled phases of pure-state trajectories in certain monitored quantum circuits. Here, we demonstrate that the LXE can distinguish distinct area-law-entangled phases of monitored circuits with symmetries, and extract universal behavior at the critical points separating these phases. We focus on (1+1)-dimensional monitored circuits with an on-site $\mathbb{Z}_{2}$ symmetry. For an appropriate choice of initial states, the LXE distinguishes the area-law-entangled spin glass and paramagnetic phases of the monitored trajectories. At the critical point, described by two-dimensional percolation, the LXE exhibits universal behavior which depends sensitively on boundary conditions, and the choice of initial states. With open boundary conditions, we show that the LXE relates to crossing probabilities in critical percolation, and is thus given by a known universal function of the aspect ratio of the dynamics, which quantitatively agrees with numerical studies of the LXE at criticality. The LXE probes correlations of other operators in percolation with periodic boundary conditions. We show that the LXE is sensitive to the richer phase diagram of the circuit model in the presence of symmmetric unitary gates. Lastly, we consider the effect of noise during the circuit evolution, and propose potential solutions to counter it., Comment: 12+6 pages, 16 figures. V2: References added

Adaptive quantum circuits, in which unitary operations, measurements, and feedback are used to steer quantum many-body systems, provide an exciting opportunity to generate new dynamical steady states. We introduce an adaptive quantum dynamics with continuous symmetry where unitary operations, measurements, and local unitary feedback are used to drive ordering. In this setting, we find a pure steady state hosting symmetry-breaking order, which is the ground state of a gapless, local Hamiltonian. We explore the dynamical properties of the approach to this steady state. We find that this steady-state order is fragile to perturbations, even those that respect the continuous symmetry., Comment: 17 pages, 10 figures

The Floquet code utilizes a periodic sequence of two-qubit measurements to realize the topological order. After each measurement round, the instantaneous stabilizer group can be mapped to a honeycomb toric code, explaining the topological feature. The code also possesses a time-crystal order - the $e-m$ transmutation after every cycle, breaking the Floquet symmetry of the measurement schedule. This behavior is distinct from the stationary topological order realized in either random circuits or time-independent Hamiltonian. Therefore, the resultant phase belongs to the overlap between the classes of Floquet enriched topological orders and measurement-induced phases. In this work, we construct a continuous path interpolating between the Floquet and toric codes, focusing on the transition between the time-crystal and stationary topological phases. We show that this transition is characterized by a divergent length scale. We also add single-qubit perturbations to the model and obtain a richer two-dimensional parametric phase diagram of the Floquet code, showing the stability of the Floquet enriched topological order., Comment: 6+8 pages, 13 figures

The coherent superposition of quantum states is an important resource for quantum information processing which distinguishes quantum dynamics and information from their classical counterparts. In this article we determine the coherence requirements to communicate quantum information in a broad setting encompassing monitored quantum dynamics and quantum error correction codes. We determine these requirements by considering hybrid circuits that are generated by a quantum information game played between two opponents, Alice and Eve, who compete by applying unitaries and measurements on a fixed number of qubits. Alice applies unitaries in an attempt to maintain quantum channel capacity, while Eve applies measurements in an attempt to destroy it. By limiting the coherence generating or destroying operations available to each opponent, we determine Alice's coherence requirements. When Alice plays a random strategy aimed at mimicking generic monitored quantum dynamics, we discover a coherence-tuned phase transitions in entanglement and quantum channel capacity. We then derive a theorem giving the minimum coherence required by Alice in any successful strategy, and conclude by proving that coherence sets an upper bound on the code distance in any stabelizer quantum error correction codes. Such bounds provide a rigorous quantification of the coherence resource requirements for quantum communication and error correction., Comment: 19+11 pages, 12+3 figures

We investigate the prospects of employing the linear cross-entropy to experimentally access measurement-induced phase transitions (MIPT) without requiring any postselection of quantum trajectories. For two random circuits that are identical in the bulk but with different initial states, the linear cross-entropy $\chi$ between the bulk measurement outcome distributions in the two circuits acts as a boundary order parameter, and can be used to distinguish the volume law from area law phases. In the volume law phase (and in the thermodynamic limit) the bulk measurements cannot distinguish between the two different initial states, and $\chi = 1$. In the area law phase $\chi < 1$. For circuits with Clifford gates, we provide numerical evidence that $\chi$ can be sampled to accuracy $\epsilon$ from $O(1/\epsilon^2)$ trajectories, by running the first circuit on a quantum simulator without postselection, aided by a classical simulation of the second. We also find that for weak depolarizing noise the signature of the MIPT is still present for intermediate system sizes. In our protocol we have the freedom of choosing initial states such that the "classical" side can be simulated efficiently, while simulating the "quantum" side is still classically hard., Comment: 7+8 pages, 6 figures. v2: 7+9 pages, 3+3 figures. Updated discussions on sample size (Fig. 2d, 2e), and new results from random Haar circuits (Fig. 3b). Accepted version

Measurements allow efficient preparation of interesting quantum many-body states with long-range entanglement, conditioned on additional transformations based on measurement outcomes. Here, we demonstrate that the so-called conformal quantum critical points (CQCP) can be obtained by performing general single-site measurements in an appropriate basis on the cluster states in $d\geq2$. The equal-time correlators of the said states are described by correlation functions of certain $d$-dimensional classical models at finite temperatures and feature spatial conformal invariance. This establishes an exact correspondence between the measurement-prepared critical states and conformal field theories of a range of critical spin models, including familiar Ising models and gauge theories. Furthermore, by mapping the long-range entanglement structure of measured quantum states into the correlations of the corresponding thermal spin model, we rigorously establish the stability condition of the long-range entanglement in the measurement-prepared quantum states deviating from the ideal setting. Most importantly, we describe protocols to decode the resulting quantum phases and transitions without post-selection, thus transferring the exponential measurement complexity to a polynomial classical computation. Therefore, our findings suggest a novel mechanism in which a quantum critical wavefunction emerges, providing new practical ways to study quantum phases and conformal quantum critical points., Comment: 37 pages, 11 figures

Quantum circuits -- built from local unitary gates and local measurements -- are a new playground for quantum many-body physics and a tractable setting to explore universal collective phenomena far-from-equilibrium. These models have shed light on longstanding questions about thermalization and chaos, and on the underlying universal dynamics of quantum information and entanglement. In addition, such models generate new sets of questions and give rise to phenomena with no traditional analog, such as new dynamical phases in quantum systems that are monitored by an external observer. Quantum circuit dynamics is also topical in view of experimental progress in building digital quantum simulators that allow control of precisely these ingredients. Randomness in the circuit elements allows a high level of theoretical control, with a key theme being mappings between real-time quantum dynamics and effective classical lattice models or dynamical processes. Many of the universal phenomena that can be identified in this tractable setting apply to much wider classes of more structured many-body dynamics., Comment: Review article for Annual Review of Condensed Matter Physics; comments welcome

We introduce an exact mapping of Clifford (stabilizer) random tensor networks (RTNs) and monitored quantum circuits, onto a statistical mechanics model. With Haar unitaries, the fundamental degrees of freedom ('spins') are permutations because all operators commuting with the action of the unitaries on a tensor product arise from permutations of the tensor factors ('Schur-Weyl duality'). For unitaries restricted to the smaller Clifford group, the set of commuting operators, the 'commutant', forming the new 'spin' degrees of freedom, will be larger. We use the recent full characterization of this commutant by Gross et al., Comm. Math. Phys. 385, 1325 (2021), to construct the Clifford statistical mechanics models for on-site Hilbert space dimensions which are powers of a prime number $p$. We show that the Boltzmann weights are invariant under a symmetry group involving orthogonal matrices with entries in the finite number field ${\bf F}_p$. This implies that the symmetry group, and consequently all universal properties of entanglement transitions in Clifford circuits and RTNs will in general depend on, and only on the prime $p$. We show that Clifford monitored circuits with on-site Hilbert space dimension $d=p^M$ are described by percolation in the limits $d \to \infty$ at (a) $p=$ fixed but $M\to \infty$, and at (b) $M= 1$ but $p \to \infty$. In the limit (a) we calculate the effective central charge, and in the limit (b) we derive the following universal minimal cut entanglement entropy $S_A =(\sqrt{3}/\pi)\ln p \ln L_A$ for $d=p$ large at the transition. We verify those predictions numerically, and present extensive numerical results for critical exponents at the transition in monitored Clifford circuits for prime number on-site Hilbert space dimension $d=p$ for a variety of different values of $p$, and find that they approach percolation values at large $p$., Comment: 23 pages, 5 figures. Abstract shortened to meet arxiv requirements, see pdf for full abstract

We explore a class of "open" quantum circuit models with local decoherence ("noise") and local projective measurements, each respecting a global Z_2 symmetry. The model supports a spin glass phase where the Z_2 symmetry is spontaneously broken (not possible in an equilibrium 1d system), a paramagnetic phase characterized by a divergent susceptibility, and an intermediate "trivial" phase. All three phases are also stable to Z_2-symmetric local unitary gates, and the dynamical phase transitions between the phases are in the percolation universality class. The open circuit dynamics can be purified by explicitly introducing a bath with its own "scrambling" dynamics, as in [Bao, Choi, Altman, arXiv:2102.09164], which does not change any of the universal physics. Within the spin glass phase the circuit dynamics can be interpreted as a quantum repetition code, with each stabilizer of the code measured stochastically at a finite rate, and the decoherences as effective bit-flip errors. Motivated by the geometry of the spin glass phase, we devise a novel decoding algorithm for recovering an arbitrary initial qubit state in the code space, assuming knowledge of the history of the measurement outcomes, and the ability of performing local Pauli measurements and gates on the final state. For a circuit with L^d qubits running for time T, the time needed to execute the decoder scales as O(L^d T) (with dimensionality d). With this decoder in hand, we find that the information of the initial encoded qubit state can be retained (and then recovered) for a time logarithmic in L for a 1d circuit, and for a time at least linear in L in 2d below a finite error threshold. For both the repetition and toric codes, we compare and contrast our decoding algorithm with earlier algorithms that map the error model to the random bond Ising model., Comment: 32+epsilon pages, 22 figures. Published version

We explore a class of random tensor network models with "stabilizer" local tensors which we name Random Stabilizer Tensor Networks (RSTNs). For RSTNs defined on a two-dimensional square lattice, we perform extensive numerical studies of entanglement phase transitions between volume-law and area-law entangled phases of the one-dimensional boundary states. These transitions occur when either (a) the bond dimension $D$ of the constituent tensors is varied, or (b) the tensor network is subject to random breaking of bulk bonds, implemented by forced measurements. In the absence of broken bonds, we find that the RSTN supports a volume-law entangled boundary state with bond dimension $D\geq3$ where $D$ is a prime number, and an area-law entangled boundary state for $D=2$. Upon breaking bonds at random in the bulk with probability $p$, there exists a critical measurement rate $p_c$ for each $D\geq 3$ above which the boundary state becomes area-law entangled. To explore the conformal invariance at these entanglement transitions for different prime $D$, we consider tensor networks on a finite rectangular geometry with a variety of boundary conditions, and extract universal operator scaling dimensions via extensive numerical calculations of the entanglement entropy, mutual information and mutual negativity at their respective critical points. Our results at large $D$ approach known universal data of percolation conformal field theory, while showing clear discrepancies at smaller $D$, suggesting a distinct entanglement transition universality class for each prime $D$. We further study universal entanglement properties in the volume-law phase and demonstrate quantitative agreement with the recently proposed description in terms of a directed polymer in a random environment., Comment: Updated to published version

Monitored quantum dynamics reveal quantum state trajectories which exhibit a rich phenomenology of entanglement structures, including a transition from a weakly-monitored volume law entangled phase to a strongly-monitored area law phase. For one-dimensional hybrid circuits with both random unitary dynamics and interspersed measurements, we combine analytic mappings to an effective statistical mechanics model with extensive numerical simulations on hybrid Clifford circuits to demonstrate that the universal entanglement properties of the volume law phase can be quantitatively described by a fluctuating entanglement domain wall that is equivalent to a "directed polymer in a random environment" (DPRE). This relationship improves upon a qualitative "mean-field" statistical mechanics of the volume-law-entangled phase [1, 2]. For the Clifford circuit in various geometries, we obtain agreement between the subleading entanglement entropies and error correcting properties of the volume-law phase (which quantify its stability to projective measurements) with predictions of the DPRE. We further demonstrate that depolarizing noise in the hybrid dynamics near the final circuit time can drive a continuous phase transition to a non-error correcting volume law phase that is not immune to the disentangling action of projective measurements. We observe this transition in hybrid Clifford dynamics, and obtain quantitative agreement with critical exponents for a "pinning" phase transition of the DPRE in the presence of an attractive interface., Comment: 22 pages, 18 figures

We propose entanglement negativity as a fine-grained probe of measurement-induced criticality. We motivate this proposal in stabilizer states, where for two disjoint subregions, comparing their "mutual negativity" and their mutual information leads to a precise distinction between bipartite and multipartite entanglement. In a measurement-only stabilizer circuit that maps exactly to two-dimensional critical percolation, we show that the mutual information and the mutual negativity are governed by boundary conformal fields of different scaling dimensions at long distances. We then consider a class of "hybrid" circuit models obtained by perturbing the measurement-only circuit with unitary gates of progressive levels of complexity. While other critical exponents vary appreciably for different choices of unitary gate ensembles at their respective critical points, the mutual negativity has scaling dimension 3 across remarkably many of the hybrid circuits, which is notably different from that in percolation. We contrast our results with limiting cases where a geometrical minimal-cut picture is available.

We study stabilizer quantum error correcting codes (QECC) generated under hybrid dynamics of local Clifford unitaries and local Pauli measurements in one dimension. Building upon 1) a general formula relating the error-susceptibility of a subregion to its entanglement properties, and 2) a previously established mapping between entanglement entropies and domain wall free energies of an underlying spin model, we propose a statistical mechanical description of the QECC in terms of "entanglement domain walls". Free energies of such domain walls generically feature a leading volume law term coming from its "surface energy", and a sub-volume law correction coming from thermodynamic entropies of its transverse fluctuations. These are most easily accounted for by capillary-wave theory of liquid-gas interfaces, which we use as an illustrative tool. We show that the information-theoretic decoupling criterion corresponds to a geometric decoupling of domain walls, which further leads to the identification of the "contiguous code distance" of the QECC as the crossover length scale at which the energy and entropy of the domain wall are comparable. The contiguous code distance thus diverges with the system size as the subleading entropic term of the free energy, protecting a finite code rate against local undetectable errors. We support these correspondences with numerical evidence, where we find capillary-wave theory describes many qualitative features of the QECC; we also discuss when and why it fails to do so., Comment: 22 pages, 8 figures. v2: fixed .bbl file, new footnote 12. v3: updated references and acknowledgements. v4: fixed typos; published version

We present random quantum circuit models for non-unitary quantum dynamics of free fermions in one spatial dimension. Numerical simulations reveal that the dynamics tends towards steady states with logarithmic violations of the entanglement area law and power law correlation functions. Moreover, starting with a short-range entangled many-body state, the dynamical evolution of entanglement and correlations quantitatively agrees with the predictions of two-dimensional conformal field theory with a space-like time direction. We argue that this behavior is generic in non-unitary free quantum dynamics with time-dependent randomness, and show that the emergent conformal dynamics of two-point functions arises out of a simple "nonlinear master equation".

We establish the emergence of a conformal field theory (CFT) in a (1+1)-dimensional hybrid quantum circuit right at the measurement-driven entanglement transition by revealing space-time conformal covariance of entanglement entropies and mutual information for various subregions at different circuit depths. While the evolution takes place in real time, the spacetime manifold of the circuit appears to host a Euclidean field theory with imaginary time. Throughout the paper we investigate Clifford circuits with several different boundary conditions by injecting physical qubits at the spatial and/or temporal boundaries, all giving consistent characterizations of the underlying "Clifford CFT." We emphasize (super)universal results that are consequences solely of the conformal invariance and do not depend crucially on the precise nature of the CFT. Among these are the infinite entangling speed as a consequence of measurement-induced quantum nonlocality and the critical purification dynamics of a mixed initial state., Comment: 31 pages, 16 figures. v2: updated references and acknowledgements. v3: new Appendix on "localizable entanglement". Published version

We introduce a procedure to systematically search for a local unitary transformation that maps a wavefunction with a non-trivial sign structure into a positive-real form. The transformation is parametrized as a quantum circuit compiled into a set of one and two qubit gates. We design a cost function that maximizes the average sign of the output state and removes its complex phases. The optimization of the gates is performed through automatic differentiation algorithms, widely used in the machine learning community. We provide numerical evidence for significant improvements in the average sign for a two-leg triangular Heisenberg ladder with next-to-nearest neighbour and ring-exchange interactions. This model exhibits phases where the sign structure can be removed by simple local one-qubit unitaries, but also an exotic Bose-metal phase whose sign structure induces "Bose surfaces" with a fermionic character and a higher entanglement that requires deeper circuits., Comment: 9 pages, 5 figures

In this paper we continue to explore "hybrid" quantum circuit models in one-dimension with both unitary and measurement gates, focussing on the entanglement properties of wavefunction trajectories at long times, in the steady state. We simulate a large class of Clifford circuits, including models with or without randomness in the unitary gates, and with or without randomness in the locations of measurement gates, using stabilizer techniques to access the long time dynamics of systems up to 512 qubits. In all models we find a volume law entangled phase for low measurement rates, which exhibits a sub-dominant logarithmic behavior in the entanglement entropy, $S_A = {\alpha} \ln |A| + s|A|$, with sub-system size $|A|$. With increasing measurement rate the volume law phase is unstable to a disentangled area law phase, passing through a single entanglement transition at a critical rate of measurement. At criticality we find a purely logarithmic entanglement entropy, $S_A = {\alpha}(p_c) \ln|A|$, a power law decay and conformal symmetry of the mutual information, with exponential decay off criticality. Various spin-spin correlation functions also show slow decay at criticality. Critical exponents are consistent across all models, indicative of a single universality class. These results suggest the existence of an effective underlying statistical mechanical model for the entanglement transition. Beyond Clifford circuit models, numerical simulations of up to 20 qubits give consistent results., Comment: 30 pages, 28 figures. Published version, w/ updated discussion, acknowledgement, and references

We introduce and explore a one-dimensional "hybrid" quantum circuit model consisting of both unitary gates and projective measurements. While the unitary gates are drawn from a random distribution and act uniformly in the circuit, the measurements are made at random positions and times throughout the system. By varying the measurement rate we can tune between the volume law entangled phase for the random unitary circuit model (no measurements) and a "quantum Zeno phase" where strong measurements suppress the entanglement growth to saturate in an area-law. Extensive numerical simulations of the quantum trajectories of the many-particle wavefunctions (exploiting Clifford circuitry to access systems up to 512 qubits) provide evidence for a stable "weak measurement phase" that exhibits volume-law entanglement entropy, with a coefficient decreasing with increasing measurement rate. We also present evidence for a novel continuous quantum dynamical phase transition between the "weak measurement phase" and the "quantum Zeno phase", driven by a competition between the entangling tendencies of unitary evolution and the disentangling tendencies of projective measurements. Detailed steady-state and dynamic critical properties of this novel quantum entanglement transition are accessed., Comment: 10 pages, 6 figures. Updated references & minor changes

Quantum indistinguishability plays a crucial role in many low-energy physical phenomena, from quantum fluids to molecular spectroscopy. It is, however, typically ignored in most high temperature processes, particularly for ionic coordinates, implicitly assumed to be distinguishable, incoherent and thus well-approximated classically. We explore chemical reactions involving small symmetric molecules, and argue that in many situations a full quantum treatment of collective nuclear degrees of freedom is essential. Supported by several physical arguments, we conjecture a "Quantum Dynamical Selection" (QDS) rule for small symmetric molecules that precludes chemical processes that involve direct transitions from orbitally non-symmetric molecular states. As we propose and discuss, the implications of the Quantum Dynamical Selection rule include: (i) a differential chemical reactivity of para- and ortho-hydrogen, (ii) a mechanism for inducing inter-molecular quantum entanglement of nuclear spins, (iii) a new isotope fractionation mechanism, (iv) a novel explanation of the enhanced chemical activity of "Reactive Oxygen Species", (v) illuminating the importance of ortho-water molecules in modulating the quantum dynamics of liquid water, (vi) providing the critical quantum-to-biochemical linkage in the nuclear spin model of the (putative) quantum brain, among others., Comment: 12 pages, 5 figures. Clarified presentation and figures

The possibility that quantum processing with nuclear spins might be operative in the brain is proposed and then explored. Phosphorus is identified as the unique biological element with a nuclear spin that can serve as a qubit for such putative quantum processing - a neural qubit - while the phosphate ion is the only possible qubit-transporter. We identify the "Posner molecule", $\text{Ca}_9 (\text{PO}_4)_6$, as the unique molecule that can protect the neural qubits on very long times and thereby serve as a (working) quantum-memory. A central requirement for quantum-processing is quantum entanglement. It is argued that the enzyme catalyzed chemical reaction which breaks a pyrophosphate ion into two phosphate ions can quantum entangle pairs of qubits. Posner molecules, formed by binding such phosphate pairs with extracellular calcium ions, will inherit the nuclear spin entanglement. A mechanism for transporting Posner molecules into presynaptic neurons during a "kiss and run" exocytosis, which releases neurotransmitters into the synaptic cleft, is proposed. Quantum measurements can occur when a pair of Posner molecules chemically bind and subsequently melt, releasing a shower of intra-cellular calcium ions that can trigger further neurotransmitter release and enhance the probability of post-synaptic neuron firing. Multiple entangled Posner molecules, triggering non-local quantum correlations of neuron firing rates, would provide the key mechanism for neural quantum processing. Implications, both in vitro and in vivo, are briefly mentioned., Comment: 8 pages, 3 figures

We propose and explore a new finite temperature phase of translationally invariant multi-component liquids which we call a "Quantum Disentangled Liquid" (QDL) phase. We contemplate the possibility that in fluids consisting of two (or more) species of indistinguishable quantum particles with a large mass ratio, the light particles might "localize" on the heavy particles. We give a precise, formal definition of this Quantum Disentangled Liquid phase in terms of the finite energy density many-particle wavefunctions. While the heavy particles are fully thermalized, for a typical fixed configuration of the heavy particles, the entanglement entropy of the light particles satisfies an area law; this implies that the light particles have not thermalized. Thus, in a QDL phase, thermal equilibration is incomplete, and the canonical assumptions of statistical mechanics are not fully operative. We explore the possibility of QDL in water, with the light proton degrees of freedom becoming "localized" on the oxygen ions. We do not presently know whether a local, generic Hamiltonian can have eigenstates of the QDL form, and if it can not, then the non-thermal behavior discussed here will exist as an interesting crossover phenomena at time scales that diverge as the ratio of the mass of the heavy to the light species also diverges., Comment: 14 pages