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Quantum Information in the Posner Model of Quantum Cognition

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Quantum Information in the Posner Model of Quantum Cognition Quantum information in quantum cognition Nicole Yunger Halpern1, 2 and Elizabeth Crosson1 1Institute for Quantum Information and Matter, Caltech, Pasadena, CA 91125, USA 2Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA (Dated: November 15, 2017) Matthew Fisher recently postulated a mechanism by which quantum phenomena could influence cognition: Phosphorus nuclear spins may resist decoherence for long times. The spins would serve as biological qubits. The qubits may resist decoherence longer when in Posner molecules. We imagine that Fisher postulates correctly. How adroitly could biological systems process quantum information (QI)? We establish a framework for answering. Additionally, we apply biological qubits in quantum error correction, quantum communication, and quantum computation. First, we posit how the QI encoded by the spins transforms as Posner molecules form. The transformation points to a natural computational basis for qubits in Posner molecules. From the basis, we construct a quantum code that detects arbitrary single-qubit errors. Each molecule encodes one qutrit. Shifting from information storage to computation, we define the model of Posner quantum computation. To illustrate the model's quantum-communication ability, we show how it can teleport information in- coherently: A state's weights are teleported; the coherences are not. The dephasing results from the entangling operation's simulation of a coarse-grained Bell measurement. Whether Posner quantum computation is universal remains an open question. However, the model's operations can efficiently prepare a Posner state usable as a resource in universal measurement-based quantum computation. The state results from deforming the Affleck-Lieb-Kennedy-Tasaki (AKLT) state and is a projected entangled-pair state (PEPS). Finally, we show that entanglement can affect molecular-binding rates (by 0.6% in an example). This work opens the door for the QI-theoretic analysis of biological qubits and Posner molecules. Fisher recently proposed a mechanism by which quan- This paper is intended for QI scientists, for chemists, tum phenomena might affect cognition [1]. Phosphorus and for biophysicists. Some readers may require back- atoms populate biochemistry. A phosphorus nucleus's ground about the QI theory behind the results. We refer spin, he argued, can store quantum information (QI) for these readers to App. A and to [7, 8]. Next, we overview long times. The nucleus has a spin quantum number this paper's contributions. 1 s = 2 . Hence the nucleus forms a qubit, a quantum two- Computational bases before and after molecule level system. The qubit is the standard unit of QI. formation: Phosphorus nuclear spins originate outside Fisher postulated physical processes that might en- Posners, in Fisher's narrative. The spins occupy ions tangle phosphorus nuclei. Six phosphorus atoms might, that join together, forming Posners. Molecular formation with other ions, form Posner molecules Ca9(PO4)6 [2{ changes how QI is encoded physically. 4].1 The molecules might protect the spins' states for Outside of molecules, phosphorus nuclear spins couple long times. Fisher also described how the QI stored in little to orbital degrees of freedom (DOFs). Spin states the spins might be read out. This QI, he conjectured, form an obvious choice of computational basis.2 In a could impact neuron firing. The neurons could partici- Posner molecule, the spins are indistinguishable. They pate in quantum cognition. occupy an antisymmetric state [1, 5]: The spins entangle These conjectures require empirical testing. Fisher with orbital DOFs. Which physical states form a useful has proposed experiments [1], including with Radzi- computational basis is not obvious. hovsky [5]. Some experiments have begun [6]. We identify such a basis. Molecule formation, we posit, Suppose that Fisher conjectures correctly. How effec- maps premolecule spin states to antisymmetric molecule arXiv:1711.04801v1 [quant-ph] 13 Nov 2017 tively could the spins process QI? We provide a frame- states deterministically. The premolecule orbital state work for answering this question, and we begin answer- determines the map. We formalize the map with a ing. We translate Fisher's physics and chemistry into in- projector-valued measure (PVM). The mapped-to anti- formation theory. The language of molecular tumbling, symmetric states form the computational basis, in terms heat, etc. is replaced with the formalism of Bloch-sphere of which Posners' QI processing should be expressed. rotations, positive operator-valued measures (POVMs), Quantum error-correcting and -detecting computational bases, etc. Additionally, we identify and codes: The basis elements may decohere quickly: Pos- quantify QI-storage, -communication, and -computation ners' geometry protects only spins. The basis elements capacities of the phosphorus nuclear spins and Posners. 2 In QI, computations are expressed in terms of a com- putational basis for the system's Hilbert space [7]. Ba- 1 Ca9(PO4)6 has been called the Posner cluster and Posner sis elements are often represented by bit strings, as in molecule. We call it the Posner, for short. fj00 ::: 0i; j00 ::: 01i;::: j11 ::: 1ig. 2 are spin-and-position entangled states. Do the dynamics satisfy τA + τB = 0 can be measured projectively. If protect any states against errors? the equation is satisfied, the twelve qubits can undergo Hamiltonians' ground spaces may form quantum error- coordinated rotations. correcting and -detecting codes (QECD codes) [8]. One Finally, hextuples can cease to correspond to geome- might hope to relate the Posner Hamiltonian HPos to a tries or to GC 's (as Posners break down into their con- 3 QECD code. Alas, HPos has not been characterized. stituent ions). Thereafter, qubits can rotate indepen- Yet HPos likely preserves two observables. One, GC , dently again, group together into new hextuples, etc. generates cyclic permutations of the spins. One such per- This model enables us to recast Fisher's narrative [1] as mutation shuffles the spins counterclockwise about the a quantum circuit. We also identify a criterion necessary molecule's symmetry axis, through an angle 2π=3. This for constructing, from Posner operations, quantum cir- permutation preserves the Posner's geometry [1{4]. The cuits of nonconstant depth: Molecules must break down zin other charge, S12:::6, is the spins' total zin-component. and form anew. Only outside molecules can qubits un- (The internal-frame axisz ^in remains fixed relative to the dergo arbitrary rotations. Only in molecules can qubits atoms' positions.) undergo entangling operations late in a computation. To The dynamics likely preserve eigenstates shared by GC alternate between rotating and entangling, qubits must zin zin and S1:::6. Yet GC shares many eigenbases with S1:::6: leave and enter Posners. The charges fail to form a complete set of commuting Entanglement generated by, and quantum- observables (CSCO). We identify a useful operator that communication application of, molecular bind- 2 2 breaks the degeneracy: S123 ⊗ S456 equals a product of ing: Two Posners, Fisher conjectures, can bind to- the spin-squared operators S2 of trios of a Posner's spins. gether [1]. Quantum-chemistry calculations support the This operator (i) respects the Posner's geometry and (ii) conjecture [9]. The binding is expected to entangle the facilitates the construction of Posner states that can fuel Posners [1]. How much entanglement does binding gen- universal quantum computation (discussed below). erate, and entanglement of what sort? zin 2 From the eigenbasis shared by GC , S1:::6, and S123 ⊗ We characterize the entanglement in two ways. First, 2 S456, we form QECD codes. A state j i in one charge we compare Posner binding to a Bell measurement [7]. A zin sector of GC and one sector of S1:::6 likely cannot trans- Bell measurement yields one of four possible outcomes| form, under the dynamics, into a state jφi in a second two bits of information. Posner binding transforms a zin sector of GC and a second sector of S1:::6. Hence j i subspace as a coarse-grained Bell measurement. A Bell and jφi suggest themselves as codewords. Charge preser- measurement is performed, and one bit is discarded, ef- vation would prevent one codeword from evolving into fectively. another. Second, we present a quantum-communication proto- We construct two quantum codes, each partially pro- col reliant on Posner binding. We define a qutrit (three- tected by charge preservation. Via one code, each Posner level) subspace of the Posner Hilbert space. A Posner P encodes one qutrit. The codewords correspond to distinct P2 may occupy a state j i = j=0 cjjji in the subspace. eigenvalues of GC . This code detects arbitrary single- 2 The coefficients jcjj form a probability distribution Q. physical-qubit errors. Via the second code, each Posner This distribution has a probability p of being teleported encodes one qubit. This repetition code corrects two bit to another Posner, P 0. Another distribution, Q~, consists flips. The codewords correspond to distinct eigenvalues j j2 ~ − zin of combinations of the cj 's. Q has a probability 1 p of of S1:::6. being teleported. Measuring P 0 in the right basis would Model of Posner quantum computation: Fisher yield an outcome distributed according to Q or according posits chemical processes, such as binding, that Pos- to Q~. ners may undergo [1]. We abstract away the chemistry, The weights of j i (or combinations of the weights) formalizing the computations effected by the processes. are teleported [10]. The coherences are not. We therefore Posner operations Pos- These effected form the model of dub the protocol incoherent teleportation. The dephasing ner quantum computation . comes from the binding's simulation of a coarse-grained The model includes the preparation of singlets Bell measurement. Bell measurements teleport QI coher- p1 (j01i − j10i). Qubits can rotate arbitrarily when the 2 ently. phosphorus atoms are outside molecules. The qubits Incoherent teleportation effects a variant of superdense 1 evolve trivially, under the identity , when Posners form.
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