Bulletin of the American Physical Society
APS March Meeting 2015
Volume 60, Number 1
Monday–Friday, March 2–6, 2015; San Antonio, Texas
Session J38: Focus Session: Entanglement and Mathematics of Quantum Information |
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Sponsoring Units: GQI Chair: Sean Carroll, California Institute of Technology Room: 212B |
Tuesday, March 3, 2015 2:30PM - 2:42PM |
J38.00001: Entanglement negativity in free-fermion systems Po-Yao Chang, Xueda Wen, Shinsei Ryu We derive a general formula of the logarithmic negativity in free-fermion systems, using the overlap matrix to construct the partially transposed reduced density matrix $\rho_{A}^{T_{A_2}}$ of a subsystem $A = A_1 \bigcup A_2$. In particular, we consider the negativity between two adjacent or disjoint regions in three systems: a homogeneous one-dimensional chain, the dimerized Su-Schrieffer-Heeger model, and the integer Quantum Hall state. For the negativity of two adjacent intervals in a homogeneous one-dimensional gas, we find agreement with the conformal field theory [P. Calabrese {\it et al.} Phys. Rev. Lett. {\bf 109}, 130502 (2012)]. On the other hand, the negativity for the integer quantum Hall states satisfies the area law. Our method is applicable to the study of the negativity in any free-fermion systems. [Preview Abstract] |
Tuesday, March 3, 2015 2:42PM - 2:54PM |
J38.00002: Fidelity of recovery and geometric squashed entanglement Kaushik Seshadreesan, Mark Wilde We define the fidelity of recovery of a tripartite quantum state on systems $A$, $B$, and $C$ as a measure of how well one can recover the full state on all three systems if system $A$ is lost and a recovery operation is performed on system $C$ alone. The surprisal of the fidelity of recovery (its negative logarithm) is an information quantity which obeys nearly all of the properties of the conditional quantum mutual information $I(A;B|C)$, including non-negativity, monotonicity under local operations, duality, and a dimension bound. We then define an entanglement measure based on this quantity, which we call the geometric squashed entanglement. We prove that the geometric squashed entanglement is an entanglement monotone, that it vanishes if and only if the state on which it is evaluated is unentangled, and that it reduces to the geometric measure of entanglement if the state is pure. We also show that it is sub-additive, continuous, and normalized on maximally entangled states. Our results for the bipartite case can easily be extended to a multipartite fidelity of recovery and a multipartite geometric squashed entanglement. [Preview Abstract] |
Tuesday, March 3, 2015 2:54PM - 3:06PM |
J38.00003: Scaling of entanglement in (2 + 1)-dimensional critical field theories Xiao Chen, Gil Young Cho, Thomas Faulkner, Eduardo Fradkin We study the universal scaling behavior of the entanglement entropy of critical theories in (2+1) dimensions. We specially consider two fermionic critical models, the Dirac and quadratic band touching models, and numerically study the two-cylinder entanglement entropy of the models on the torus. We find that the entanglement entropy satisfies the area law and has a finite subleading term which is a scaling function of the length scales of the torus. To find the analytic form of the scaling function, we test three possible scaling functions derived from the quasi-one-dimensional conformal field theory, the bosonic quantum Lifshitz model, and the holographic AdS/CFT correspondence. To high precision, we find that the subleading term is consistent with the scaling form of the quantum Lifshitz model for both the fermionic models and that of the AdS/CFT correspondence for the Dirac model. Based on this observation and the previous studies on the subleading terms of entanglement entropy of bosonic critical theories which were found consistent with the scaling subleading term of the quantum Lifshitz model, we propose a universal scaling form of the subleading term for the entanglement entropy of (2+1)-dimensional critical models on the torus. [Preview Abstract] |
Tuesday, March 3, 2015 3:06PM - 3:18PM |
J38.00004: Generalized Entanglement Entropy and Space-Time Geometry of Quantum System Zhao Yang, Patrick Hayden, Xiaoliang Qi Entanglement entropy plays a key role in relating quantum information with quantum gravity and condensed matter physics. Based on the proposal of arxiv:1309.6282, we would like to use quantum entanglement between two regions of a quantum system as a measure of the geometrical distance between them. However, since entanglement entropy can only be defined between space-like separated regions, we are forced to treat space and time inhomogeneously. In this work, we propose a generalized entanglement entropy (GEE) which is defined between two generic regions of the system which do not have to be space-like separated. We study the generalized mutual information defined using this generalized entanglement entropy, and demonstrate for several different systems that this provides a reasonable measure of space-time distance. The generalized mutual information is complex-valued and the space-like distance and time-like one are determined by the amplitude and the phase of the generalized mutual information, correspondingly. We study the properties of this generalized entropy and generalized mutual information, and apply this framework to the exact holographic mapping of free fermions in various conditions. [Preview Abstract] |
Tuesday, March 3, 2015 3:18PM - 3:30PM |
J38.00005: Entanglement Spectrum of a Random Partition: Connection with the Anderson Transition Sagar Vijay, Liang Fu The entanglement spectrum of a topologically-ordered ground-state that is obtained by partitioning the system under consideration into two subsystems which extend throughout the bulk, has been recently shown to be a probe of the quantum critical behavior of the topological phase at the transition to a direct-product state [1]. Here, we generalize this notion of a bulk entanglement spectrum to extract universal information about disorder-driven topological phase transitions, by performing an extensive, random partition into two subsystems with probability $p\in[0,1]$. We apply our random partitioning procedure to a one-dimensional topological superconductor (TSC), and demonstrate that the phase diagram of the resulting entanglement Hamiltonian describes disorder-driven transitions to a Griffiths phase. [1] T. H. Hsieh and L. Fu, Phys. Rev. Lett. {\bf 113}, 106801 (2014). [Preview Abstract] |
Tuesday, March 3, 2015 3:30PM - 3:42PM |
J38.00006: The existence of maximally multipartite entangled states of N particles may depend on their spin Jay Lawrence, Mario Gaeta, Andrei Klimov Maximally multipartite entangled states (MMES) are defined [1] as pure states of N particles for which all subsystems consisting of up to half the particles (k=[N/2]) are maximally mixed ($\rho_k \sim I$). Such states exist for two- or three-particle systems (Bell states for N=2 and GHZ states for N=3), and this holds for any spin. The situation changes for four particles, where MMES states do not exist for spin-1/2 (dimension d=2), but they do exist for all odd prime dimensions d [or spins S = (d-1)/2]. The latter systems exhibit three types of graph states, the GHZ and cluster states accessible to qubits, which are not MMES, but also a third type, called P states [2], that are MMES but are not accessible to qubits. We show how the P states succeed while GHZ and cluster states fail by comparing (i) the reduced states of subsystems, and (ii) the measurement-induced pathways which project Bell states of any two particles. We discuss the possibilities that similar transitions exist for larger systems, for which it is known [1] that MMES do not exist for eight or more qubits. 1. L. Arnaud and N.J. Cerf, Phys. Rev. A {\bf 87}, 012319 (2013), 2. J. Lawrence, Phys. Rev. A {\bf 84}, 022338 (2011). [Preview Abstract] |
Tuesday, March 3, 2015 3:42PM - 4:18PM |
J38.00007: Entanglement and coherence in many-body dipolar systems Invited Speaker: Susanne Yelin The presence of dipoles in atoms and molecules can nonlinearly change the coherent dynamics and entanglement structure in many-body situations. I will present examples for atomic and molecular systems. [Preview Abstract] |
Tuesday, March 3, 2015 4:18PM - 4:30PM |
J38.00008: Entanglement-Assisted Transformations of W-Type States Jiayang Xiao, Eric Chitambar In multipartite systems, it is usually impossible to transform one entangled state into another via local operations and classical communication (LOCC). However, the transformation may become possible with the help of some extra entanglement. This kind of transformation is called entanglement-assisted LOCC (eLOCC). Beyond the bipartite setting, very little is known about eLOCC. We prove the optimal eLOCC probability for transforming a tripartite W-type state ($\sqrt{x_1}\{100\}+\sqrt{x_2}\{010\}+\sqrt{x_3}\{001\}$) into a GHZ state $\sqrt{1/2}(\{000\}+\{111\})$ when any two of the parties share a resource EPR state. Interestingly, this is the same optimal probability of converting the given state into an EPR pair with no entanglement resource. Finally, we consider the eLOCC transformation of a more general W-type state ($\sqrt{x_0}\{000\}+\sqrt{x_1}\{100\}+\sqrt{x_2}\{010\}+\sqrt{x_3}\{001\}$) into GHZ and compare this with EPR distillation rates for a variety of different protocols. [Preview Abstract] |
Tuesday, March 3, 2015 4:30PM - 4:42PM |
J38.00009: Ground state entanglement constrains low-energy excitations Isaac Kim, Benjamin Brown For a general quantum many-body system, we show that its ground state entanglement imposes a fundamental constraint on the low-energy excitations. For two-dimensional systems, our result implies that any system that supports anyons must have a nonvanishing topological entanglement entropy. We demonstrate the generality of this argument by applying it to three-dimensional quantum many-body systems, and showing that there is a pair of ground state topological invariants that are associated to their physical boundaries. From the pair, one can determine whether the given boundary can or cannot absorb point-like or line-like excitations. [Preview Abstract] |
Tuesday, March 3, 2015 4:42PM - 4:54PM |
J38.00010: Jordan Algebraic Quantum Categories Matthew Graydon, Howard Barnum, Cozmin Ududec, Alexander Wilce State cones in orthodox quantum theory over finite dimensional complex Hilbert spaces enjoy two particularly essential features: homogeneity and self-duality. Orthodox quantum theory is not, however, unique in that regard. Indeed, all finite dimensional formally real Jordan algebras --- arenas for generalized quantum theories with close algebraic kinship to the orthodox theory --- admit homogeneous self-dual positive cones. We construct categories wherein these theories are unified. The structure of composite systems is cast from universal tensor products of the universal C$^{*}$-algebras enveloping ambient spaces for the constituent state cones. We develop, in particular, a notion of composition that preserves the local distinction of constituent systems in quaternionic quantum theory. More generally, we explicitly derive the structure of hybrid quantum composites with subsystems of arbitrary Jordan algebraic type. [Preview Abstract] |
Tuesday, March 3, 2015 4:54PM - 5:06PM |
J38.00011: Jacobi -- type identities and the underlying quantal algebra Samir Lipovaca Reference [1] introduced two identities written in terms of single commutators and anticommutators for any three elements of an arbitrary associative algebra where one is a consequence of other which is called the fundamental identity. From the fundamental identity a set of four other identities, represented in terms of double commutators and anticommutators is derived. We will show that three of these identities are in fact the defining identities of the quantal algebra. In this light, [1] did actually derive an underlying quantal algebra for an arbitrary associative algebra from the fundamental identity. Remarkably, the proof of the Theorem 2 showed that the existence of this underlying quantal algebra is equivalent to the associativity condition. A generalization to the super case of the quantal algebra is, in essence, derived in the section 3 of [1].\\[4pt] [1] \underline {arXiv:1304.5050v2} [math-ph] [Preview Abstract] |
Tuesday, March 3, 2015 5:06PM - 5:18PM |
J38.00012: Quantum Bochner's theorem for phase spaces built on projective representations Ninnat Dangniam, Christopher Ferrie Bochner's theorem gives the necessary and sufficient conditions on a characteristic function such that it corresponds to a true probability density function. In the Wigner phase space picture, quantum Bochner's theorem gives the necessary and sufficient conditions on the quantum characteristic function such that it corresponds to a valid quantum state and such that its Fourier transform is a true probability density. We extend this theorem to discrete phase space representations which possess enough symmetry to define a generalized Fourier transform. [Preview Abstract] |
Tuesday, March 3, 2015 5:18PM - 5:30PM |
J38.00013: Galois-unitary operators that cycle mutually-unbiased bases Hoan Dang, Marcus Appleby, Ingemar Bengtsson Wigner's theorem states that probability-preserving transformations of quantum states must be either unitary or anti-unitary. However, if we restrict ourselves to a subspace of a Hilbert space, it is possible to generalize the notion of anti-unitaries. Such transformations were recently constructed in search of Symmetric Informationally-Complete (SIC) states. They are called Galois-unitaries (g-unitaries for short), as they are unitaries composed with Galois automorphisms of a chosen number field extension. Despite certain bizarre behaviors of theirs, we show that g-unitaries are indeed useful in the theory of Mutually-Unbiased Bases (MUBs), as they help solve the MUB-cycling problem and provide a construction of MUB-balanced states. [Preview Abstract] |
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