Mutual information area laws for thermal free fermions

We provide a rigorous and asymptotically exact expression of the mutual information of translationally invariant free fermionic lattice systems in a Gibbs state. In order to arrive at this result, we introduce a novel frameworkfor computing determinants of Toeplitz operators with smooth symbols, and for treating Toeplitz matrices with system size dependent entries. The asymptotically exact mutual information for a partition of the one-dimensional lattice satisfies an area law, with a prefactor which we compute explicitly. As examples, we discuss the fermionic XX model in one dimension and free fermionic models on the torus in higher dimensions in detail. Special emphasis is put onto the discussion of the temperature dependence of the mutual information, scaling like the logarithm of the inverse temperature, hence confirming an expression suggested by conformal field theory. We also comment on the applicability of the formalism to treat open systems driven by quantum noise. In the appendix, we derive useful bounds to the mutual information in terms of purities. Finally, we provide a detailed error analysis for finite system sizes. This analysis is valuable in its own right for the abstract theory of Toeplitz determinants.

[1]  S. Rachel,et al.  General relation between entanglement and fluctuations in one dimension , 2010, 1002.0825.

[2]  U. Grenander,et al.  Toeplitz Forms And Their Applications , 1958 .

[3]  E. Rico,et al.  Majorana modes in driven-dissipative atomic superfluids with a zero Chern number. , 2012, Physical review letters.

[4]  Germany,et al.  Quantum states and phases in driven open quantum systems with cold atoms , 2008, 0803.1482.

[5]  J Eisert,et al.  Solving frustration-free spin systems. , 2010, Physical review letters.

[6]  U. Vazirani,et al.  Improved one-dimensional area law for frustration-free systems , 2011, 1111.2970.

[7]  S. Michalakis,et al.  Stability of the Area Law for the Entropy of Entanglement , 2012, 1206.6900.

[8]  Statistics dependence of the entanglement entropy. , 2006, Physical review letters.

[9]  V. Korepin,et al.  Universality of entropy scaling in one dimensional gapless models. , 2003, Physical Review Letters.

[10]  Density-matrix spectra for integrable models , 1998, cond-mat/9810174.

[11]  Ulrich Schollwoeck,et al.  Entanglement scaling in critical two-dimensional fermionic and bosonic systems , 2006 .

[12]  J. Eisert,et al.  Noise-driven quantum criticality , 2010, 1012.5013.

[13]  Lee,et al.  Quantum source of entropy for black holes. , 1986, Physical review. D, Particles and fields.

[14]  Barbara M. Terhal,et al.  The power of noisy fermionic quantum computation , 2012, 1208.5334.

[15]  B. Swingle,et al.  Area law violations in a supersymmetric model , 2012, 1202.2367.

[16]  F. Verstraete,et al.  Quantum computation and quantum-state engineering driven by dissipation , 2009 .

[17]  Half the entanglement in critical systems is distillable from a single specimen , 2005, quant-ph/0509023.

[18]  J. Eisert,et al.  Entanglement-area law for general bosonic harmonic lattice systems (14 pages) , 2005, quant-ph/0505092.

[19]  F. Verstraete,et al.  Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems , 2008, 0907.2796.

[20]  J. Eisert,et al.  Multiparty entanglement in graph states , 2003, quant-ph/0307130.

[21]  J Eisert,et al.  Precisely timing dissipative quantum information processing. , 2012, Physical review letters.

[22]  G. Vidal,et al.  Entanglement in quantum critical phenomena. , 2002, Physical review letters.

[23]  Michael M Wolf Violation of the entropic area law for fermions. , 2006, Physical review letters.

[24]  Density-matrix spectra for two-dimensional quantum systems , 2000, cond-mat/0004222.

[25]  K. Audenaert,et al.  Entanglement properties of the harmonic chain , 2002, quant-ph/0205025.

[26]  Fernando G. S. L. Brandão,et al.  Exponential Decay of Correlations Implies Area Law , 2012, Communications in Mathematical Physics.

[27]  J. Cirac,et al.  Topological and entanglement properties of resonating valence bond wave functions , 2012, 1202.0947.

[28]  Quantum entanglement in states generated by bilocal group algebras (9 pages) , 2005, quant-ph/0504049.

[29]  M. Srednicki,et al.  Entropy and area. , 1993, Physical review letters.

[30]  J. Cirac,et al.  Resonating valence bond states in the PEPS formalism , 2012, 1203.4816.

[31]  O. Bratteli Operator Algebras And Quantum Statistical Mechanics , 1979 .

[32]  Brian Swingle,et al.  Rényi entropy, mutual information, and fluctuation properties of Fermi liquids , 2010, 1007.4825.

[33]  V. Alba,et al.  Entanglement spectrum of the Heisenberg XXZ chain near the ferromagnetic point , 2012, 1206.0131.

[34]  U. Schollwoeck The density-matrix renormalization group , 2004, cond-mat/0409292.

[35]  A. Lefevre,et al.  Entanglement spectrum in one-dimensional systems , 2008, 0806.3059.

[36]  B. Nachtergaele,et al.  An area law for the bipartite entanglement of disordered oscillator systems , 2013, 1301.3116.

[37]  Matthew B Hastings,et al.  Area laws in quantum systems: mutual information and correlations. , 2007, Physical review letters.

[38]  Dimitri Gioev,et al.  Entanglement entropy of fermions in any dimension and the Widom conjecture. , 2006, Physical review letters.

[39]  R. Bhatia Matrix Analysis , 1996 .

[40]  J Eisert,et al.  Entropy, entanglement, and area: analytical results for harmonic lattice systems. , 2005, Physical review letters.

[41]  J. Eisert,et al.  Correlations, spectral gap and entanglement in harmonic quantum systems on generic lattices , 2005, quant-ph/0509167.

[42]  Vladimir E. Korepin,et al.  Quantum Spin Chain, Toeplitz Determinants and the Fisher—Hartwig Conjecture , 2004 .

[43]  Single-copy entanglement in critical quantum spin chains (5 pages) , 2005, quant-ph/0506250.

[44]  R. Kress Numerical Analysis , 1998 .

[45]  Vladimir E. Korepin,et al.  The Fisher-Hartwig Formula and Entanglement Entropy , 2009 .

[46]  F. Wilczek,et al.  Geometric and renormalized entropy in conformal field theory , 1994, hep-th/9403108.

[47]  A. Böttcher,et al.  Introduction to Large Truncated Toeplitz Matrices , 1998 .

[48]  Hui Li,et al.  Entanglement spectrum as a generalization of entanglement entropy: identification of topological order in non-Abelian fractional quantum Hall effect states. , 2008, Physical review letters.

[49]  J. Eisert,et al.  Area laws for the entanglement entropy - a review , 2008, 0808.3773.

[50]  I. Peschel,et al.  Reduced density matrices and entanglement entropy in free lattice models , 2009, 0906.1663.