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Entropy of entanglement
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<h2 id="bipartite-entanglement-entropy">Bipartite entanglement entropy</h2> <p>Suppose that a quantum system consists of N {\displaystyle N} particles. A bipartition of the system is a partition which divides the system into two parts A {\displaystyle A} and B {\displaystyle B} , containing k {\displaystyle k} and l {\displaystyle l} particles respectively with k + l = N {\displaystyle k+l=N} . Bipartite entanglement entropy is defined with respect to this bipartition. </p> <h3>Von Neumann entanglement entropy</h3> <p>The bipartite von Neumann entanglement entropy S {\displaystyle S} is defined as the <a href="/facts/Von_Neumann_entropy/nzeJKowg">von Neumann entropy</a> of either of its reduced states, since they are of the same value (can be proved from Schmidt decomposition of the state with respect to the bipartition); the result is independent of which one we pick. That is, for a pure state ρ A B = | Ψ ⟩ ⟨ Ψ | A B {\displaystyle \rho _{AB}=|\Psi \rangle \langle \Psi |_{AB}} , it is given by: </p> S ( ρ A ) = − Tr [ ρ A log ρ A ] = − Tr [ ρ B log ρ B ] = S ( ρ B ) {\displaystyle {\mathcal {S}}(\rho _{A})=-\operatorname {Tr} [\rho _{A}\operatorname {log} \rho _{A}]=-\operatorname {Tr} [\rho _{B}\operatorname {log} \rho _{B}]={\mathcal {S}}(\rho _{B})} <p>where ρ A = Tr B ( ρ A B ) {\displaystyle \rho _{A}=\operatorname {Tr} _{B}(\rho _{AB})} and ρ B = Tr A ( ρ A B ) {\displaystyle \rho _{B}=\operatorname {Tr} _{A}(\rho _{AB})} are the <a href="/facts/Quantum_entanglement/afzslwN0">reduced density matrices</a> for each partition. </p><p>The entanglement entropy can be expressed using the singular values of the <a href="/facts/Schmidt_decomposition/t7CYaV0F">Schmidt decomposition</a> of the state. Any pure state can be written as | Ψ ⟩ = ∑ i = 1 m α i | u i ⟩ A ⊗ | v i ⟩ B {\displaystyle |\Psi \rangle =\sum _{i=1}^{m}\alpha _{i}|u_{i}\rangle _{A}\otimes |v_{i}\rangle _{B}} where | u i ⟩ A {\displaystyle |u_{i}\rangle _{A}} and | v i ⟩ B {\displaystyle |v_{i}\rangle _{B}} are orthonormal states in subsystem A {\displaystyle A} and subsystem B {\displaystyle B} respectively. The entropy of entanglement is simply: </p><p> − ∑ i α i 2 log ( α i 2 ) {\textstyle -\sum _{i}\alpha _{i}^{2}\log(\alpha _{i}^{2})} </p><p>This form of writing the entropy makes it explicitly clear that the entanglement entropy is the same regardless of whether one computes partial trace over the A {\displaystyle A} or B {\displaystyle B} subsystem. </p><p> Many entanglement measures reduce to the entropy of entanglement when evaluated on pure states. Among those are: </p> <ul><li>Distillable entanglement</li> <li>Entanglement cost</li> <li><a href="/facts/Entanglement_of_formation/00SqC89K">Entanglement of formation</a></li> <li><a href="/facts/Quantum_relative_entropy/cdoXwgDh">Relative entropy of entanglement</a></li> <li><a href="/facts/Squashed_entanglement/JEoFbxzA">Squashed entanglement</a></li></ul> <p>Some entanglement measures that do not reduce to the entropy of entanglement are: </p> <ul><li><a href="/facts/Negativity_(quantum_mechanics)/dblRlmk9">Negativity</a></li> <li><a href="/facts/Logarithmic_negativity/dblRlmk9">Logarithmic negativity</a></li> <li>Robustness of entanglement<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li></ul> <h3>Renyi entanglement entropies</h3> <p>The Renyi entanglement entropies S α {\displaystyle {\mathcal {S}}_{\alpha }} are also defined in terms of the reduced density matrices, and a Renyi index α ≥ 0 {\displaystyle \alpha \geq 0} . It is defined as the <a href="/facts/R%C3%A9nyi_entropy/SlrvqKNv">Rényi entropy</a> of the reduced density matrices: </p> S α ( ρ A ) = 1 1 − α log tr ( ρ A α ) = S α ( ρ B ) {\displaystyle {\mathcal {S}}_{\alpha }(\rho _{A})={\frac {1}{1-\alpha }}\operatorname {log} \operatorname {tr} (\rho _{A}^{\alpha })={\mathcal {S}}_{\alpha }(\rho _{B})} <p>Note that in the limit α → 1 {\displaystyle \alpha \rightarrow 1} , The Renyi entanglement entropy approaches the Von Neumann entanglement entropy. </p> <h2 id="example-with-coupled-harmonic-oscillators">Example with coupled harmonic oscillators</h2> <p>Consider two coupled <a href="/facts/Quantum_harmonic_oscillator/JYA5J0h5">quantum harmonic oscillators</a>, with positions q A {\displaystyle q_{A}} and q B {\displaystyle q_{B}} , momenta p A {\displaystyle p_{A}} and p B {\displaystyle p_{B}} , and system Hamiltonian </p> H = ( p A 2 + p B 2 ) / 2 + ω 1 2 ( q A 2 + q B 2 ) / 2 + ω 2 2 ( q A − q B ) 2 / 2 {\displaystyle H=(p_{A}^{2}+p_{B}^{2})/2+\omega _{1}^{2}(q_{A}^{2}+q_{B}^{2})/{2}+{\omega _{2}^{2}(q_{A}-q_{B})^{2}}/{2}} <p>With ω ± 2 = ω 1 2 + ω 2 2 ± ω 2 2 {\displaystyle \omega _{\pm }^{2}=\omega _{1}^{2}+\omega _{2}^{2}\pm \omega _{2}^{2}} , the system's pure ground state density matrix is ρ A B = | 0 ⟩ ⟨ 0 | {\displaystyle \rho _{AB}=|0\rangle \langle 0|} , which in position basis is ⟨ q A , q B | ρ A B | q A ′ , q B ′ ⟩ ∝ exp ( − ω + ( q A + q B ) 2 / 2 − ω − ( q A − q B ) 2 / 2 − ω + ( q A ′ + q B ′ ) 2 / 2 − ω − ( q A ′ − q B ′ ) 2 / 2 ) {\displaystyle \langle q_{A},q_{B}|\rho _{AB}|q_{A}',q_{B}'\rangle \propto \exp \left(-{\omega _{+}(q_{A}+q_{B})^{2}}/{2}-{\omega _{-}(q_{A}-q_{B})^{2}}/{2}-{\omega _{+}(q'_{A}+q'_{B})^{2}}/{2}-{\omega _{-}(q'_{A}-q'_{B})^{2}}/{2}\right)} . Then <a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p><p> ⟨ q A | ρ A | q A ′ ⟩ ∝ exp ( 2 ( ω + − ω − ) 2 q A q A ′ − ( 8 ω + ω − + ( ω + − ω − ) 2 ) ( q A 2 + q A ′ 2 ) 8 ( ω + + ω − ) ) {\displaystyle \langle q_{A}|\rho _{A}|q_{A}'\rangle \propto \exp \left({\frac {2(\omega _{+}-\omega _{-})^{2}q_{A}q_{A}'-(8\omega _{+}\omega _{-}+(\omega _{+}-\omega _{-})^{2})(q_{A}^{2}+q_{A}'^{2})}{8(\omega _{+}+\omega _{-})}}\right)} </p><p>Since ρ A {\displaystyle \rho _{A}} happens to be precisely equal to the density matrix of a single quantum harmonic oscillator of frequency ω ≡ ω + ω − {\displaystyle \omega \equiv {\sqrt {\omega _{+}\omega _{-}}}} at <a href="/facts/Thermal_Equilibrium/1D7nGm0d">thermal equilibrium</a> with <a href="/facts/Temperature/QyGe4gmj">temperature</a> T {\displaystyle T} ( such that ω / k B T = cosh − 1 ( 8 ω + ω − + ( ω + − ω − ) 2 ( ω + − ω − ) 2 ) {\displaystyle \omega /k_{B}T=\cosh ^{-1}\left({\frac {8\omega _{+}\omega _{-}+(\omega _{+}-\omega _{-})^{2}}{(\omega _{+}-\omega _{-})^{2}}}\right)} where k B {\displaystyle k_{B}} is the <a href="/facts/Boltzmann_constant/5DIejyR3">Boltzmann constant</a>), the eigenvalues of ρ A {\displaystyle \rho _{A}} are λ n = ( 1 − e − ω / k B T ) e − n ω / k B T {\displaystyle \lambda _{n}=(1-e^{-\omega /k_{B}T})e^{-n\omega /k_{B}T}} for nonnegative integers n {\displaystyle n} . The Von Neumann Entropy is thus </p> − ∑ n λ n ln ( λ n ) = ω / k B T e ω / k B T − 1 − ln ( 1 − e − ω / k B T ) {\displaystyle -\sum _{n}\lambda _{n}\ln(\lambda _{n})={\frac {\omega /k_{B}T}{e^{\omega /k_{B}T}-1}}-\ln(1-e^{-\omega /k_{B}T})} . <p>Similarly the Renyi entropy S α ( ρ A ) = ( 1 − e − ω / k B T ) α 1 − e − α ω / k B T / ( 1 − α ) {\displaystyle S_{\alpha }(\rho _{A})={\frac {(1-e^{-\omega /k_{B}T})^{\alpha }}{1-e^{-\alpha \omega /k_{B}T}}}/(1-\alpha )} . </p> <h2 id="area-law-of-bipartite-entanglement-entropy">Area law of bipartite entanglement entropy</h2> <p>A quantum state satisfies an <i>area law</i> if the leading term of the entanglement entropy grows at most proportionally with the boundary between the two partitions. Area laws are remarkably common for ground states of local gapped quantum many-body systems. This has important applications, one such application being that it greatly reduces the complexity of quantum many-body systems. The <a href="/facts/Density_matrix_renormalization_group/06NNgwsH">density matrix renormalization group</a> and <a href="/facts/Matrix_product_state/amohgody">matrix product states</a>, for example, implicitly rely on such area laws.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> <h2 id="further-reading">Further reading</h2> <ul><li>Dominik J (2009). "Entropy of Entanglement". In Greenberger D, Hentschel K, Weinert F (eds.). <a href="https://archive.org/details/compendiumquantu00gree_195"><i>Compendium of Quantum Physics</i></a>. Springer. pp. <a href="https://archive.org/details/compendiumquantu00gree_195/page/n267">205</a>–209. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-540-70626-7_66">10.1007/978-3-540-70626-7_66</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-70626-7.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Bombelli L, Koul RK, Lee J, Sorkin RD (July 1986). "Quantum source of entropy for black holes". Physical Review. D, Particles and Fields. 34 (2): 373–383. Bibcode:1986PhRvD..34..373B. doi:10.1103/PhysRevD.34.373. PMID 9957156. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Solodukhin SN (December 2011). "Entanglement Entropy of Black Holes". Living Reviews in Relativity. 14 (1): 8. arXiv:1104.3712. Bibcode:2011LRR....14....8S. doi:10.12942/lrr-2011-8. PMC 5255889. PMID 28179833. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5255889" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5255889</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Jacobson T (August 1995). "Thermodynamics of spacetime: The Einstein equation of state". Physical Review Letters. 75 (7): 1260–1263. arXiv:gr-qc/9504004. Bibcode:1995PhRvL..75.1260J. doi:10.1103/PhysRevLett.75.1260. PMID 10060248. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Sakharov AD (1968). "Vacuum Fluctuations in Curved Space and the Theory of Gravitation". Doklady Akademii Nauk SSSR. 177: 64–66. doi:10.1142/9789812815941_0006. ISBN 978-981-02-3606-9. <a href="978-981-02-3606-9" target="_blank">978-981-02-3606-9</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Anonymous (2015-10-23). "Entropy of entanglement". Quantiki. Retrieved 2019-10-17. <a href="https://www.quantiki.org/wiki/entropy-entanglement-0" target="_blank">https://www.quantiki.org/wiki/entropy-entanglement-0</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Srednicki M (1993). "Entropy and area". Physical Review Letters. 71 (5): 666–669. arXiv:hep-th/9303048. Bibcode:1993PhRvL..71..666S. doi:10.1103/PhysRevLett.71.666. PMID 10055336. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Eisert J, Cramer M, Plenio MB (February 2010). "Colloquium: Area laws for the entanglement entropy". Reviews of Modern Physics. 82 (1): 277–306. arXiv:0808.3773. Bibcode:2010RvMP...82..277E. doi:10.1103/RevModPhys.82.277. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> </ol>