A remarkable feature of typical ground states of strongly-correlated many-body systems is that the entanglement entropy is not an extensive quantity. In one dimension, there exists a proof that a finite correlation length sets a constant upper-bound on the entanglement entropy, called the area law. However, the known bound exists only in a hypothetical limit, rendering its physical relevance highly questionable. In this paper, we give a simple proof of the area law for entanglement entropy in one dimension under the condition of exponentially decaying correlations. Our proof dramatically reduces the previously known bound on the entanglement entropy, bringing it, for the first time, into a realistic regime. The proof is composed of several simple and straightforward steps based on elementary quantum information tools. We discuss the underlying physical picture, based on a renormalization-like construction underpinning the proof, which transforms the entanglement entropy of a continuous region into a sum of mutual informations in different length scales and the entanglement entropy at the boundary.A remarkable feature of typical ground states of strongly-correlated many-body systems is that the entanglement entropy is not an extensive quantity. In one dimension, there exists a proof that a finite correlation length sets a constant upper-bound on the entanglement entropy, called the area law. However, the known bound exists only in a hypothetical limit, rendering its physical relevance highly questionable. In this paper, we give a simple proof of the area law for entanglement entropy in one dimension under the condition of exponentially decaying correlations. Our proof dramatically reduces the previously known bound on the entanglement entropy, bringing it, for the first time, into a realistic regime. The proof is composed of several simple and straightforward steps based on elementary quantum information tools. We discuss the underlying physical picture, based on a renormalization-like construction underpinning the proof, which transforms the entanglement entropy of a continuous region into a sum of mutual informations in different length scales and the entanglement entropy at the boundary.

Jun 28, 20170 citations

Bound States in One Dimension

S. Brandt, Hans Dieter Dahmen

Springer eBooksJan 1, 19941 citation

Random MERA states and the tightness of the Brandao-Horodecki entropy bound

Matthew B. Hastings

We construct a random MERA state with a bond dimension that varies with the level of the MERA. This causes the state to exhibit a very different entanglement structure from that usually seen in MERA, with neighboring intervals of length l exhibiting a mutual information proportional to epsilon l for some constant epsilon, up to a length scale exponentially large in epsilon. We express the entropy of a random MERA in terms of sums over cuts through the MERA network, with the entropy in this case controlled by the cut minimizing bond dimensions cut through. One motivation for this construction is to investigate the tightness of the Brandao-Horodecki[8] entropy bound relating entanglement to correlation decay. Using the random MERA, we show that at least part of the proof is tight: there do exist states with the required property of having linear mutual information between neighboring intervals at all length scales. We conjecture that this state has exponential correlation decay and that it demonstrates that the Brandao-Horodecki bound is tight (at least up to constant factors), and we provide some numerical evidence for this as well as a sketch of how a proof of correlation decay might proceed.We construct a random MERA state with a bond dimension that varies with the level of the MERA. This causes the state to exhibit a very different entanglement structure from that usually seen in MERA, with neighboring intervals of length l exhibiting a mutual information proportional to epsilon l for some constant epsilon, up to a length scale exponentially large in epsilon. We express the entropy of a random MERA in terms of sums over cuts through the MERA network, with the entropy in this case controlled by the cut minimizing bond dimensions cut through. One motivation for this construction is to investigate the tightness of the Brandao-Horodecki[8] entropy bound relating entanglement to correlation decay. Using the random MERA, we show that at least part of the proof is tight: there do exist states with the required property of having linear mutual information between neighboring intervals at all length scales. We conjecture that this state has exponential correlation decay and that it demonstrates that the Brandao-Horodecki bound is tight (at least up to constant factors), and we provide some numerical evidence for this as well as a sketch of how a proof of correlation decay might proceed.

arXiv:1505.06468Quantum Information and ComputationOct 1, 20167 citations

Bound States in One Dimension

S. Brandt, Hans Dieter Dahmen

Springer eBooksJan 1, 19950 citations

Bound States in One Dimension

S. Brandt, Hans Dieter Dahmen

Springer eBooksJan 1, 19920 citations

From Kraus Operators to the Stinespring Form of Quantum Maps: An Alternative Construction for Infinite Dimensions

Frederik vom Ende

We present an alternative (constructive) proof of the statement that for every completely positive, trace-preserving map $Φ$ there exists an auxiliary Hilbert space $\mathcal K$ in a pure state $|ψ\rangle\langleψ|$ as well as a unitary operator $U$ on system plus environment such that $Φ$ equals $\operatorname{tr}_{\mathcal K}(U((\cdot)\otimes|ψ\rangle\langleψ|)U^*)$ . The main tool of our proof is Sz.-Nagy's dilation theorem applied to isometries defined on a subspace. In our construction, the environment consists of a system of dimension "Kraus rank of $Φ$ " together with a qubit, the latter only acting as a catalyst. In contrast, the original proof of Hellwig & Kraus given in the 70s yields an auxiliary system of dimension "Kraus rank plus one". We conclude by providing an example which illustrates how the constructions differ from each other.We present an alternative (constructive) proof of the statement that for every completely positive, trace-preserving map $Φ$ there exists an auxiliary Hilbert space $\mathcal K$ in a pure state $|ψ\rangle\langleψ|$ as well as a unitary operator $U$ on system plus environment such that $Φ$ equals $\operatorname{tr}_{\mathcal K}(U((\cdot)\otimes|ψ\rangle\langleψ|)U^*)$. The main tool of our proof is Sz.-Nagy's dilation theorem applied to isometries defined on a subspace. In our construction, the environment consists of a system of dimension "Kraus rank of $Φ$" together with a qubit, the latter only acting as a catalyst. In contrast, the original proof of Hellwig & Kraus given in the 70s yields an auxiliary system of dimension "Kraus rank plus one". We conclude by providing an example which illustrates how the constructions differ from each other.

arXiv:2301.05488Jan 13, 20230 citations

We present an alternative (constructive) proof of the statement that for every completely positive, trace-preserving map

Φ

there exists an auxiliary Hilbert space

\mathcal K

in a pure state

|ψ\rangle\langleψ|

as well as a unitary operator

U

on system plus environment such that

Φ

equals

\operatorname{tr}_{\mathcal K}(U((\cdot)\otimes|ψ\rangle\langleψ|)U^*)

. The main tool of our proof is Sz.-Nagy's dilation theorem applied to isometries defined on a subspace. In our construction, the environment consists of a system of dimension "Kraus rank of

Φ

" together with a qubit, the latter only acting as a catalyst. In contrast, the original proof of Hellwig & Kraus given in the 70s yields an auxiliary system of dimension "Kraus rank plus one". We conclude by providing an example which illustrates how the constructions differ from each other.