arXiv:1309.7607

Translation invariant pure state on $\otimes_{\IZ}/!M_d(\IC)$ and Haag duality

Anilesh Mohari

Published Sep 29, 2013• 0 citations

Abstract

We prove Haag duality property of any translation invariant pure state on $\clb = \otimes_{\IZ}/!M_d(\IC), \;d \ge 2$, where $/!M_d(\IC)$ is the set of $d \times d$ dimensional matrices over the field of complex numbers. We also prove a necessary and sufficient condition for a translation invariant factor state to be pure on $\clb$.

DOI: 10.48550/arxiv.1309.7607

Related papers

Translation invariant pure state and its split property

Anilesh Mohari

arXiv:0904.2104Apr 14, 20090 citations

ON HAAG DUALITY FOR PURE STATES OF QUANTUM SPIN CHAINS

Michael Keyl, Taku Matsui, Dirk-M. Schlingemann +1 more

In this note, we consider quantum spin chains and their translationally invariant pure states. We prove Haag duality for quasilocal observables localized in semi-infinite intervals (-∞ , 0] and [1, ∞) when the von Neumann algebra generated by observables localized in [0, ∞) is non-type I.

arXiv:math-ph/0703013Reviews in Mathematical PhysicsJul 1, 20087 citations

An entropic characterization of Haag duality

Ruizhi Liu, Lauritz van Luijk

quant-pharXiv:2609.13550Sep 15, 20260 citations

Translation invariant pure states and its split property

Anilesh Mohari, Cit Campus

Apr 14, 20094 citations

Essential Duality and Quantitative Channel Composability in Algebraic Quantum Field Theory

Hassan Nasreddine

Let $O\mapsto A(O)$ be a local Haag--Kastler net and let $A^d(O)=A(O')'$ be its dual net. Normal unital completely positive maps on $B(H)$ that fix $A(O')$ pointwise are exactly the channels with Kraus operators in $A^d(O)$. Thus Haag duality at $O$ is equivalent to equality with the $A(O)$-inner channels, while essential duality is equivalent to order-independence of the corresponding causal-complement channel classes at spacelike separation. For self-adjoint generators $a\in M$ and $b\in N$, the small-parameter order defect has coefficient $\operatorname{diam}σ(-i[a,b])$. Uniform optimization gives $Γ(M,N)=2Δ_{\mathrm{sa}}(M,N)$, and reversible inner-channel cb balls recover the same invariant through $\lim_{\varepsilon\downarrow0}Ω^{\mathrm{rev}}_\varepsilon(M,N)/\varepsilon^2=Γ(M,N)/4$. The full inner-channel class admits a normalized coefficient with the same commutativity zero set. With a positive reference observable $G$, the mean-input-energy-constrained coefficient $Γ_{G,E}$ has the same zero criterion and increases to $Γ$ as $E\to\infty$, while no model-independent recovery rate follows from the general von Neumann-algebraic hypotheses. For bosonic second-quantization nets the spacelike defect is either $0$ or $4$; generalized free-field examples realize both branches. In the translation-covariant case with positive one-particle time generator, failure of essential duality admits extremal bounded witness flows that are energy-limited relative to the second-quantized time-translation energy.

quant-pharXiv:2605.00075May 1, 20260 citations

Translation invariant pure state on $\clb=\otimes_{\IZ}M_d(\IC)$ and its split property

Anilesh Mohari

We prove that a real lattice symmetric reflection positive translation invariant pure state of $\clb=\otimes_{\IZ}M_d(\IC)$ is a split state if its two points spatial correlations functions decay exponentially.

arXiv:1310.1886Oct 7, 20130 citations

arXiv:1309.7607

Translation invariant pure state on $\otimes_{\IZ}/!M_d(\IC)$ and Haag duality

Anilesh Mohari

Published Sep 29, 2013• 0 citations

arXiv HTML PDF

Abstract

We prove Haag duality property of any translation invariant pure state on

\clb = \otimes_{\IZ}/!M_d(\IC), \;d \ge 2

, where

/!M_d(\IC)

is the set of

d \times d

dimensional matrices over the field of complex numbers. We also prove a necessary and sufficient condition for a translation invariant factor state to be pure on

\clb

DOI: 10.48550/arxiv.1309.7607

Related papers

Translation invariant pure state and its split property

Anilesh Mohari

We prove Haag duality property of any translation invariant pure state on $\clb = \otimes_{\IZ}M_d(C), \;d \ge 2$, where $M_d(C)$ is the set of $d \times d$ dimensional matrices over field of complex numbers. We also prove a necessary and sufficient condition for a translation invariant factor state to be pure on $\clb$. This result makes it possible to study such a pure state with additional symmetry. We prove that exponentially decaying two point spacial correlation function of a real lattice symmetric reflection positive translation invariant pure state is a split state. Further there exists no translation invariant pure state on $\clb$ that is real, lattice symmetric, refection positive and $su(2)$ invariant when $d$ is an even integer. This in particular says that Heisenberg iso-spin anti-ferromagnets model for 1/2-odd integer spin degrees of freedom admits spontaneous symmetry breaking at it's ground statesWe prove Haag duality property of any translation invariant pure state on $\clb = \otimes_{\IZ}M_d(C), \;d \ge 2$, where $M_d(C)$ is the set of $d \times d$ dimensional matrices over field of complex numbers. We also prove a necessary and sufficient condition for a translation invariant factor state to be pure on $\clb$. This result makes it possible to study such a pure state with additional symmetry. We prove that exponentially decaying two point spacial correlation function of a real lattice symmetric reflection positive translation invariant pure state is a split state. Further there exists no translation invariant pure state on $\clb$ that is real, lattice symmetric, refection positive and $su(2)$ invariant when $d$ is an even integer. This in particular says that Heisenberg iso-spin anti-ferromagnets model for 1/2-odd integer spin degrees of freedom admits spontaneous symmetry breaking at it's ground states

arXiv:0904.2104Apr 14, 20090 citations

ON HAAG DUALITY FOR PURE STATES OF QUANTUM SPIN CHAINS

Michael Keyl, Taku Matsui, Dirk-M. Schlingemann +1 more

arXiv:math-ph/0703013Reviews in Mathematical PhysicsJul 1, 20087 citations

An entropic characterization of Haag duality

Ruizhi Liu, Lauritz van Luijk

Motivated by Haag duality in quantum spin systems, we show that Haag duality for a pair of commuting factors generated by increasing sequences of finite-dimensional matrix algebras is equivalent to the asymptotic vanishing of a suitable conditional mutual information. For spin systems, let an annulus separate a finite region from the exterior. Haag duality then holds for a region $A$ if and only if the mutual information of the inner region and the exterior, conditioned on the part of $A$ inside the annulus, vanishes as the outer radius becomes large. As a corollary, assuming a strict area law with subleading corrections, Haag duality holds for single cones, and it holds for unions of two or more disjoint cones precisely when the topological entanglement entropy vanishes. For translation-invariant pure states on spin chains, half-chain Haag duality is equivalent to vanishing entropy density.Motivated by Haag duality in quantum spin systems, we show that Haag duality for a pair of commuting factors generated by increasing sequences of finite-dimensional matrix algebras is equivalent to the asymptotic vanishing of a suitable conditional mutual information. For spin systems, let an annulus separate a finite region from the exterior. Haag duality then holds for a region $A$ if and only if the mutual information of the inner region and the exterior, conditioned on the part of $A$ inside the annulus, vanishes as the outer radius becomes large. As a corollary, assuming a strict area law with subleading corrections, Haag duality holds for single cones, and it holds for unions of two or more disjoint cones precisely when the topological entanglement entropy vanishes. For translation-invariant pure states on spin chains, half-chain Haag duality is equivalent to vanishing entropy density.

quant-pharXiv:2609.13550Sep 15, 20260 citations

Translation invariant pure states and its split property

Anilesh Mohari, Cit Campus

A translation invariant state in quantum spin chain is determined uniquely upto isomorphism by a Markov map on the support projection of an associated Cuntz’s state. We prove that Kolmogorov’s property of the Markov map is a necessary and sufficient condition for such a state to be pure. Kolmogorov’s property naturally give rise to a Mackey’s system of imprimitivity for the group of integers. A duality argument originated from non-commutative probability theory is employed to prove an elegant alternative necessary and sufficient condition for pureness. Main result of this theory made it possible to prove Haag duality property of any translation invariant lattice symmetric pure state. Further such a real state is split if special correlation function decays exponentially. The last statement proves T Matsui’s conjecture on split property for a translation invariant real lattice symmetric pure state.A translation invariant state in quantum spin chain is determined uniquely upto isomorphism by a Markov map on the support projection of an associated Cuntz’s state. We prove that Kolmogorov’s property of the Markov map is a necessary and sufficient condition for such a state to be pure. Kolmogorov’s property naturally give rise to a Mackey’s system of imprimitivity for the group of integers. A duality argument originated from non-commutative probability theory is employed to prove an elegant alternative necessary and sufficient condition for pureness. Main result of this theory made it possible to prove Haag duality property of any translation invariant lattice symmetric pure state. Further such a real state is split if special correlation function decays exponentially. The last statement proves T Matsui’s conjecture on split property for a translation invariant real lattice symmetric pure state.

Apr 14, 20094 citations

Essential Duality and Quantitative Channel Composability in Algebraic Quantum Field Theory

Hassan Nasreddine

Let $O\mapsto A(O)$ be a local Haag--Kastler net and let $A^d(O)=A(O')'$ be its dual net. Normal unital completely positive maps on $B(H)$ that fix $A(O')$ pointwise are exactly the channels with Kraus operators in $A^d(O)$ . Thus Haag duality at $O$ is equivalent to equality with the $A(O)$ -inner channels, while essential duality is equivalent to order-independence of the corresponding causal-complement channel classes at spacelike separation. For self-adjoint generators $a\in M$ and $b\in N$ , the small-parameter order defect has coefficient $\operatorname{diam}σ(-i[a,b])$ . Uniform optimization gives $Γ(M,N)=2Δ_{\mathrm{sa}}(M,N)$ , and reversible inner-channel cb balls recover the same invariant through $\lim_{\varepsilon\downarrow0}Ω^{\mathrm{rev}}_\varepsilon(M,N)/\varepsilon^2=Γ(M,N)/4$ . The full inner-channel class admits a normalized coefficient with the same commutativity zero set. With a positive reference observable $G$ , the mean-input-energy-constrained coefficient $Γ_{G,E}$ has the same zero criterion and increases to $Γ$ as $E\to\infty$ , while no model-independent recovery rate follows from the general von Neumann-algebraic hypotheses. For bosonic second-quantization nets the spacelike defect is either $0$ or $4$ ; generalized free-field examples realize both branches. In the translation-covariant case with positive one-particle time generator, failure of essential duality admits extremal bounded witness flows that are energy-limited relative to the second-quantized time-translation energy.Let $O\mapsto A(O)$ be a local Haag--Kastler net and let $A^d(O)=A(O')'$ be its dual net. Normal unital completely positive maps on $B(H)$ that fix $A(O')$ pointwise are exactly the channels with Kraus operators in $A^d(O)$. Thus Haag duality at $O$ is equivalent to equality with the $A(O)$-inner channels, while essential duality is equivalent to order-independence of the corresponding causal-complement channel classes at spacelike separation. For self-adjoint generators $a\in M$ and $b\in N$, the small-parameter order defect has coefficient $\operatorname{diam}σ(-i[a,b])$. Uniform optimization gives $Γ(M,N)=2Δ_{\mathrm{sa}}(M,N)$, and reversible inner-channel cb balls recover the same invariant through $\lim_{\varepsilon\downarrow0}Ω^{\mathrm{rev}}_\varepsilon(M,N)/\varepsilon^2=Γ(M,N)/4$. The full inner-channel class admits a normalized coefficient with the same commutativity zero set. With a positive reference observable $G$, the mean-input-energy-constrained coefficient $Γ_{G,E}$ has the same zero criterion and increases to $Γ$ as $E\to\infty$, while no model-independent recovery rate follows from the general von Neumann-algebraic hypotheses. For bosonic second-quantization nets the spacelike defect is either $0$ or $4$; generalized free-field examples realize both branches. In the translation-covariant case with positive one-particle time generator, failure of essential duality admits extremal bounded witness flows that are energy-limited relative to the second-quantized time-translation energy.

quant-pharXiv:2605.00075May 1, 20260 citations

Translation invariant pure state on $\clb=\otimes_{\IZ}M_d(\IC)$ and its split property

Anilesh Mohari

arXiv:1310.1886Oct 7, 20130 citations

We prove Haag duality property of any translation invariant pure state on

\clb = \otimes_{\IZ}/!M_d(\IC), \;d \ge 2

, where

/!M_d(\IC)

is the set of

d \times d

dimensional matrices over the field of complex numbers. We also prove a necessary and sufficient condition for a translation invariant factor state to be pure on

\clb

We prove Haag duality property of any translation invariant pure state on $\clb = \otimes_{\IZ}M_d(C), \;d \ge 2$, where

M_d(C)

is the set of

d \times d

dimensional matrices over field of complex numbers. We also prove a necessary and sufficient condition for a translation invariant factor state to be pure on $\clb$. This result makes it possible to study such a pure state with additional symmetry. We prove that exponentially decaying two point spacial correlation function of a real lattice symmetric reflection positive translation invariant pure state is a split state. Further there exists no translation invariant pure state on $\clb$ that is real, lattice symmetric, refection positive and

su(2)

invariant when

d

is an even integer. This in particular says that Heisenberg iso-spin anti-ferromagnets model for 1/2-odd integer spin degrees of freedom admits spontaneous symmetry breaking at it's ground states

A

if and only if the mutual information of the inner region and the exterior, conditioned on the part of

A

inside the annulus, vanishes as the outer radius becomes large. As a corollary, assuming a strict area law with subleading corrections, Haag duality holds for single cones, and it holds for unions of two or more disjoint cones precisely when the topological entanglement entropy vanishes. For translation-invariant pure states on spin chains, half-chain Haag duality is equivalent to vanishing entropy density.

Let

O\mapsto A(O)

be a local Haag--Kastler net and let

A^d(O)=A(O')'

be its dual net. Normal unital completely positive maps on

B(H)

that fix

A(O')

pointwise are exactly the channels with Kraus operators in

A^d(O)

. Thus Haag duality at

O

is equivalent to equality with the

A(O)

-inner channels, while essential duality is equivalent to order-independence of the corresponding causal-complement channel classes at spacelike separation. For self-adjoint generators

a\in M

and

b\in N

, the small-parameter order defect has coefficient

\operatorname{diam}σ(-i[a,b])

. Uniform optimization gives

Γ(M,N)=2Δ_{\mathrm{sa}}(M,N)

, and reversible inner-channel cb balls recover the same invariant through

\lim_{\varepsilon\downarrow0}Ω^{\mathrm{rev}}_\varepsilon(M,N)/\varepsilon^2=Γ(M,N)/4

. The full inner-channel class admits a normalized coefficient with the same commutativity zero set. With a positive reference observable

G

, the mean-input-energy-constrained coefficient

Γ_{G,E}

has the same zero criterion and increases to

Γ

E\to\infty

, while no model-independent recovery rate follows from the general von Neumann-algebraic hypotheses. For bosonic second-quantization nets the spacelike defect is either

0

4

; generalized free-field examples realize both branches. In the translation-covariant case with positive one-particle time generator, failure of essential duality admits extremal bounded witness flows that are energy-limited relative to the second-quantized time-translation energy.