This paper is generally concerned with understanding how the uncertainty principle arises in formulations of quantum mechanics, such as the decoherent histories approach, whose central goal is the assignment of probabilities to histories. We first consider histories characterized by position or momentum projections at two moments of time. Both exact and approximate (Gaussian) projections are studied. Shannon's information is used as a measure of the uncertainty expressed in the probabilities for these histories. We derive a number of inequalities in which the uncertainty principle is expressed as a lower bound on the information of phase space distributions derived from the probabilities for two-time histories. We go on to consider histories characterized by position samplings at $n$ moments of time. We derive a lower bound on the information of the joint probability for $n$ position samplings. Similar bounds are derived for histories characterized by samplings of other variables. All lower bounds on the information of histories have the general form $\mathrm{ln}(\frac{{V}_{H}}{{V}_{S}})$ , where ${V}_{H}$ is a volume element of history space, which we define, and ${V}_{S}$ is the volume of that space probed by the projections. We thus obtain a concise and general form of the uncertainty principle referring directly to the histories description of the system, and making no reference to notions of phase space.This paper is generally concerned with understanding how the uncertainty principle arises in formulations of quantum mechanics, such as the decoherent histories approach, whose central goal is the assignment of probabilities to histories. We first consider histories characterized by position or momentum projections at two moments of time. Both exact and approximate (Gaussian) projections are studied. Shannon's information is used as a measure of the uncertainty expressed in the probabilities for these histories. We derive a number of inequalities in which the uncertainty principle is expressed as a lower bound on the information of phase space distributions derived from the probabilities for two-time histories. We go on to consider histories characterized by position samplings at $n$ moments of time. We derive a lower bound on the information of the joint probability for $n$ position samplings. Similar bounds are derived for histories characterized by samplings of other variables. All lower bounds on the information of histories have the general form $\mathrm{ln}(\frac{{V}_{H}}{{V}_{S}})$, where ${V}_{H}$ is a volume element of history space, which we define, and ${V}_{S}$ is the volume of that space probed by the projections. We thus obtain a concise and general form of the uncertainty principle referring directly to the histories description of the system, and making no reference to notions of phase space.

arXiv:gr-qc/9304039Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsSep 15, 199342 citations

Extremal elements of a sublattice of the majorization lattice and approximate majorization

C Massri, G Bellomo, F Holik +1 more

Abstract Given a probability vector x with its components sorted in non-increasing order, we consider the closed ball B ϵ p ( x ) with p ⩾ 1 formed by the probability vectors whose ℓ p -norm distance to the center x is less than or equal to a radius ϵ . Here, we provide an order-theoretic characterization of these balls by using the majorization partial order. Unlike the case p = 1 previously discussed in the literature, we find that the extremal probability vectors, in general, do not exist for the closed balls B ϵ p ( x ) with 1 < p < ∞. On the other hand, we show that B ϵ ∞ ( x ) is a complete sublattice of the majorization lattice. As a consequence, this ball also has extremal elements. In addition, we give an explicit characterization of those extremal elements in terms of the radius and the center of the ball. This allows us to introduce some notions of approximate majorization and discuss its relation with previous results of approximate majorization given in terms of the ℓ 1 -norm. Finally, we apply our results to the problem of approximate conversion of resources within the framework of quantum resource theory of nonuniformity.Abstract Given a probability vector x with its components sorted in non-increasing order, we consider the closed ball B ϵ p ( x ) with p ⩾ 1 formed by the probability vectors whose ℓ p -norm distance to the center x is less than or equal to a radius ϵ . Here, we provide an order-theoretic characterization of these balls by using the majorization partial order. Unlike the case p = 1 previously discussed in the literature, we find that the extremal probability vectors, in general, do not exist for the closed balls B ϵ p ( x ) with 1 < p < ∞. On the other hand, we show that B ϵ ∞ ( x ) is a complete sublattice of the majorization lattice. As a consequence, this ball also has extremal elements. In addition, we give an explicit characterization of those extremal elements in terms of the radius and the center of the ball. This allows us to introduce some notions of approximate majorization and discuss its relation with previous results of approximate majorization given in terms of the ℓ 1 -norm. Finally, we apply our results to the problem of approximate conversion of resources within the framework of quantum resource theory of nonuniformity.

arXiv:2001.08766Journal of Physics A Mathematical and TheoreticalApr 3, 20205 citations

Analysis of quantum probability theory. I

James van Aken

Journal of Philosophical LogicAug 1, 19856 citations

Chiral Central Charge from Real-Space Twist Operator Correlator

Yunlong Zang

Recent years have witnessed substantial progress in the entanglement-based characterization of quantum phases of matter, with growing interest in real-space formulas due to their theoretical and experimental relevance. For two-dimensional gapped quantum systems, we propose a universal real-space formula for the chiral central charge based on the twist-operator correlator $Z_{g,h}=\langleψ^{(N)}\rvert\mathcal{T}_A(g)\mathcal{T}_B(h)\mathcal{T}_C(g)\mathcal{T}_C(h)\lvertψ^{(N)}\rangle$ , where $\lvertψ^{(N)}\rangle=\lvertψ\rangle^{\otimes N}$ is the $N$ -copy state and $g,h\in S_N$ are permutations. The twist operator $\mathcal{T}_R(g)$ permutes the $N$ replicas within region $R$ according to $g$ . For pairs $(g,h)$ satisfying the spherical condition, we show that the phase of the correlator is determined by the chiral central charge $\mathfrak{c}_{-}$ through $Z_{g,h}/\left\lvert Z_{g,h}\right\rvert=(θ_gθ_h/θ_{gh})^{\mathfrak{c}_{-}}$ , where $θ_g^{\mathfrak{c}_{-}}$ is the topological spin of the twist defect labeled by $g\in S_N$ . The construction applies to both bosonic and fermionic systems. We derive the formula within a field-theoretic framework under suitable assumptions and further test it numerically in microscopic lattice models. Our results establish a universal connection between real-space entanglement structures and the chiral central charge, providing a systematic framework for characterizing chiral topological phases directly on the lattice.Recent years have witnessed substantial progress in the entanglement-based characterization of quantum phases of matter, with growing interest in real-space formulas due to their theoretical and experimental relevance. For two-dimensional gapped quantum systems, we propose a universal real-space formula for the chiral central charge based on the twist-operator correlator $Z_{g,h}=\langleψ^{(N)}\rvert\mathcal{T}_A(g)\mathcal{T}_B(h)\mathcal{T}_C(g)\mathcal{T}_C(h)\lvertψ^{(N)}\rangle$, where $\lvertψ^{(N)}\rangle=\lvertψ\rangle^{\otimes N}$ is the $N$-copy state and $g,h\in S_N$ are permutations. The twist operator $\mathcal{T}_R(g)$ permutes the $N$ replicas within region $R$ according to $g$. For pairs $(g,h)$ satisfying the spherical condition, we show that the phase of the correlator is determined by the chiral central charge $\mathfrak{c}_{-}$ through $Z_{g,h}/\left\lvert Z_{g,h}\right\rvert=(θ_gθ_h/θ_{gh})^{\mathfrak{c}_{-}}$, where $θ_g^{\mathfrak{c}_{-}}$ is the topological spin of the twist defect labeled by $g\in S_N$. The construction applies to both bosonic and fermionic systems. We derive the formula within a field-theoretic framework under suitable assumptions and further test it numerically in microscopic lattice models. Our results establish a universal connection between real-space entanglement structures and the chiral central charge, providing a systematic framework for characterizing chiral topological phases directly on the lattice.

arXiv:2610.08788Oct 6, 20260 citations

Analysis of quantum probability theory. II

James van Aken

Journal of Philosophical LogicAug 1, 19861 citation

n

moments of time. We derive a lower bound on the information of the joint probability for

n

position samplings. Similar bounds are derived for histories characterized by samplings of other variables. All lower bounds on the information of histories have the general form

\mathrm{ln}(\frac{{V}_{H}}{{V}_{S}})

, where

{V}_{H}

is a volume element of history space, which we define, and

{V}_{S}

is the volume of that space probed by the projections. We thus obtain a concise and general form of the uncertainty principle referring directly to the histories description of the system, and making no reference to notions of phase space.

Z_{g,h}=\langleψ^{(N)}\rvert\mathcal{T}_A(g)\mathcal{T}_B(h)\mathcal{T}_C(g)\mathcal{T}_C(h)\lvertψ^{(N)}\rangle

, where

\lvertψ^{(N)}\rangle=\lvertψ\rangle^{\otimes N}

is the

N

-copy state and

g,h\in S_N

are permutations. The twist operator

\mathcal{T}_R(g)

permutes the

N

replicas within region

R

according to

g

. For pairs

(g,h)

satisfying the spherical condition, we show that the phase of the correlator is determined by the chiral central charge

\mathfrak{c}_{-}

through

Z_{g,h}/\left\lvert Z_{g,h}\right\rvert=(θ_gθ_h/θ_{gh})^{\mathfrak{c}_{-}}

, where

θ_g^{\mathfrak{c}_{-}}

is the topological spin of the twist defect labeled by

g\in S_N

. The construction applies to both bosonic and fermionic systems. We derive the formula within a field-theoretic framework under suitable assumptions and further test it numerically in microscopic lattice models. Our results establish a universal connection between real-space entanglement structures and the chiral central charge, providing a systematic framework for characterizing chiral topological phases directly on the lattice.