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arXiv:1810.12845

Constraints on Multipartite Quantum Entropies

Christian Majenz

Published Oct 30, 2018• 2 citations

Abstract

The von Neumann entropy plays a vital role in quantum information theory. The von Neumann entropy determines, e.g., the capacities of quantum channels. Also, entropies of composite quantum systems are important for future quantum networks, and their characterization is related to the quantum marginal problem. Furthermore, they play a role in quantum thermodynamics. In this thesis the set of quantum entropies of multipartite quantum systems is studied. The problem of its characterization is not new -- however, progress has been sparse, indicating that the problem might be hard and that new methods might be needed. Here, a variety of different and complementary approaches are taken. First, I look at global properties. It is known that the von Neumann entropy region -- just like its classical counterpart -- forms a convex cone. I describe the symmetries of this cone and highlight geometric similarities and differences to the classical entropy cone. In a different approach, I utilize the local geometric properties of extremal rays of a cone. I show that quantum states whose entropy lies on such an extremal ray of the quantum entropy cone have a very simple structure. As the set of all quantum states is very complicated, I look at a simple subset called stabilizer states. I improve on previously known results by showing that under a technical condition on the local dimension, entropies of stabilizer states respect an additional class of information inequalities that is valid for random variables from linear codes. In a last approach I find a representation-theoretic formulation of the classical marginal problem simplifying the comparison with its quantum mechanical counterpart. This novel correspondence yields a simplified formulation of the group characterization of classical entropies (IEEE Trans. Inf. Theory, 48(7):1992-1995, 2002) in purely combinatorial terms.

DOI: 10.48550/arxiv.1810.12845

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The von Neumann entropy plays a vital role in quantum information theory. The von Neumann entropy determines, e.g., the capacities of quantum channels. Also, entropies of composite quantum systems are important for future quantum networks, and their characterization is related to the quantum marginal problem. Furthermore, they play a role in quantum thermodynamics. In this thesis the set of quantum entropies of multipartite quantum systems is studied. The problem of its characterization is not new -- however, progress has been sparse, indicating that the problem might be hard and that new methods might be needed. Here, a variety of different and complementary approaches are taken. First, I look at global properties. It is known that the von Neumann entropy region -- just like its classical counterpart -- forms a convex cone. I describe the symmetries of this cone and highlight geometric similarities and differences to the classical entropy cone. In a different approach, I utilize the local geometric properties of extremal rays of a cone. I show that quantum states whose entropy lies on such an extremal ray of the quantum entropy cone have a very simple structure. As the set of all quantum states is very complicated, I look at a simple subset called stabilizer states. I improve on previously known results by showing that under a technical condition on the local dimension, entropies of stabilizer states respect an additional class of information inequalities that is valid for random variables from linear codes. In a last approach I find a representation-theoretic formulation of the classical marginal problem simplifying the comparison with its quantum mechanical counterpart. This novel correspondence yields a simplified formulation of the group characterization of classical entropies (IEEE Trans. Inf. Theory, 48(7):1992-1995, 2002) in purely combinatorial terms.

Relations among von Neumann entropies of different parts of an N-partite quantum system have direct impact on our understanding of diverse situations ranging from spin systems to quantum coding theory and black holes. Best formulated in terms of the set Σ_{N}^{*} of possible vectors comprising the entropies of the whole and its parts, the famous strong subaddivity inequality constrains its closure Σ[over ¯]_{N}^{*}, which is a convex cone. Further homogeneous constrained inequalities are also known. In this Letter we provide (nonhomogeneous) inequalities that constrain Σ_{N}^{*} near the apex (the vector of zero entropies) of Σ[over ¯]_{N}^{*}, in particular showing that Σ_{N}^{*} is not a cone for N≥3. Our inequalities apply to vectors with certain entropy constraints saturated and, in particular, they show that while it is always possible to upscale an entropy vector to arbitrary integer multiples it is not always possible to downscale it to arbitrarily small size, thus answering a question posed by Winter.

Information inequalities govern the ultimate limitations in information theory and as such play a pivotal role in characterizing what values the entropy of multipartite states can take. Proving an information inequality, however, quickly becomes arduous when the number of involved parties increases. For classical systems, [Yeung, IEEE Trans. Inf. Theory (1997)] proposed a framework to prove Shannon-type inequalities via linear programming. Here, we derive an analogous framework for quantum systems, based on the strong sub-additivity and weak monotonicity inequalities for the von-Neumann entropy. Importantly, this also allows us to handle constrained inequalities, which - in the classical case - served as a crucial tool in proving the existence of non-standard, so-called non-Shannon-type inequalities [Zhang & Yeung, IEEE Trans. Inf. Theory (1998)]. Our main contribution is the Python package qITIP, for which we present the theory and demonstrate its capabilities with several illustrative examples.

The idea that information can be localized is pervasive in physics. In thermodynamics, entropy flows from one reservoir to another, and in quantum gravity, the entropy content of some spatial regions is bounded. In quantum information theory, information is transmitted from one place to another, and entanglement can provide an advantage when there are constraints on such transmission. Shannon entropy quantifies localizable information in the sense that marginalization defines an outer measure on the set of parts of a classical multipartite system. In contrast, von Neumann entropy does not quantify localizable information in this sense. However, a variant of von Neumann entropy, which originates in noncommutative geometry, does. We prove that this adjusted von Neumann entropy is the minimum quantum entropy that quantifies localizable information. We also prove that this is the unique quantum entropy that characterizes Bell states in terms of redundant information, directly generalizing the classical case. We work with finite-dimensional

C^*

-algebras throughout, modeling finite quantum systems that may have superselection sectors.

We investigate whether the presence or absence of correlations between subsystems of an N-partite quantum system is solely constrained by the non-negativity and monotonicity of mutual information. We argue that this relatively simple question is in fact very deep because it is sensitive to the structure of the set of N-partite states. It can be informed by inequalities satisfied by the von Neumann entropy, but has the advantage of being more tractable. We exemplify this by deriving the explicit solution for N=4, despite having limited knowledge of the entropic inequalities. Furthermore, we describe how this question can be tailored to the analysis of more specialized classes of states such as classical probability distributions, stabilizer states, and geometric states in the holographic gauge/gravity duality.

Entanglement entropies of multipartite pure states satisfy linear inequalities; for stabilizer states, the backbone of quantum error correction, the achievable entropy vectors generate a convex cone. We show that, after a linear lift, the closure of this cone is exactly the self-dual slice of a classical object on one more element than there are parties: the closed cone generated by the rank functions of arrangements of subspaces over

\mathrm{GF}(d)

for a prime local dimension

d

, and of subgroups of abelian groups of exponent dividing

d

in general. This correspondence brings two features of rank inequalities into play. First, the purifier acts as an extra variable: already for five parties the stabilizer cone is strictly smaller than the quantum linear rank cone, with which earlier evidence suggested it might coincide; an explicit six-variable inequality witnesses the gap. Second, the characteristic of the field enters, so the cone depends on the local dimension. It does not for up to four parties. With enough parties, the cone of

\mathbb{Z}_d

qudits is contained in that of

\mathbb{Z}_{d'}

qudits if and only if

d

divides

d'

, and already for six parties the logical zero state of Steane's code lies outside the stabilizer cone of every odd local dimension. For this last result we show that the seven-variable characteristic-dependent inequalities we checked all follow from Shannon inequalities on the self-dual slice, and instead transfer one of them through a family of subgroups derived from the Fano plane.

We show via explicit construction that all the extreme rays of both the three-party quantum entropy cone and the four-party stabilizer entropy cone can be obtained from subsystem coarse grainings of specific higher-party quantum states, namely, extreme states characterized by saturating a (nontrivial) maximal set of instances of subadditivity. This suggests that the study of the ``subadditivity cone,'' and the set of its extreme rays realizable in quantum mechanics, provides a powerful tool for deriving inner bounds for the quantum and stabilizer entropy cones, as well as constraints on new inequalities for the von Neumann entropy.