On the Lattice Structure of Probability Spaces in Quantum Mechanics

Abstract

Let $\mathcal{C}$ be the set of all possible quantum states. We study the convex subsets of $\mathcal{C}$ with attention focused on the lattice theoretical structure of these convex subsets and, as a result, find a framework capable of unifying several aspects of quantum mechanics, including entanglement and Jaynes’ Max-Ent principle. We also encounter links with entanglement witnesses, which leads to a new separability criteria expressed in lattice language. We also provide an extension of a separability criteria based on convex polytopes to the infinite dimensional case and show that it reveals interesting facets concerning the geometrical structure of the convex subsets. It is seen that the above mentioned framework is also capable of generalization to any statistical theory via the so-called convex operational models’ approach. In particular, we show how to extend the geometrical structure underlying entanglement to any statistical model, an extension which may be useful for studying correlations in different generalizations of quantum mechanics.

Appendices

Appendix A: Basic Mathematical Concepts Used in the Text

• 1. A function is surjective (onto) if every possible image is mapped to by at least one argument. In other words, every element in the codomain has non-empty preimage. Equivalently, a function is surjective if its image is equal to its codomain. A surjective function is a surjection.

• 2. A linear functional (also called a one-form or covector) is a linear map from a vector space to its field of scalars K. In general, if V is a vector space over a field K, then a linear functional f is a function from V to K, which is linear. Linear functionals are particularly important in quantum mechanics. Quantum mechanical systems are represented by Hilbert spaces, which are anti-isomorphic to their own dual spaces. A state of a quantum mechanical system can be identified with a linear functional.

• 3. Suppose that K is a field (for example, the real numbers) and V is a vector space over K. If v ₁,…,v _n are vectors and a ₁,…,a _n are scalars, then the linear combination of those vectors with those scalars as coefficients is, of course, $\sum_{i=1}^{n}a_{i}v_{i}$. By restricting the coefficients used in linear combinations, one can define the related concepts of affine combination, conical combination, and convex combination, together with the associated notions of sets closed under these operations. If $\sum_{i=1}^{n} a_{i}=1$, we have an affine combination, its span being an affine subspace while the model space is an hyperplane. If all a _i ≥0, we have instead a conical combination, a convex cone and a quadrant, respectively. Finally, if all a _i ≥0 plus $\sum_{i=1}^{n}a_{i}=1$, we have now a convex combination, a convex set and a simplex, respectively.

• 4. By a σ-algebra one means a collection of sets that satisfy certain properties, used in the definition of measures: it is the collection of sets over which a measure is defined. The concept is important in probability theory, being there interpreted as the collection of events which can be assigned probabilities. Such an algebra, over a set X, is a nonempty collection S of subsets of X (including X itself) that is closed under complementation and countable unions of its members. It is an algebra of sets, completed to include countably infinite operations. The pair (X,S) is also a field of sets, called a measurable space.

• 5. A quotient space (also called an identification space) is, intuitively speaking, the result of identifying certain points of a given space. The points to be identified are specified by an equivalence relation. This is commonly done in order to construct new spaces from given ones. Let (X,τ _X ) be a topological space, and let R be an equivalence relation on X. The quotient space Y=X/R is defined to be the set of equivalence classes of elements of X:

  $$Y= \bigl\{[x]: x \in X\bigr\}=\bigl\{\{v\in X: v R x\}: x \in X\bigr\}, $$

  equipped with the topology where the open sets are defined to be those sets of equivalence classes whose unions are open sets in X. Equivalently, we can define them to be those sets with an open pre-image under the quotient map which sends a point in X to the equivalence class containing it.

• 6. Banach spaces are vector spaces V with a norm ∥.∥ such that every Cauchy sequence (with respect to the metric d(x,y)=∥x−y∥ in V) has a limit in V (with respect to the topology induced by that metric). As for general vector spaces, a Banach space over the real numbers is called a real Banach space, and a Banach space over the complex numbers is called a complex Banach space.

• 7. Algebras: general vector spaces do not possess a multiplication between vectors. A vector space equipped with an additional bilinear operator defining the multiplication of two vectors is an algebra over a field. Many algebras stem from functions on some geometrical object: since functions with values in a field can be multiplied, these entities form algebras.

• 8. In functional analysis, a Banach algebra is an associative algebra A over the real or complex numbers which at the same time is also a Banach space The algebra multiplication and the Banach space norm are required to be related by the following inequality: ∀x,y∈A:∥xy∥ ≤∥x∥∥y∥ (i.e., the norm of the product is less than or equal to the product of the norms). This ensures that the multiplication operation is continuous. This property is found in the real and complex numbers; for instance.

• 9. A C ^∗-algebra is a Banach algebra with an antiautomorphic involution ∗ which satisfies (x ^∗)^∗=x (1); x ^∗ y ^∗=(yx)^∗ (2); x ^∗+y ^∗=(x+y)^∗ (3); and (cx)^∗=c ^∗  x ^∗ (4), where c ^∗ is the complex conjugate of c, and whose norm satisfies ∥xx ^∗∥=∥x∥².

• 10. C ^∗-algebras are an important area of research in functional analysis. An outstanding example is the complex algebra of linear operators on a complex Hilbert space with two additional properties:

  It is a topologically closed set in the norm topology of operators and is closed under the operation of taking adjoints of operators.

  It is generally believed that these algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables, beginning with Werner Heisenberg’s matrix mechanics and developed further by Pascual Jordan circa 1933. Afterwards, John von Neumann established a general framework for them which culminated in papers on rings of operators, considered as a special class of C ^∗-algebras known as von Neumann algebras.

• 11. It is now generally accepted that the description of quantum mechanics in which all self-adjoint operators represent observables is untenable. For this reason, observables are identified to elements of an abstract C ^∗-algebra A (that is one without a distinguished representation as an algebra of operators) and states are positive linear functionals on A. However, by using the GNS construction, we can recover Hilbert spaces which realize A as a subalgebra of operators. Geometrically, a pure state on a C ^∗-algebra A is a state which is an extreme point of the set of all states on A. By properties of the GNS construction these states correspond to irreducible representations of A. The states of the C ^∗-algebra of compact operators $K(\mathcal{H})$ correspond exactly to the density operators and therefore the pure states of $K(\mathcal{H})$ are exactly the pure states in the sense of quantum mechanics. The C ^∗-algebraic formulation can be seen to include both classical and quantum systems. When the system is classical, the algebra of observables become an abelian C ^∗-algebra. In that case the states become probability measures.

• 12. In functional analysis, given a C ^∗-algebra A, the Gelfand-Naimark-Segal (GNS) construction establishes a correspondence between cyclic ∗-representations of A and certain linear functionals on A (called states). The correspondence is shown by an explicit construction of the ∗-representation from the state.

• 13. A ∗-representation of a C ^∗-algebra A on a Hilbert space $\mathcal{H}$ is a mapping π from A into the algebra of bounded operators on $\mathcal{H}$.

• 14. Point-wise convergence is one of various senses in which a sequence of functions can converge to a particular function. Suppose {f _n } is a sequence of functions sharing the same domain and codomain. The sequence {f _n } converges pointwise to f, often written as lim _n→∞ f _n =f point wise iff for every x in the domain one has lim _n→∞ f _n (x)=f(x).

• 15. Every subset Q of a vector space is contained within a smallest convex set (called the convex hull of Q), namely the intersection of all convex sets containing Q.

• 16. A set with a binary relation R on its elements that is reflexive (for all a in the set, aRa), antisymmetric (if aRb and bRa, then a=b) and transitive (if aRb and bRc, then aRc) is described as a partially ordered set or poset.

• 17. Let X be a space. Its dual space X ^∗ consists of all linear functions from X into the base field K which are continuous with respect to the prevailing topology.

• 18. The weak topology on X is the coarsest topology (the topology with the fewest open sets) such that each element of X ^∗ is a continuous function.

• 19. The predual of a space D is a space D′ whose dual space is D. For example, the predual of the space of bounded operators $\mathcal{B}(\mathcal{H})$ is the space of trace class operators.

• 20. The ultraweak topology, also called the weak-∗ topology, on the set $\mathcal{B}(\mathcal{H})$ is the weak-topology obtained from the trace class operators on $\mathcal{H}$. In other words it is the weakest topology such that all elements of the predual are continuous (when considered as functions on $\mathcal{H}$).

• 21. A partially-ordered group is a group (G,+) equipped with a partial order “⊢” that is translation-invariant. That is, “⊢” has the property that, for all a, b, and g in G, if a⊢b then a+g⊢b+g and g+a⊢g+b.

• 22. An element x of G is called positive element if 0⊢x. The set of elements 0⊢x is often denoted with G+, and it is called the positive cone of G. So we have a⊢b if and only if −a+b∈G+.

• 23. For the general group G, the existence of a positive cone specifies an order on G. A group G is a partially-ordered group if and only if there exists a subset J (which is G+) of G such that: 0∈J; if a∈J and b∈J then a+b∈J; if a∈J then −x+a+x∈J for each x of G; if a∈J and −a∈J then a⊢0.

• 24. In linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. There are many different matrix decompositions; each finds use among a particular class of problems. The Cholesky decomposition is applicable to any square, symmetric, positive definite matrix A in the form A=U ^T U, where U is upper triangular with positive diagonal entries. The Cholesky decomposition is a special case of the symmetric LU decomposition, with L=U ^T. The Cholesky decomposition is unique and also applicable for complex hermitian positive definite matrices. The singular value decomposition is applicable to m times n matrix A in the fashion A=UDV ^†, where D is a nonnegative diagonal matrix while U, V are unitary matrices, and V ^† denotes the conjugate transpose of V (or simply the transpose, if V contains real numbers only). The diagonal elements of D are called the singular values of A.

• 25. The orthogonal complement W ^⊥ of a subspace W of an inner product space V is the set of all vectors in V that are orthogonal to every vector in W, i.e.,

  $$W^\bot= \bigl\{x\in V : \langle x, y \rangle= 0 \mbox{ for all } y\in W \bigr\}. $$

• 26. A topological space is called separable if it contains a countable dense subset. In other words, there exists a sequence $\{ x_{n} \}_{n=1}^{\infty}$ of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

• 27. A cover of a set X is a collection of sets whose union contains X as a subset.

• 28. A topological space X is called compact if each of its open covers has a finite subcover. Otherwise it is called non-compact.

• 29. A relatively compact subspace (or relatively compact subset) Y of a topological space X is a subset whose closure is compact.

• 30. T is a compact operator on Hilbert’s space if the image of each bounded set under T is relatively compact.

  Compact operators on Hilbert spaces are a direct extensions of matrices. In such spaces they are the closure of finite-rank operators. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. In contrast, the study of general operators on infinite dimensional spaces often requires a genuinely different approach. For example, the spectral theory of compact operators on Banach spaces takes a form that is very similar to the Jordan canonical form of matrices. In the context of Hilbert spaces, a square matrix is unitarily diagonalizable if and only if it is normal. A corresponding result holds for normal compact operators on Hilbert spaces.

• 31. A simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n+1 vertices. A 2-simplex is a triangle, a 3-simplex is a tetrahedron, and a 4-simplex is a pentachoron. A single point may be considered a 0-simplex, and a line segment may be considered a 1-simplex. A simplex may be defined as the smallest convex set containing the given vertices.

Appendix B: Lattices

A lattice $\mathcal{L}$ (also called a poset) is a partially ordered set (also called a poset) in which any two elements a and b have a unique supremum (the elements’ least upper bound “a∨b”; called their join) and an infimum (greatest lower bound “a∧b”; called their meet). Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order (>, <) theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These “lattice-like” structures all admit order-theoretic as well as algebraic descriptions.

A bounded lattice has a greatest (or maximum) and least (or minimum) element, denoted 1 and 0 by convention (also called top and bottom, respectively). Any lattice can be converted into a bounded lattice by adding a greatest and least element, and every non-empty finite lattice is bounded. For any set A, the collection of all subsets of A (called the power set of A) can be ordered via subset inclusion to obtain a lattice bounded by A itself and the null set. Set intersection and union represent the operations meet and join, respectively.

A poset is called a complete lattice if all its subsets have both a join and a meet. In particular, every complete lattice is a bounded lattice. While bounded lattice homomorphisms in general preserve only finite joins and meets, complete lattice homomorphisms are required to preserve arbitrary joins and meets.

Any quantum system represented by an N-dimensional Hilbert space $\mathcal{H}$ has associated a lattice formed by all its convex subspaces ${\mathcal{L}}_{v\mathcal{N}}({\mathcal{H}})= \langle{\mathcal{P}}({\mathcal{H}}), \cap, \oplus, \neg, 0, 1\rangle$, where 0 is the empty set ∅, 1 is the total space $\mathcal{H}$, ⊕ the closure of the sum, and ¬(S) is the orthogonal complement of a subspace S [43]. This lattice was called “Quantum Logic” by Birkhoff and von Neumann. One refers to this lattice as the von Neumann-lattice $\mathcal{L}_{v\mathcal{N}}(\mathcal{H})$ [43].

Let $\mathcal{L}$ be a bounded lattice with greatest element 1 and least element 0. Two elements x and y of the lattice are complements of each other if and only if: x∨y=1 and x∧y=0. In the case the complement is unique, we write ¬x=y and equivalently, ¬y=x. A bounded lattice for which every element has a complement is called a complemented lattice. The corresponding unitary operation over the lattice, called complementation, introduces an analogue of logical negation into lattice theory. The complement is not necessarily unique, nor does it have a special status among all possible unitary operations over $\mathcal{L}$.

Distributive lattices are lattices for which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenario completely. A complemented lattice that is also distributive is a Boolean algebra. For a distributive lattice, the complement of x, when it exists, is unique.

The concept of lattice’s atom is of great physical importance. If $\mathcal{L}$ has a null element 0, then an element x of $\mathcal{L}$ is an atom if 0<x and there exists no element y of $\mathcal{L}$ such that 0<y<x. One says that $\mathcal{L}$ is:

• (i) Atomic, if for every nonzero element x of $\mathcal{L}$, there exists an atom a of $\mathcal{L}$ such that a=x.

• (ii) Atomistic, if every element of $\mathcal{L}$ is a supremum of atoms.

A modular lattice is one that satisfies the following self-dual condition (modular law) x≤b implies x∨(a∧b)=(x∨a)∧b, where ≤ is the partial order, and ∨ and ∧ (join and meet, respectively) are the operations of the lattice.

Modular lattices arise naturally in algebra and in many other areas of mathematics. For example, the subspaces of a vector space (and more generally the submodules of a module over a ring) form a modular lattice. Every distributive lattice is modular. In a not necessarily modular lattice, there may still be elements b for which the modular law holds in connection with arbitrary elements a and x (≤b). Such an element is called a modular element. Even more generally, the modular law may hold for a fixed pair (a,b). Such a pair is called a modular pair, and there are various generalizations of modularity related to this notion and to semi-modularity.

For $a,b \in\mathcal{L}$, to assert that a is orthogonal to b (a⊥b) implies a∧b=0. Equivalently, in “order” terms, one says that a≤b ^⊥. Now, $\mathcal{L}$ is an orthocomplemented lattice if whenever a⊥b then b≤a ^⊥. ⊥ is a symmetric relation.

For any $a \in\mathcal{L}$, define $M(a): =\{c\in\mathcal{L} \vert c\mathbin{\bot} a, \mbox{ and } 1=c\vee a\}$. An element of M(a) is called an orthogonal complement of a. We have a ^⊥∈M(a), and any orthogonal complement of a is a complement of a. If we replace the unity in M(a) by an arbitrary element b≥a, then we have the set $M(a,b):=\{c\in\mathcal{L}\vert c \vee a \mbox{ and } b=c\vee a\}$. An element of M(a,b) is called an orthogonal complement of a relative to b. Clearly, M(a)=M(a,1). Also, for a≤cb, c∈M(a,b), iff a∈M(c,b). As a result, we can define still another symmetric binary operator ⊕ on [0,b], given by b=a⊕c iff c∈M(a,b). Note that b=b⊕0. A final operation is the “difference” b−a=b∧a. Some properties: (1) a−a=0, a−0=a, 0−a=0, a−1=0 , and 1−a=a ^⊥; (2) b−a=a−b; (3) if a≤b, then a∧(b−a) and a⊕(b−a)≤b.

Definition

A lattice $\mathcal{L}$ is called an orthomodular lattice if (i) $\mathcal{L}$ is orthocomplemented, and (orthomodular law) (ii) if x≤y, then y=x⊕(y−x). The orthomodular law can be recasted as follows: if x≤y , then y=x∨(y∧x ^⊥). Equivalently, x≤y implies y=(y∧x)∨(y∧x ^⊥). Such relation is automatically true in an arbitrary distributive lattice, even without the assumption that x≤y. For example, the lattice $\mathcal{L}(H)$ of closed subspaces of a Hilbert space H is orthomodular. $\mathcal{L}(H)$ is modular iff H is finite dimensional. In addition, if we give the set $\mathcal{P}_{p}(H)$ of (bounded) projection operators on H an ordering structure by defining P≤Q iff $\mathcal{P}(H) \le\mathcal{Q}(H)$, then $\mathcal{P}_{p}(H)$is lattice isomorphic to $\mathcal{L}(H)$, and hence orthomodular [12].

Appendix C: Faces of a Convex Set

We define here a convex set’s face in a real vector space of finite dimension. Let $\mathcal{C}$ be a convex subset of ℝⁿ and let us introduce the auxiliary notions of oriented hyperplanes and supporting hyperplanes. Given n,p∈ℝⁿ let us define the hyperplane H(n,p) via

$$H( \mathbf{n}, \mathbf{p}) = \bigl\{\mathbf{x} \in\mathbb{R}^n: \mathbf{n} \cdot(\mathbf{x} - \mathbf{p})=0\bigr\}. $$

If n=0 it is equal to ℝⁿ and we call it degenerate. As long as H(n,p) is nondegenerate, its removal disconnects ℝⁿ. The upper halfspace of ℝⁿ determined by H(n,p) is H(n,p)⁺={x∈ℝⁿ:n⋅(x−p)≥0}. A hyperplane H(n,p) is a supporting hyperplane for $\mathcal{C}$ if its upper halfspace contains $\mathcal{C}$, that is, if $\mathcal{C} \subset H( \mathbf{n}, \mathbf{p})^{+}$.

Using this terminology, we can define a face of a convex set $\mathcal{C}$ to be the intersection of $\mathcal{C}$ with a supporting hyperplane of $\mathcal{C}$. Notice that we still get both the empty set and $\mathcal{C}$ itself as improper faces of $\mathcal{C}$. For the definition of a face in the infinite dimensional case we extend the definition of a supporting hyperplane to a real Hilbert space $\mathcal{H}$. Given $\mathbf{n,p}\in\mathcal{H}$, we say that H(n,p),

$$H( \mathbf{n},\mathbf{p})=\bigl\{\mathbf{x} \in\mathcal{H}: \langle\mathbf{n} ,\mathbf{x}-\mathbf{p}\rangle=0\bigr\}, $$

is a supporting hyperplane if $\mathcal{C} \subset H( \mathbf{n}, \mathbf{p})^{+}$. Note that H(n,p) is closed and using Riesz representation theorem, for every closed hyperplane H there exists $\mathbf{n,p}\in\mathcal{H}$ such that H=H(n,p).

In the general case (in a Banach space) we say that F is a face of $\mathcal{C}$ if there exist a closed hyperplane H such that $F=\mathcal{C}\cap H$. A closed hyperplane is given by a continuous lineal functional.

Remarks

Let $\mathcal{C}$ be a convex set. Then:

• If $F_{1}=\mathcal{C}\cap H( \mathbf{n}_{1}, \mathbf{p}_{1})$ and $F_{2}=\mathcal{C}\cap H( \mathbf{n}_{2}, \mathbf{p}_{2})$ are faces of $\mathcal{C}$ intersecting at a point p then H(n ₁+n ₂,p) is a supporting hyperplane of $\mathcal{C}$ and F1∩F2=C∩H(n ₁+n ₂,p). This shows that the faces of $\mathcal{C}$ form a meet-semilattice.

• Since each proper face lies on the base of the upper halfspace of some supporting hyperplane, each such face must lie on the relative boundary of $\mathcal{C}$.

An extreme point of a convex set $\mathcal{C}$ in a real vector space is a point in $\mathcal{C}$ which does not lie in any open line segment joining two points of $\mathcal{C}$. Intuitively, an extreme point is a “corner” of $\mathcal{C}$. The Krein-Milman theorem states that if $\mathcal{C}$ is convex and compact in a locally convex space, then $\mathcal{C}$ is the closed convex hull of its extreme points. In particular, such a set has extreme points.