Majorization–Lattice Theorem in Quantum Information

• Majorization–lattice theorem is a structure where finite, nonincreasing probability vectors are ordered by cumulative sums, essential for comparing quantum states.
• It establishes clear lattice operations (meet and join), providing optimal common resources for entanglement transformations via local operations and classical communication.
• This framework underpins probabilistic protocols in entanglement transformation, allowing 'greedy' and 'thrifty' strategies to effectively convert quantum states.

The majorization–lattice theorem asserts that the set of finite probability vectors, each sorted in nonincreasing order and normalized to sum to one, forms a lattice under the majorization partial order. This structure underpins the theoretical characterization of permissible transformations between pure bipartite entangled states under local operations and classical communication, as utilized in quantum resource theory. The notions of meet ($\wedge$) and join ($\vee$) in this lattice yield operationally relevant constructs, such as the optimal common resource and optimal common product states, which are instrumental in formulating probabilistic protocols for state conversion (Deside et al., 2023).

1. The Majorization Partial Order

Given $x = (x_1,\dots,x_n)$ and $y = (y_1,\dots,y_n)$, which are real vectors with entries sorted in nonincreasing order and each summing to one, the majorization relation $x \prec y$ holds if and only if their partial sums satisfy $X_k \leq Y_k$ for all $k = 1,\dots,n-1$ and $X_n=Y_n=1$, where $X_k = \sum_{i=1}^k x_i$ and $Y_k = \sum_{i=1}^k y_i$. The vector $x$ is therefore said to be more disordered than $y$ in the sense of majorization. This relation defines a partial order on the set

$$\mathcal P_n = \Bigl\{\,p \in \mathbb R^n : p_1 \geq \cdots \geq p_n \geq 0, \, \sum_i p_i = 1 \Bigr\}.$$

2. Lattice Operations: Meet and Join

Within $(\mathcal P_n, \prec)$, any two elements possess a greatest lower bound (meet) and a least upper bound (join).

Meet ($x \wedge y$): For $x, y \in \mathcal P_n$ with $X_0 = Y_0 = 0$, the meet $m = x \wedge y$ is defined componentwise:

$$m_i = \min\{X_i, Y_i\} - \min\{X_{i-1}, Y_{i-1}\}, \quad i = 1, \dots, n.$$

This yields the property that for all $k$,

$\sum_{i=1}^{k} m_i = \min\{X_k, Y_k\}.$

$m$ is thus a valid probability vector and the greatest lower bound of $x$ and $y$.

Join ($x \vee y$): Analogously, define $C_k = \max\{X_k, Y_k\}$ with $C_0 = 0$ and set

$d_i = C_i - C_{i-1}.$

The resulting $d = (d_1, \dots, d_n)$ need not be sorted, so one reorders it in nonincreasing order to obtain $x \vee y \in \mathcal P_n$. The partial sums then satisfy

$\sum_{i=1}^k (x \vee y)_i = C_k = \max\{X_k, Y_k\}$

for all $k$, making $x \vee y$ the least upper bound.

3. Normalization and Boundary Elements

All vectors in $\mathcal P_n$ are required to be probability distributions (sorted nonincreasingly and summing to one). The top element under $\prec$ is the uniform distribution $u = (1/n, \dots, 1/n)$, representing maximal disorder. The bottom element is the "pure" distribution $e = (1, 0, \dots, 0)$, representing maximal order. These boundaries define the extremal points of the lattice structure (Deside et al., 2023).

4. Proof Sketch of Lattice Properties

The meet construction ensures that for any $r \prec x, y$, the prefix sums satisfy $\sum_{i=1}^{k} r_i \leq \min\{X_k, Y_k\}$, and hence $r \prec m$. By the definition of $m$, it is the unique vector satisfying these inequalities as equalities, establishing it as the greatest lower bound. The join is constructed analogously with maxima, and after reordering has the required properties of the least upper bound. These arguments confirm the lattice property of $(\mathcal P_n,\prec)$ without pathologies.

5. Operational Examples

Distinct behaviors emerge based on whether vectors are comparable under $\prec$.

Case	$p$	$q$	$p \wedge q$	$p \vee q$
Comparable	(0.5, 0.3, 0.2)	(0.6, 0.2, 0.2)	(0.5, 0.3, 0.2)	(0.6, 0.2, 0.2)
Incomparable	(0.5, 0.4, 0.1)	(0.6, 0.2, 0.2)	(0.5, 0.3, 0.2)	(0.6, 0.3, 0.1)

In the incomparable case, the meet is constructed via the pointwise minimum of the cumulative sums, producing a new vector. The join is obtained by pointwise maxima followed by reordering to maintain nonincreasing order.

6. Implications for Quantum Information Theory

The majorization lattice provides a framework for entanglement transformation protocols under local operations and classical communication. For pure bipartite states, the lattice grants a rigorous method to define the optimal common resource (meet) and the optimal common product state (join). These structures enable the definition of two probabilistic protocols—named "greedy" and "thrifty"—that allow for the (single-copy) conversion of incomparable bipartite pure states. When initial and final states are comparable, both protocols reduce to Vidal's protocol. Otherwise, the thrifty protocol yields a more entangled residual state under failure while both succeed with the same optimal probability (Deside et al., 2023). A plausible implication is the applicability of these constructions in scenarios involving multiple initial or final states, where the generalization proceeds through the recursive use of meet and join.

7. Key Lemma on Prefix Sums

A fundamental lemma states that if $m = x \wedge y$, then for every $k$,

$$\sum_{i=1}^k m_i = \min\left\{ \sum_{i=1}^k x_i, \sum_{i=1}^k y_i \right\}.$$

This telescoping property provides a concise verification of the correctness and optimality of the meet (and, analogously, the join via maxima and reordering) in operational protocols and in related theoretical analyses.

These elements constitute the core of the majorization–lattice theorem and underpin its centrality to the resource-theoretic approach to quantum information and entanglement transformation (Deside et al., 2023).