[Paper Review] Dichotomic probability representation of quantum states

Margarita A. Man’ko, V. I. Manʹko+3 more|arXiv (Cornell University)|May 25, 2019

Statistical Mechanics and Entropy31 references4 citations

TL;DR

This paper establishes a systematic dichotomic probability representation for qudit density matrices using $d(d-1)$ sets of binary random variables, proving that any $d \times d$ density matrix can be fully parameterized by $d^2 - 1$ dichotomic probability distributions. The method enables new entropic and determinant inequalities for matrix elements, with explicit derivations for qubit and qutrit states, providing a classical-like statistical framework for quantum state tomography and reconstruction accuracy control.

ABSTRACT

We present systematic proofs of statements about probability representations of qudit density states in terms of standard probability distributions of dichotomic random variables. New relations and new entropic-information inequalities are derived. The examples of 3- and 4- level states are explicitly worked out.

Motivation & Objective

• To establish a rigorous probability representation of qudit density matrices using only dichotomic random variables.
• To prove that any $d \times d$ density matrix can be fully parameterized by $d^2 - 1$ probability distributions of binary outcomes.
• To derive new entropic and determinant inequalities for matrix elements of density operators based on this representation.
• To enable a reduction of $d=nm$-dimensional states into $n \times n$ and $m \times m$ sub-states preserving quantum state properties.
• To provide a classical-like statistical framework for quantum tomography and experimental reconstruction accuracy control.

Proposed method

• The method constructs $d(d-1)$ complex planes within $\mathbb{C}^d$, each associated with a $\mathfrak{u}(2)$ Lie algebra isomorphic to $U(2)$, enabling tomographic description via dichotomic probabilities.
• For each $\mathfrak{u}(2)$ subalgebra, the density matrix elements are mapped to probabilities $p_a^{(jk)}$ of binary outcomes through projectors onto two-dimensional subspaces.
• The representation relies on the Silvester criterion to ensure nonnegativity of the density matrix, guaranteeing physical validity of the probability distributions.
• The approach uses trace-preserving reductions to decompose a $d=nm$-dimensional state into $n \times n$ and $m \times m$ reduced density matrices with preserved Hermiticity, trace, and positivity.
• Entropic inequalities are derived using Tsallis relative entropy and logarithmic functions of probabilities, leading to bounds on matrix elements.
• Explicit computations are performed for qubit ($d=2$) and qutrit ($d=3$) states, demonstrating the method’s consistency and applicability.

Experimental results

Research questions

• RQ1 Can every $d \times d$ density matrix be exactly parameterized by $d^2 - 1$ dichotomic probability distributions?
• RQ2 What are the necessary and sufficient conditions on dichotomic probabilities to ensure the resulting matrix is a valid density operator (Hermitian, trace-one, positive semi-definite)?
• RQ3 How can entropic inequalities involving matrix elements of a qudit density matrix be derived from this probability representation?
• RQ4 Can a $d=nm$-dimensional qudit state be decomposed into smaller $n \times n$ and $m \times m$ density matrices while preserving quantum state properties?
• RQ5 What new inequalities for eigenvalues and determinants of reduced density matrices emerge from this representation?

Key findings

• A $d \times d$ density matrix can be exactly parameterized by $d^2 - 1$ dichotomic probability distributions, as proven in Theorem 2.1.
• The matrix elements $\rho_{jk}$ are expressed in terms of probabilities $p_a^{(jk)}$, satisfying the Silvester criterion for nonnegativity.
• For qubits, the matrix elements are explicitly related to probabilities via $p_1 = \frac{1}{2} + \mathrm{Re}\,\rho_{12}$, $p_2 = \frac{1}{2} - \mathrm{Im}\,\rho_{12}$, and $p_3 = \rho_{11}$.
• New entropic inequalities are derived using Tsallis relative entropy, such as $ (1-q)^{-1}\left\{ \left(\frac{1}{2}+\mathrm{Re}\rho_{12}\right)^q\left(\frac{1}{2}-\mathrm{Im}\rho_{12}\right)^{1-q} + \cdots - 1 \right\} \geq 0 $.
• The method allows iterative reduction of $d=nm$-dimensional states into $n \times n$ and $m \times m$ sub-states, preserving $\rho^\dagger = \rho$, $\mathrm{Tr}\,\rho = 1$, and $\rho \geq 0$, with new determinant and eigenvalue relations.
• The probability representation enables a classical-like decomposition of quantum states using orthogonal projections on edges of a simplex, offering a new geometric interpretation for quantum state evolution and tomography.