Isoholonomic inequalities and speed limits for cyclic quantum systems

Abstract

Quantum speed limits set fundamental lower bounds on the time required for a quantum system to evolve between states. Traditional bounds, such as those by Mandelstam-Tamm and Margolus-Levitin, rely on state distinguishability and become trivial for cyclic evolutions where the initial and final states coincide. In this work, we explore an alternative approach based on isoholonomic inequalities, which bound the length of closed trajectories in the state space in terms of their holonomy. Building on a gauge-theoretic framework for mixed-state geometric phases, we extend the concept of isoholonomic inequalities to closed curves of isospectral and isodegenerate density operators. This allows us to derive a new quantum speed limit that remains nontrivial for cyclic evolutions. Our results reveal deep connections between the temporal behavior of cyclic quantum systems and holonomy.

Figures (6)

• Figure 1: The total space $\mathcal{W}(\boldsymbol{\Lambda})$ of the principal bundle $\eta$ over the space $\mathcal{D}(\boldsymbol{m})$ of density operators on $\mathcal{H}$ with degeneracy spectrum $\boldsymbol{m}$ consists of all linear maps $W : \mathcal{K} \to \mathcal{H}$ such that $W^\dagger W$ is a density operator on $\mathcal{K}$ with eigenprojectors given by $\boldsymbol{\Lambda}$. The complementary product $WW^\dagger$ belongs to $\mathcal{D}(\boldsymbol{m})$, and the bundle projection $\eta$ maps $W$ to $WW^\dagger$. The gauge group $\mathrm{U}(\mathcal{K};\boldsymbol{\Lambda})$, acting on $\mathcal{W}(\boldsymbol{\Lambda})$ from the right via operator composition, consists of the unitary operators on $\mathcal{K}$ that commute with every projector in $\boldsymbol{\Lambda}$. This action is free and transitive on the fibers of the bundle, so each fiber is diffeomorphic to $\mathrm{U}(\mathcal{K};\boldsymbol{\Lambda})$.
• Figure 2: The vertical space at $W$ consists of the tangent vectors at $W$ annihilated by $d\eta$ and is shown here as a vertical line. Its dimension is $m_1^2 + m_2^2 + \cdots + m_l^2$, equal to that of the fiber of $\eta$ through $W$, which is diffeomorphic to the gauge group. The horizontal space at $W$, orthogonal to the vertical space, has the same dimension as $\mathcal{D}(\boldsymbol{m})$; see Eq. \ref{['dim of D(m)']}. It is defined as the kernel of the connection form $\mathcal{A}$ at $W$. Thus, a tangent vector $\dot{W}$ at $W$ is horizontal if $\mathcal{A}(\dot W)=0$.
• Figure 3: Let $\rho_t$ be a closed curve in $\mathcal{D}(\boldsymbol{m})$ based at $\rho$, and let $W$ be an amplitude of $\rho$. The horizontal lift $W_t$ of $\rho_t$ starting at $W$ ends in the fiber above $\rho$ but need not return to $W$. This phenomenon is known as holonomy. The parallel transport operator maps $W$ to the endpoint of the horizontal lift: $\Gamma[\rho_t](W) = W_\tau$. If $U$ belongs to the gauge group, the horizontal lift starting at $WU$ is given by $W_t U$, so right multiplication by $U$ transfers the horizontal lift from $W$ to $WU$. Consequently, the parallel transport operator satisfies $\Gamma[\rho_t](WU) = \Gamma[\rho_t](W) U$, meaning it commutes with the gauge group action. The parallel transport of $WU$ can also be written as $U \Gamma[\rho_t;W] U^\dagger$, where $\Gamma[\rho_t;W]$ is the unique gauge group element transforming $W$ to $\Gamma[\rho_t](W)$.
• Figure 4: The concatenation of two closed curves, $\rho_{1;t}$ and $\rho_{2;t}$, at $\rho$ defines a new closed curve $\rho_{1;t} \ast \rho_{2;t}$ at $\rho$. Its horizontal lift to the amplitude $W$ is obtained by concatenating the horizontal lift $W_{1;t}$ of $\rho_{1;t}$ starting at $W$ with the horizontal lift $W_{2;t}$ of $\rho_{2;t}$ starting at $\Gamma[\rho_{1;t}](W)$. The resulting horizontal lift terminates at $\Gamma[\rho_{2;t}](\Gamma[\rho_{1;t}](W))$, which establishes the composition law \ref{['eq: composition law']} for the parallel transport operator.
• Figure 5: The metric on $\mathcal{D}(\boldsymbol{m})$ is defined by lifting tangent vectors $\dot\rho_1$ and $\dot\rho_2$ at $\rho$ to horizontal vectors $\dot W_1$ and $\dot W_2$ at any amplitude $W$ of $\rho$, and evaluating their inner product $G(\dot W_1, \dot W_2)$. Because the gauge group acts by isometries, the result is independent of the choice of horizontal space used for the lift: the differential of the right action by any $U$ in $\mathrm{U}(\mathcal{K};\boldsymbol{\Lambda})$ maps $\dot W_1$ and $\dot W_2$ to $\dot W_1 U$ and $\dot W_2 U$, respectively, which still project to $\dot\rho_1$ and $\dot\rho_2$, and $G(\dot W_1 U, \dot W_2 U) = G(\dot W_1, \dot W_2)$.

Theorems & Definitions (16)

• Example 1
• Remark 1
• Example 2
• Example 3
• Example 4
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Example 5