Conditional Erasure and the Landauer Limit

Abstract

Quantum-dynamical proofs of dissipation bounds for Landauer erasure are presented, with emphasis on the crucial connection between conditioning in erasure protocols and fundamental limits on erasure costs. Bounds on erasure costs for conditional and unconditional erasure protocols are shown to follow from a very general and ecumenical physical description of Landauer erasure, a straightforward accounting of its energetic cost, refined definitions of what it means for physical system states to bear known and unknown data, and a transparent application of quantum dynamics and entropic inequalities. These results generalize and support the results of Landauer and Bennett for unconditional and conditional erasure, respectively, and do so using a theoretical methodology that sidesteps or otherwise withstands methodological objections that have been leveled against thermodynamic proofs and other theoretical arguments in the literature. The dissipation bounds obtained here coincide with those obtained elsewhere from an even more general approach that is based on a thoroughly physical conception of information and that clearly distinguishes information from entropy. This connection may help to clarify central issues in the debate over Landauer’s Principle, since the more general approach bounds dissipative costs of irreversible information loss in a range of scenarios that are both broader and less idealized than those typically considered in explorations of the Landauer limit.

Appendix

Here, for convenience, we catalog several established properties of von Neumann entropy, unitary transformations, and trace operations that have been used in this work.

• 1. von Neumann Entropy is Subadditive: For any state $\hat {\rho }^{\mathcal {SE}}$,

  $$\displaystyle \begin{aligned}S(\hat{\rho}^{\mathcal{SE}})\leq S(\hat{\rho}^{\mathcal{S}})+S(\hat{\rho}^{\mathcal{E}})\end{aligned}$$

  where

  $$\displaystyle \begin{aligned}\hat{\rho}^{\mathcal{S}}=Tr_{\mathcal{E}}[\hat{\rho}^{\mathcal{SE}}]\qquad \hat{\rho}^{\mathcal{E}}=Tr_{\mathcal{S}}[\hat{\rho}^{\mathcal{SE}}].\end{aligned}$$

  Equality is achieved in this bound when $\hat {\rho }^{\mathcal {SE}}$ is separable ($\hat {\rho }^{\mathcal {SE}}=\hat {\rho }^{\mathcal {S}}\otimes \hat {\rho }^{\mathcal {E}}$).

• 2. Global von Neumann Entropy is Invariant under Unitary-Similarity Transformations: For any state $\hat {\rho }^{\mathcal {SE}}$ and any unitary operator $\hat {U}$,

  $$\displaystyle \begin{aligned}S(\hat{\rho}^{\mathcal{SE}'})=\hat{U}\hat{\rho}^{\mathcal{SE}}\hat{U}^{\dagger}=S(\hat{\rho}^{\mathcal{SE}}).\end{aligned}$$

• 3. Partovi’s Inequality: For unitary evolution

  $$\displaystyle \begin{aligned}\hat{\rho}^{\mathcal{SE}'}=\hat{U}(\rho^{\mathcal{S}}\otimes \hat{\rho}_{\mathrm{th}}^{\mathcal{E}})\hat{U}^{\dagger}\end{aligned}$$

  of a system initially in any state $\hat {\rho }^{\mathcal {S}}$ and an environment initially in a thermal state $\hat {\rho }_{\mathrm{th}}^{\mathcal {S}}$ at temperature T, Partovi [32] showed that¹⁶

  $$\displaystyle \begin{aligned}\varDelta\langle E^{\mathcal{E}}\rangle\geq k_BT\ln (2) \varDelta S^{\mathcal{E}}\end{aligned}$$

  where

  $$\displaystyle \begin{aligned}\varDelta\langle E^{\mathcal{E}}\rangle=\langle E^{\mathcal{E}'}\rangle - \langle E^{\mathcal{E}}\rangle=Tr[\hat{\rho}^{\mathcal{E}'}\hat{H}_{\mathrm{self}}^{\mathcal{E}}]-Tr[\hat{\rho}^{\mathcal{E}}\hat{H}_{\mathrm{self}}^{\mathcal{E}}].\end{aligned}$$

• 4. Linearity of Unitary-Similarity Transformations: For a unitary operator $\hat {U}$ and sum $\sum _ip_i\hat {\rho }_i$ of operators $\hat {\rho }_i$,

  $$\displaystyle \begin{aligned}\hat{U}\left(\sum_ip_i\hat{\rho}_i\right)\hat{U}^{\dagger}=\sum_ip_i(\hat{U}\hat{\rho}_i\hat{U}^{\dagger}).\end{aligned}$$

• 5. Grouping Property of von Neumann Entropy: For a set $\hat {\rho }_i$ of density operators with support on orthogonal subspaces, the von Neumann entropy of the convex combination

  $$\displaystyle \begin{aligned}\hat{\rho}=\sum_ip_i\hat{\rho}_i\end{aligned}$$

  is¹⁷

  $$\displaystyle \begin{aligned}S(\hat{\rho})=H(\{p_i\})+\sum_ip_iS_i(\hat{\rho})\end{aligned}$$

  where

  $$\displaystyle \begin{aligned}H(\{p_i\})=-\sum_ip_i\log _2 p_i.\end{aligned}$$

• 6. Unitary Evolution Preserves Orthogonality: Consider a unitary $\hat {U}$ and two density operators $\hat {\rho }_i^{\mathcal {SE}}$ and $\hat {\rho }_{i'}^{\mathcal {SE}}$. If $\hat {\rho }_i^{\mathcal {SE}}$ and $\hat {\rho }_{i'}^{\mathcal {SE}}$ are orthogonal, i.e. if

  $$\displaystyle \begin{aligned}\hat{\rho}_i^{\mathcal{SE}}\hat{\rho}_{i'}^{\mathcal{SE}}=0\end{aligned}$$

  then

  $$\displaystyle \begin{aligned}\hat{\rho}_i^{\mathcal{SE}'}\hat{\rho}_{i'}^{\mathcal{SE}'}=0\end{aligned}$$

  where

  $$\displaystyle \begin{aligned}\hat{\rho}_{i}^{\mathcal{SE}'}=\hat{U}\hat{\rho}_i^{\mathcal{SE}}\hat{U}^{\dagger}\qquad \hat{\rho}_{i'}^{\mathcal{SE}'}=\hat{U}\hat{\rho}_{i'}^{\mathcal{SE}}\hat{U}^{\dagger}.\end{aligned}$$

  Note that this global preservation of orthogonality on $\mathcal {SE}$ does not imply local preservation of orthogonality on $\mathcal {S}$ and/or $\mathcal {E}$.