Free encoding capacity: A universal unit for quantum resources

TL;DR

This work introduces the Free Encoding Capacity (FEC) as a universal unit of quantum resources for arbitrary quantum resource theories, by restricting encodings to free operations over a perfect quantum channel. It proves that for pointed resource theories, FEC is a faithful resource measure, and that the optimal free-encoding strategy can be achieved using extreme free operations. The proofs establish boundedness, strong monotonicity, and convexity of FEC, and derive a key corollary bounding stochastic state transformation probabilities. As a case study, the resource theory of Activity shows that passive qubits have a fixed FEC around $0.322$ bits and that a scaled version $\tilde{\mathcal{C}}_{\text{activity}}$ can faithfully quantify resourcefulness for active states, illustrating practical impact in resource-cost accounting for classical information processing via quantum channels.

Abstract

A perfect d-dimensional quantum channel can convey log d-bits of classical information by encoding messages in d-orthogonal quantum states. Alternatively, for every quantum state at the senders end, there exist d-encoding operations which produce d-orthogonal quantum states. Transmitting which via a d-level perfect quantum channel it is possible to communicate log d-bits of classical information. But what if the set of encoding operations is restricted only within a physically constrained class? Here, we consider such a class of encoding operations to be the set of free operations for any quantum resource theory and show that the constrained capacity - namely, the free encoding capacity (FEC) emerged as a unit of the corresponding quantum resource. Moreover, we show that for the pointed resource theories - a resource theory admitting only a single free state - FEC becomes a faithful resource measure also. We also discuss the implications of FEC in the question of resource-theoretic state transformations and the possibility of extending its faithfulness for general quantum resource theories.

Figures (4)

• Figure 1: (Color online) The FEC for pointed QRTs. (a) In a pointed QRT $\mathcal{R}$, Alice can use a resource state $\rho$ (the clean, white ball) to encode classical information $x\in X$ by applying different free operations (different coloring), and by decoding it (seeing the color of the ball), Bob can obtain $y$ (the same color here). (b) On the other hand, the free state $\rho_F$ (the muddy, gray ball) is unable to communicate any information by remaining invariant (the same gray color) under all possible free operations (different coloring).
• Figure 2: (Color online)Strong monotonicity of FEC under stochastic free operations. The input quantum state $\rho$, after the isometry $\mathcal{V}$ and the post-selection of the $k^{\th}$ outcome on the ancilla becomes $L_k\rho L_k^{\dagger}\propto\sigma_k^{\rho}$. Then the input information $x\in X$ is encoded on $\sigma_k^{\rho}$ by a free operation $\mathcal{N}_x$ and sent through the quantum channel $\mathcal{I}_d$. As supplementary information, the classical index $k$ is transmitted through the classical channel $\text{I}_{d^2}$, shown with the dotted line. The receiver, having the classical index $k$ in hand, can extract optimally $\mathcal{C}_{\mathcal{R}}(\sigma_k^{\rho})$ amount of information from the quantum channel.
• Figure 3: (Color online) $\mathcal{C}^*_{\text{activity}}$ for qubit states coherent in energy eigen basis. The leftmost image shows that the bit-flip operations $\mathcal{N}_{bp}$ dephase any quantum state $\rho$ on the respective version $\rho_{\mu}$ state on the X-axis, and specifically every passive state to the maximally mixed $\frac{\mathbb{I}}{2}$. Then by applying three free operations $\{\mathcal{I},\mathcal{Z},\mathcal{N}_0\}$, with probabilities $\{0.2,0.2,0.6\}$ respectively, Alice effectively communicates the state $\overline{\rho_{\mu}}$, denoted as a star symbol. Note that the star is in the same position as in Fig \ref{['f2']} (a).
• Figure 4: (Color online) $\mathcal{C}^*_{\text{activity}}$ for qubit states incoherent in energy eigen basis. (a) Any passive state $\rho_{\text{passive}}$ achieves its FEC with two free operations $\{\mathcal{N}_0,\mathcal{N}_1\}$, respectively with probabilities $\{0.6,0.4\}$. This produces an effective state denoted as a star symbol on the $Z$-axis. (b) For any active state incoherent in the energy eigenbasis, Alice adopts two free operations $\{\mathcal{N}_0,\mathcal{I}\}$, again with probabilities $\{0.6, 0.4\}$. This effectively produces the state, denoted by a star symbol. Since the distance $x$ between the states $\rho_{\nu}$ and $|0\rangle \langle 0|$ is greater than $1$, it is trivial that the star state in figure (b) possesses higher entropy than that of figure (a).

Theorems & Definitions (16)

• Theorem 1
• Corollary 1
• Lemma 1
• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Corollary 2
• Proposition 5
• Theorem 2