Mixed State Entanglement and Quantum Error Correction

TL;DR

This work develops a comprehensive framework linking mixed-state entanglement to reliable quantum communication by connecting entanglement purification protocols (EPP) and quantum error-correcting codes (QECC). It establishes a fundamental equivalence between one-way EPP on channel-induced mixed states and QECCs on the corresponding quantum channels, and shows how two-way communication can surpass one-way strategies in both distillation and coding. The authors introduce and analyze hashing, breeding, and recurrence purification schemes, proving the existence of channels (e.g., 50% depolarizing) that admit two-way but not one-way quantum communication, and they present a simple five-qubit code achieving single-qubit error correction. The results unify state-distillation, channel coding, and teleportation, and they reveal deep bounds and conditions (Knill-Laflamme-type) that constrain quantum capacities, with implications for practical quantum communication and error correction.

Abstract

Entanglement purification protocols (EPP) and quantum error-correcting codes (QECC) provide two ways of protecting quantum states from interaction with the environment. In an EPP, perfectly entangled pure states are extracted, with some yield D, from a mixed state M shared by two parties; with a QECC, an arbi- trary quantum state $|ξ\rangle$ can be transmitted at some rate Q through a noisy channel $χ$ without degradation. We prove that an EPP involving one- way classical communication and acting on mixed state $\hat{M}(χ)$ (obtained by sharing halves of EPR pairs through a channel $χ$) yields a QECC on $χ$ with rate $Q=D$, and vice versa. We compare the amount of entanglement E(M) required to prepare a mixed state M by local actions with the amounts $D_1(M)$ and $D_2(M)$ that can be locally distilled from it by EPPs using one- and two-way classical communication respectively, and give an exact expression for $E(M)$ when $M$ is Bell-diagonal. While EPPs require classical communica- tion, QECCs do not, and we prove Q is not increased by adding one-way classical communication. However, both D and Q can be increased by adding two-way com- munication. We show that certain noisy quantum channels, for example a 50% depolarizing channel, can be used for reliable transmission of quantum states if two-way communication is available, but cannot be used if only one-way com- munication is available. We exhibit a family of codes based on universal hash- ing able toachieve an asymptotic $Q$ (or $D$) of 1-S for simple noise models, where S is the error entropy. We also obtain a specific, simple 5-bit single- error-correcting quantum block code. We prove that {\em iff} a QECC results in high fidelity for the case of no error the QECC can be recast into a form where the encoder is the matrix inverse of the decoder.

Figures (17)

• Figure 1: Typical scenario for creation of entangled quantum states. At some early time and at location $I$, two quantum systems $A$ and $B$ interact foot1, then become spatially separated, one going to Alice and the other to Bob. The joint system's state lies in a Hilbert space ${\cal H}= {\cal H}_A\otimes{\cal H}_B$ that is the tensor product of the spaces of the subsystems, but the state itself is not expressible as a product of states of the subsystems: $\Upsilon\neq \Upsilon_A\otimes\Upsilon_B$. State $\Upsilon$, its pieces acted upon separately by noise processes $N_A$ and $N_B$, evolves into mixed state $M$.
• Figure 2: Entanglement purification protocol involving two-way classical communication (2-EPP). In the basic step of 2-EPP, Alice and Bob subject the bipartite mixed state to two local unitary transformations $U_1$ and $U_2$. They then measure some of their particles $\cal M$, and interchange the results of these measurements (classical data transmission indicated by double lines). After a number of stages, such a protocol can produce a pure, near-maximally-entangled state (indicated by *'s).
• Figure 3: One-way Entanglement Purification Protocol (1-EPP). In 1-EPP there is only one stage; after unitary transformation $U_1$ and measurement $\cal M$, Alice sends her classical result to Bob, who uses it in combination with his measurement result to control a final transformation $U_3$. The unidirectionality of communication allows the final, maximally-entangled state (*) to be separated both in space and in time.
• Figure 4: If the 1-EPP of Fig. \ref{['1way']} is used as a module for creating time-separated EPR pairs (*), then by using quantum teleportationteleportation, an arbitrary quantum state $|\xi\rangle$ may be recovered exactly after $U_4$, despite the presence of intervening noise. This is the desired effect of a quantum error correcting code (QECC).
• Figure 7: Effect on the fidelity of Werner states of one step of purification, using the recurrence protocol. $F$ is the initial fidelity of the Werner state (Eq. (\ref{['WF']})), $F'$ is the final fidelity of the "passed" pairs after one iteration. Also shown is the fraction $p_{pass}/2$ of pairs remaining after one iteration (cf. Eq. (\ref{['ppass']})).