Feedback Based On Multi-Qubit Measurements, Multiple-Particle Interference And Quantum Error Correction, Theory Of Quantum Error-Correcting Codes, Theory Of Quantum Error-Correcting Codes-2, Theory Of Quantum Error-Correcting Codes-3, New Class Of Quantum Error-Correcting Codes For A Bosonic Mode, New Class Of Quantum Error-Correcting Codes For A Bosonic Mode, Bosonic Quantum Memories, Photonic Quantum Communication, And Optical-To-Microwave Up- And Down-Conversion, Mixed State Entanglement And Quantum Error Correction, Mixed State Entanglement And Quantum Error Correction-2, Mixed State Entanglement And Quantum Error Correction-3, Universal Operator Theoretic Framework For Quantum Fault Tolerance, Efficiently Controllable Graphs. Foregoing system and its connected properties are studied in detail. The full paper: http://www.iiste.org/PDFshare/APTA-PAGENO-555138-561343.pdf
The group of transversal gates and the group of local symmetries are important features of quantum error correcting codes and pure quantum states, respectively; the former provides fault-tolerant operations on a code while the latter tells us about a state's reachability via stochastic local operations with classical communication. We prove that there exists a bijection between local unitary (LU) orbits of absolutely maximally entangled (AME) states in
(CD)⊗n where
n is even, also known as perfect tensors, and LU orbits of
((n−1,D,n/2))D quantum error correcting codes. Furthermore, there is a close connection between the local symmetries of an AME state and the transversal gates of its corresponding quantum error correcting code. We explore in detail the 4-qutrit AME state
∣Φ⟩ and its corresponding
((3,3,2))3 qutrit code
C. We show that
∣Φ⟩ and
C are both unique up to the action of the LU group. We find generators of the local symmetry group of
∣Φ⟩ and the group of transversal gates on
C. Our proofs rely on prior results by Huber and Grassl (2020), Hebenstreit et al. (2016), and Rather et al. (2023). We use Vinberg's theory of Lie algebras to study the special case of
∣Φ⟩ and
C.
Entanglement purification protocols (EPPs) and quantum error-correcting codes (QECCs) 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 arbitrary quantum state |\ensuremath{\xi}〉 can be transmitted at some rate Q through a noisy channel \ensuremath{\chi} without degradation. We prove that an EPP involving one-way classical communication and acting on mixed state M^(\ensuremath{\chi}) (obtained by sharing halves of Einstein-Podolsky-Rosen pairs through a channel \ensuremath{\chi}) yields a QECC on \ensuremath{\chi} 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
D1(M) and
D2(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 communication, 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 communication. 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 communication is available. We exhibit a family of codes based on universal hashing able to achieve 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 iff a QECC results in high fidelity for the case of no error then the QECC can be recast into a form where the encoder is the matrix inverse of the decoder. \textcopyright{} 1996 The American Physical Society.