If the observers are allowed also to exchange quantum information (via qubits or other non-local quantum operations) then the joint system may be described by states which are not separable. If the observers are allowed to manipulate the states by local quantum operations and classical communication, the states of the total system which are achievable in this way are said to be separable. Any such representation of T is called a Stinespring representation and the. Similar to Kraus representations, the term Stinespring representation is often. In order to explain, if only briefly, the important notion of entanglement, consider a system composed of initially independent subsystems, with an associated observer who can prepare a quantum state. To circumvent this obstacle the theory of quantum error correction and fault. current active theory of quantum error correction, which is the subject. Peter Harremoës, Flemming Topsøe, in Philosophy of Information, 2008 2.3 Entanglement The distance d of an (( n, K)) QECC is the smallest weight of a nontrivial Pauli operator E ∈ P n s.t. We get a similar result in the case where the noise is a general quantum operation on each qubit which differs from the identity by something of size O( ε).
#Steinspring quantum error correction code#
If we have a channel which causes errors independently with probability O( ε) on each qubit in the QECC, then the code will allow us to decode a correct state except with probability O( ε t+1), which is the probability of having more than t errors.
Then the Pauli operators of weight t or less form a basis for the set of all errors acting on t or fewer qubits, so a QECC which corrects these Pauli operators corrects all errors acting on up to t qubits. The quantum error correction code must coherently map the 2D space spanned by 0 L and 1 L into the 2D Hilbert spaces required to correct the three types of errors (bit-flip, phase-flip, and a combination of both, which we discuss in the next section) for each of the n physical qubits. The weight wt( P) of a Pauli operator P ∈ P n is the number of qubits on which it acts as X, Y, or Z (i.e., not as the identity). I = ( 1 0 0 1 ), X = ( 0 1 1 0 ) Y = ( 0 − i i 0 ), Z = ( 1 0 0 − 1 )ĭefine the Pauli group P n as the group consisting of tensor products of I, X, Y, and Z on n qubits, with an overall phase of ☑ or ±i. Thus, it is sufficient in general to check that the error-correction conditions hold for a basis of errors. Consider two quantum systems, Q ɛ HQ and R ɛ H R, and let automatically corrects S as well. The partial trace is a quantum operation. We will describe our quantum error correction. Through an application of the Stinespring dilation theorem Stinespring.
Marinescu, in Classical and Quantum Information, 2012 Example. Suppose a quantum system Q is subjected to a dynamical evolution, which may represent the transmission of. Moreover, we demonstrate that the quantum error correction condition from the.