Encoding and decoding

Fig.5.1(a) shows an example of a standard encoding/decoding scheme, where the sender (Alice) wants to transmit the quantum state|ψi of a qubit to the receiver, Bob. In order to protect the quantum information against errors in the transmission, Alice encodes it in a quantum code using a [[5,1,3]] quantum error correcting code. This is done by an encoding circuit (see Fig.5.2) which performs the unitary operationUenc on the transmitted qubit

and some ancilla qubits. The ancilla qubits are initially in the states |aii,

withai ∈ {0,1}(in the computational basis). It is possible for Alice and Bob

to agree on the values ai beforehand, e, g. ai = 0 for all i; in this case, no

classical communication is required at all. However, if for some reason Alice and Bob did not agree on these values, Alice has to tell Bob which values she chooses. In the figure, this classical one-way communication channel is indicated by the thick line.

Upon receiving all qubits, Bob performs the decoding operation Udec =

U−1

encon the qubits and measures the ancilla qubits in the computational basis.

The measurement results, together with the initial values ai of the ancilla

qubits, allow Bob to infer a correction operation Ocorr. If there have not

been too many errors in the transmission of the qubits (the numbers depend on the quantum code used), the correction operation will restore the initial state |ψi of the transmitted qubit.

[[n, k, d]]-purification protocols

In Fig.5.1(b), the setting is different: maximally entangled pairs of qubits in the state|Φ+_{i= 1/}√_{2(|00i+|11i) are created somewhere between Alice and}

Bob; one qubit of each pair is sent to Alice, the other to Bob. Note that the in this figure, there are two time axes: On Bob’s side, time increases from left to right, while on Alice’s side, time increases from right to left.

Alice performs the transpose of the encoding operation of (a), and mea- sures the “ancilla”-qubits in the computational basis (measurement results

ai). The remaining qubit could be measured in an arbitrary basis (e. g., the σz- or the σx-basis), thereby projecting it and its partner onto some state |ψi. However, as we will see below, it will be useful to leave this qubit

5.1 Creating purification protocols from coding circuits 91 ! " # # ! " # # $ $ $ % % %

Figure 5.1: The equivalence between a quantum coding/decoding scheme (a) and an entanglement purification protocol (b). Note that on the right-hand side of the dotted line, both figures are identical.

92 5. Entanglement purification protocols from quantum codes PSfrag replacements

|a

i

|a

i

|a

i

|a

i

|ψi

H

H

H

H

X

X

X

X

Y

Y

Z

Z

Z

Figure 5.2: An example of an encoding circuitUencfor the code [[5,1,3]] (from [38]).

unmeasured.

Since Alice’s qubits were initially entangled with Bob’s, the unitary op- erations and measurements on Alice’s side project Bob’s qubits into some state. In order to calculate this state, we use theUU∗_{-invariance of the state}

|Φ+_i⊗n

AB, i. e. UA⊗UB∗ |Φ+i⊗ABn =|Φ+i ⊗n

AB, where UA and UB is the same uni-

tary operation, performed on Alice’s and Bob’s qubits, respectively. Note that this invariance implies UA|Φ+i⊗ABn = UAUA−1UBt |Φ+i

⊗n

AB = UBt |Φ+i ⊗n AB.

For Bob’s state, we thus get

|ψi_B =Aha|Udec∗ ,A|Φ+i⊗ABn =Aha|Uenct ,A|Φ+i⊗ABn =Uenc,B|aiB, (5.1)

where|ai_B=|a1, . . .i ⊗ |ψi.

In other words, in Fig. 5.1(b), Alice can prepare a posteriori the same type of state which she had prepared in Fig. 5.1(a), so that on the right- hand side of the dotted line, both parts of the figure show the very same situation; after applying the correction operation, Bob’s qubit would be in the state |ψi, if Alice had measured her remaining qubit earlier. However, if Alice and Bob do not measure their qubits of the last pair, they are left with a pair which shows perfectσx- andσz-correlations, if there were not too

many errors in the transmission of all qubits: Alice and Bob have created one perfect EPR-pair in the state|Φ+_{ifrom several “noisy” pairs. Fig. 5.1(b) is}

thus an entanglement purification protocol, which has been derived from a quantum code.

5.1 Creating purification protocols from coding circuits 93

It should be noted that Bob may choose to discard his qubit (as a special case of a correction operation). In this case, he has to tell Alice to also dis- card her remaining qubit, and the protocol becomes a two-way entanglement purification protocol.

In general, quantum codes do not only protect one qubit, as in the exam- ple of Fig. 5.1; rather, a quantum error correcting code [[n, k, d]] encodes the state of k ≥1 logical qubits inton > k physical qubits. Such quantum codes will be converted into entanglement purification protocols, which create k

more entangled pairs from n less entangled pairs. We call such protocols [[n, k, d]] entanglement purification protocols.