The 5-EPR-pair single-error-correcting code

Suppose there exists a finite block-size 1-EPP which distills one good pair of spins in a specific Bell state from a block of five pairs, and no more than one of the five pairs is subjected to noise. When this 1-EPP is combined with a teleportation protocol, two parties, Alice and Bob, can transmit quantum states reliably from one to the other. The combination of the 1-EPP and teleportation protocol therefore is equivalent to a QECC.

The 1-EPP considered herein is schematically depicted in Fig. 7.1. Suppose Alice is the encoder, Bob the decoder, and the Bell state Φ⁺ = (|00i + |11i)/√

2 is the good state to be purified. Alice and Bob are supposed to be provided with five pairs of spins in the state Φ⁺by a quantum source (QS). However, they actually share five Bell states in which generic errors have or have not occurred on at most one Bell state due to the presence of noise NB in the quantum channel via which the pairs are transmitted. The noise models

ENTANGLEMENT PURIFICATION

QS

U

U

m

m

U

N

Alice

Bob

: the channel for the 1^st pairs of entangled qubits : the channel for the 2^nd~ 5^th pairs of entangled qubits

" ⁺ w ⁽ⁱ⁾

e r (i)

v

= v

" v

(i)

Figure 7.1: The 1-EPP with notations used in the context. Alice performs U₁ and m and then sends her classical result (vA) to Bob. Bob performs U2 and m, and then combines his own result (vB) and Alice’s to control a final operation U₃⁽ⁱ⁾.

ENTANGLEMENT PURIFICATION

are assumed to be one-sided [74] and can cause the good Bell state Φ⁺ to become one of the incorrect Bell states

Φ⁻ = 1

√2(|00i − |11i), Ψ^±= 1

√2(|01i ± |10i). (7.1)

The good Bell state Φ⁺ can become one of the erroneous Bell states expressed in (1) if it is subjected to either a phase error (Φ⁺ → Φ⁻), an amplitude error (Φ⁺ → Ψ⁺), or both (Φ⁺ → Ψ⁻) [78, 142]. When performing the 1-EPP, Alice and Bob have a total of 16 error syndromes to deal with. The collection of error syndromes includes the case that none of the five pairs has been subjected to errors and the 15 cases in which one of the five pairs has been subjected to one of the three types of error. The strategy of Alice and Bob is to perform a sequence of unilateral and bilateral unitary operations (as shown in Fig. 7.1, U1 and U2 performed by Alice and Bob, respectively) to transform the collection of the 16 error syndromes to another collection that can provide information about the errors subjected by their particles. Suppose the state of the first pair in the block is to be recovered. After performing the sequence of their operations (U1 and U2

respectively), Alice and Bob, should then perform local measurements on their respective halves of the second to fifth pairs. Alice sends her result via classical channels to Bob who then performs the Pauli operation U3 to recover the original state of the first pair conditionally on both Alice’s and his results. The ultimate requirement of these results of final measurement is that each and every of them should be distinguishable from the others. In other words, there should be 16 distinct measurements obtained from the aforementioned transformation of the error syndrome. The main issue now is that the sequence of unilateral and bilateral unitary operations performed by the two parties to transform the error syndrome should be well designed so the requirement just mentioned can be fulfilled.

To arrange the sequence of operations, basic concepts of linear algebra are used. The

ENTANGLEMENT PURIFICATION

Table 7.1: The correspondence among the error syndrome e⁽ⁱ⁾r (Er⁽ⁱ⁾), the codeword w⁽ⁱ⁾ (W⁽ⁱ⁾), the measurement result v⁽ⁱ⁾, and the Pauli operation U₃⁽ⁱ⁾ controlled by the mea-surement result in the restricted 1-EPP (five-qubit QECC) applying the encoder-decoder circuit shown in Fig. 7.3.2 (Fig. 7.4)

i e⁽ⁱ⁾r , Er⁽ⁱ⁾ w⁽ⁱ⁾, W⁽ⁱ⁾ v⁽ⁱ⁾ U₃⁽ⁱ⁾

four Bell states Φ^± and Ψ^± are first labeled by two classical bits, namely,

Φ⁺ = 00, Φ⁻= 10, Ψ⁺= 01, Ψ⁻ = 11. (7.2)

The right, low-order or amplitude bit identifies the Φ/Ψ property of the Bell state, while the left, high-order or phase bit identifies the +/− property. Note that the combined result of the local measurements obtained by Alice and Bob on a Bell state is revealed by the Bell state’s low or amplitude bit. In the representation of the high-low bits, each error syndrome thus is expressed as a ten-bit codeword, e.g., the error syndrome Φ⁺Ψ⁻Φ⁺Φ⁺Φ⁺ is written as 00 11 00 00 00. Codewords of the error syndrome, denoted by e⁽ⁱ⁾r , i = 0, 1, ..., 15, are listed in Table 7.1. The effect of the sequence of unilateral and bilateral unitary operations performed by Alice and Bob is to map the codewords e⁽ⁱ⁾r

onto another collection of ten-bit codewords w⁽ⁱ⁾. If both the codewords e⁽ⁱ⁾r and w⁽ⁱ⁾ are

ENTANGLEMENT PURIFICATION

written as column vectors in the ten-dimensional Boolean-valued (∈ {0, 1}) space, then the mapping e⁽ⁱ⁾r → w⁽ⁱ⁾ can be simply expressed by a matrix equation

w⁽ⁱ⁾= Me⁽ⁱ⁾_r , (7.3)

provided that the mapping is confined to w⁽⁰⁾ = e⁽⁰⁾r (= 00 00 00 00 00). The four error syndromes, e^(3k)r , e^(3k−1)r , e^(3k−2)r , and e⁽⁰⁾r , corresponding to a common erroneous pair, form a group and are characterized by

e^(3k−2)_r ⊕ e^(3k−1)r = e^(3k)_r , k = 1, 2, ..., 5, (7.4)

where k enumerates the erroneous pair and ⊕ is the addition modulo 2. Accordingly, the 16 codewords w⁽ⁱ⁾ should be subdivided into five corresponding groups, each of which has w^(3k), w^(3k−1), w^(3k−2), and w⁽⁰⁾, and holds the relation

w^(3k−2)⊕ w^(3k−1) = w^(3k), k = 1, 2, ..., 5. (7.5)

Therefore the matrix M can be simply expressed by a 10 × 10 matrix, such as

M=

w⁽¹⁾w⁽²⁾w⁽⁴⁾w⁽⁵⁾w⁽⁷⁾w⁽⁸⁾w⁽¹⁰⁾w⁽¹¹⁾w⁽¹³⁾w⁽¹⁴⁾

, (7.6)

in accordance with the arrangement of error syndromes listed in Table 7.1. The first two rows of M represent the states of the pair to be recovered, and the 4th, 6th, 8th,and 10th rows represent the low bits of the second to fifth Bell states and thus construct the four-bit codewords for the measurement results v⁽ⁱ⁾. The measurement result v⁽ⁱ⁾ of course is also characterized by

v^(3k−2)⊕ v^(3k−1) = v^(3k), k = 1, 2, ..., 5, (7.7)

in accordance with relations (7.4) and (7.5). In the language of linear algebra, the action of

ENTANGLEMENT PURIFICATION

the sequence of unilateral and bilateral unitary operations that accounts for the mapping e⁽ⁱ⁾r → w⁽ⁱ⁾ is to perform a sequence of elementary row operations on the 10 × 10 identity matrix 1 to reduce it to the matrix M. In this spirit, Bennett et al. [74] have undertaken a Monte Carlo numerical search program to find out suitable solutions for matrix M and their corresponding encoder-decoder networks. Basically, the approach implemented by Bennett et al. is a tedious numerical method of trial and error performing the transfor-mation 1 → M subjected to a forward sequence of local operations. In this work, we will present an analytical method for creating M implemented in the present QECC. The present method will be described in detail in the next section.

7.3 Analytical technique for simplification of the