My collaborators on the work contained in this thesis have taught me an awful lot. This thesis provides limits on the performance of quantum error correcting codes when used for quantum communication and quantum key distribution.
The Repetition Code
Linear Codes
Random Linear Codes
Since the probability is less than 1, there must be at least one [n, k] code with distance at least d.
Noisy Channel Coding Theorem
In the next chapter we will study the quantum analogues of the Gilbert-Varshamov connection and the noisy channel coding theorem. However, the quantum analogue of mutual information subadditivity will fail and the rates found are not optimal.
The Pauli Group
Because our encoding and decoding operations will be linear (as any quantum operation must be), we can focus on correcting errors in the formXuZv, with alfaiu1v1 in Eq. 2.9) plays the role of the probability (or more precisely the amplitude) of the error Xu1Zv1. Note that the inclusion of the phase factors together with the relations Y =iXZ, Z=iY X and X =iZY and the fact that any one of X,Y and Zantipendulum with the other two ensure that this set is in fact closed under multiplication.
A Quantum Code
By doing so, we will be able to fix a logicalZ error on one of the three blocks that make up our X code. If there is an error on one of the nine qubits, the first level of codes can correct the amplitude part of the error, but any phase error will be propagated to the next level as a logic Z.
Stabilizer Codes
When this happens, the other two blocks will be error-free (since we are only concerned with correcting single qubit errors) and the external repeat code will be able to correct this. Degenerate codes that also allow E1†E2 ∈ S (and must have many such pairs to behave significantly differently from a non-degenerate code) are more poorly understood and will be the subject of Chapters 4 and 6.
Quantum Gilbert-Varshamov Bound
There we will see degenerate codes that can tolerate noise levels for which all non-degenerate codes will fail. so we will have a non-zero probability of having all error pairs outside the stabilizer for that long. 2.40) Since E(2) is the set of all products of error pairs with weight up to, we have Quantum Gilbert-Varshamov bound] For sufficiently large there are stabilizer codes for relative distance δrel =nd and rateR for all.
Probabilistic Quantum Errors
Such a code can be used to correct arbitrary errors affecting up to a ≈δrel/2 fraction of qubits in a block. Since the elements of Etype will all occur with roughly equal probability, the average (over S) of the error probability of the code defined by S is no greater than.
Quantum Noisy Channel Coding Theorem
By only asking for arbitrarily high reliability and allowing the sender and receiver to use a secret key with a logarithmic length in the number of qubits transmitted, we achieve a dramatic improvement over QGV speeds. Furthermore, to improve reliability, a random subcode is chosen based on a secret key unknown to the adversary.
Background and Definitions
Note that a single level of random code could also be used, but the secret key required would be O(n2) bits. One could also achieve a speedup of 1-H(p)-plog 3 by using secret key to determine a permutation of thenchannel applications (see e.g. [SP00] or [Ren05]), at a cost of O (nlogn) bits that , unfortunately also lead to a divergent secret key rate. In a way, our list-code construction can be seen as a derandomization of these key-inefficient protocols, achieving the same result with a much shorter secret key.
Quantum List Codes
If we adapt the above counting argument to the present case, it is. 3.16) sets of generators for stabilizers where all n−k generators commute withE0†Ej for allj.
Coding Strategy
After each addition, we get a subcode of the previous code, and the number of coded qubits decreases. Since we use the list code [n, Rn, pn, L], the adversary's power is reduced to choosing the probability distribution of fors and the corresponding superposition of the list. So, if for all list code syndromes the probability (given the choice of T1, · · ·, TK) that there is a pair of list elements having the same commutation relations with stabilizers Tj is less than², the fidelity of the decoded state with the original will be at least 1 −².
Discussion
It seems very likely that this is equal to the capacity of the depolarizing channel with error rate p, which would be analogous to the classical result of [Lan04]. It was Shannon [Sha48] who, by means of an arbitrary coding argument, discovered the beautiful fact that the capacity of a noisy channel is equal to the maximum mutual information between an input variable, X, and its image under the influence of the channel:. Although the optimal iteration length and basis vary, as does the magnitude of the benefit, it is a general fact that using such a code is beneficial in the regime where the hashing rate is close to zero.
Cat Codes for Pauli Channels
We first provide an explicit formula for the rate achieved over Npby anm-qubit repetition code coupled with a random stabilizer code, and find a channel for which the advantage of degenerate codes over the hashing rate is dramatic - its hashing rate goes to zero atpx +py+ pz=p≈0.274 while repeat codes allow non-zero rates up to above≈0.295. Because the repetition code is highly symmetric, we can find explicit formulas for both Pr(r) and Nr, and thus a fairly compact expression for Ic(ρABm). In general, forpx≥pz≥py it is a good rule of thumb to use aZ repeating code of length m≈1/pz, with the largest increase in rate for fairly asymmetric channels (eg Fig. 4.1).
The Almost Bitflip Channel
The optimal repeat length minimizes the entropy in the logical qubits based on r, given by. However, because the number of Zerrors will have roughly a Poisson distribution with a mean of 1 for m ≈1/qz À1, the probabilities of 0, 1, and 2 errors are all quite similar, even though the expected number of Z errors is 1, and they remain well below a bit of entropy in the logical Z error. The probability of an a bit.
Concatenated Repetition Codes
Note that essentially all the entropy in the X errors is removed by the best code, with the optimal length determined by a trade-off between the reduction of entropy in the This kind of trade-off also determines the optimal repetition code length for a general Pauli channel.
A Special Channel
The bottom line shows the hashing threshold, the middle line for the 5-cat code, and the top line for the concatenated 5-in-5-cat code. The lower curve is the threshold of a 3 inm concatenated cat code as a function of m, while the upper curve shows the threshold for a 5 in m concatenated code. This suggests that the capacity of Nptp is exactly 1−H(p )−p is, and in light of its non-decomposability we hope that a proof of this conjecture will point to a new sufficient criterion for the additivity of coherent information.
Discussion
We provide a Shor-Preskill-type security proof for the preparation and measurement of quantum key distribution protocols using noisy preprocessing and one-sided key postprocessing. The twist operator ensures that, while Alice and Bob may not share a maximally entangled state, Eve's reduced state is independent of the key value. In the following, we will show that a QKD preparation and measurement scheme with noisy preprocessing and biased postprocessing is reliable precisely when a coupled twisted-state distillation protocol succeeds with high fidelity.
Twisted State Distillation
This step is the same as the usual analysis, as all bit errors must be corrected in the final key, regardless of their source. In the classic description of the protocol, this is equivalent to encrypting the syndromes with a one-time pad before transmission, preventing information leakage to Eve. Since this encryption requires a key, which in the quantum description is a twisted state, Alice and Bob generally collect the parity syndromes in the key subsystems of twisted states, not maximally entangled states.
Detailed Analysis
A subtlety arises in the use of the Neumark expansion in that we have moved beyond the usual definition of a twisted state. However, the privacy of the key is uncompromised: while Eve may have knowledge of the shield system, as long as Alice and Bob hold the key and the shield, the fact that they may be untethered implies that Eve is unaware of the key. When they are not, randomly choosing quantities Vs of size 2n(S(σ|u)−δ), where S(σ|u) is the conditional entropy of σ given u, will lead to an exponentially small average probability of decoding error for PGM and the rest of the argument remains unchanged (see e.g. [Lo01]).
Achievable Key Rates
Putting it all together, using a random code Alice and Bob can choose a subset of sizes. Then, with probability exponentially close to unity, it is possible to construct an unwinding operation from a reasonably good metric, thus ensuring that the final key² is secure. In the six-state protocol, all Pauli errors occur at the same rate, from which we find a marginal error rate of 14.1%.
Discussion
On the contrary, the security of the key generated by a QKD protocol depends only on fundamental laws of physics. An important property of any QKD protocol is the amount of noise that can be tolerated without compromising the security of the resulting key. Degenerate codes have been used in security proofs before; in particular, the noise threshold of the six-state protocol was improved from 12.6 to 12.7 percent [Lo01].
Analytic Key Rate Expression
Alice adds independent noise with error rateq to the A register so that the state of the Alice-Bob-Eve system can be described as. Note that Eve's system is determined by the fact that, in the worst case, she keeps the purification of the state after it comes out of the channel. We turn to the second term in Eq. 6.11), we want to find the mutual information about the state obtained by tracing Bob's systems.
Numerical Evaluation of Key Rates
Moreover, because the E1 system is not correlated with the rest, the mutual information between Alice and Eve can be easily calculated, which is quite beneficial. Due to the permutation invariance of the state ρ⊗p,qm, it can be compactly expressed as a direct sum of states on the SU(2) irreducible representations (irreps). In general, larger m allow us to reach higher thresholds, with the optimal value of q≈0.3 increasing slowly with m (e.g. Figure 6.1).
Discussion
We present an upper bound for the quantum channel capacity that is additive and convex. In addition, we will use capacity-assisted cloning to find new upper limits of depolarization channel capacity. In particular, the performance using the clone we find will not be an upper bound for the performance using two-way classical communication.
Properties of Q ca
The other ingredient we need is the following expression for the clone-supported capacity, followed by standard arguments (e.g. [Dev05]). To see that the ca capacitance is not less than the right side, note that for any|φABnU Vi that is symmetric under the exchange of U andV, the velocity. To prove the converse, fix ², let C ⊂ A˜nS be a (n, ²) code of rate R using a symmetric broadcast channel of output dimension d2V and let |φCi be a state maximally entangled with C .
Implications for Unassisted Quantum Capacities
The performance of a degradable channel is given by a one-letter maximization of coherent information, as shown in [DS05]. This means that the maximum value of the left side of Eq. 7.26) is not greater than the maximum on the right side. This immediately implies Qca(N1/4) = 0, since otherwise Bob and Eve could reconstruct the coded state with high accuracy, which would result in a violation of the no-cloning theorem.
Discussion
Smolin and A.V.Thapliyal, Entanglement-assisted capacity of a quantum channel and the inverse Shannon theorem, IEEE Trans. Shor, A quantum channel's capacity for simultaneous transmission of classical and quantum information, Communications in Mathematical Physics arXiv quant-ph/0311131. Lo, Proof of security for quantum key distribution with two-way classical communication, IEEE Trans.
