This thesis includes a survey of the results known for private and approximate private quantum channels. We develop the best known upper bound for $ε$-randomizing maps, $n+2\log(1/ε)+c$ bits required to $ε$-randomize an arbitrary $n$-qubit state by improving a scheme of Ambainis and Smith \cite{AS04} based on small bias spaces \cite{NN90, AGHP92}. We show by a probabilistic argument that in fact the great majority of random schemes using slightly more than this many bits of key are also $ε$-randomizing. We provide the first known non-trivial lower bound for $ε$-randomizing maps, and develop several conditions on them which we hope may be useful in proving stronger lower bounds in the future.
This thesis includes a survey of the results known for private and approximate private quantum channels. We develop the best known upper bound for ε-randomizing maps, n+2log(1/ε)+c bits required to ε-randomize an arbitrary n-qubit state by improving a scheme of Ambainis and Smith \cite{AS04} based on small bias spaces \cite{NN90, AGHP92}. We show by a probabilistic argument that in fact the great majority of random schemes using slightly more than this many bits of key are also ε-randomizing. We provide the first known non-trivial lower bound for ε-randomizing maps, and develop several conditions on them which we hope may be useful in proving stronger lower bounds in the future.
We addressed the question of optimality of private quantum channels. We have shown that the Shannon entropy of the classical key necessary to securely transfer the quantum information is lower bounded by the entropy exchange of the private quantum channel E and von Neumann entropy of the ciphertext state ϱ(0). Based on these bounds we have shown that decomposition of private quantum channels into orthogonal unitaries (if exists) is optimizing the entropy. For non-ancillary single qubit PQC we have derived the optimal entropy for arbitrary set of plaintexts. In particular, we have shown that except when the (closure of the) set of plaintexts contains all states, one bit key is sufficient. We characterized and analyzed all the possible single qubit private quantum channels for arbitrary set of plaintexts. For the set of plaintexts consisting of all qubit states we have characterized all possible approximate private quantum channels and we have derived the relation between the security parameter and the corresponding minimal entropy.
It is well known that n bits of entropy are necessary and sufficient to perfectly encrypt n bits (one-time pad). Even if we allow the encryption to be approximate, the amount of entropy needed does not asymptotically change. However, this is not the case when we are encrypting quantum bits. For the perfect encryption of n quantum bits, 2n bits of entropy are necessary and sufficient (quantum one-time pad), but for approximate encryption one asymptotically needs only n bits of entropy. In this paper, we provide the optimal trade-off between the approximation measure ϵ and the amount of classical entropy used in the encryption of single quantum bits. Then, we consider n-qubit encryption schemes which are a composition of independent single-qubit ones and provide the optimal schemes both in the 2- and the ∞-norm. Moreover, we provide a counterexample to show that the encryption scheme of Ambainis-Smith [Proceedings of RANDOM ’04, pp. 249–260] based on small-bias sets does not work in the ∞-norm.
We investigate how a classical private key can be used by two players, connected by an insecure one-way quantum channel, to perform private communication of quantum information. In particular we show that in order to transmit n qubits privately, 2n bits of shared private key are necessary and sufficient. This result may be viewed as the quantum analogue of the classical one-time pad encryption scheme. From the point of view of the eavesdropper, this encryption process can be seen as a randomization of the original state. We thus also obtain strict bounds on the amount of entropy necessary for randomizing n qubits.
We investigate the randomized and quantum communication complexities of the well-studied Equality function with small error probability ϵ, getting the optimal constant factors in the leading terms in a number of different models.
In the randomized model,
1) we give a general technique to convert public-coin protocols to private-coin protocols by incurring a small multiplicative error, at a small additive cost. This is an improvement over Newman's theorem [Inf. Proc. Let.'91] in the dependence on the error parameter.
2) Using this we obtain a (log(n/ϵ2)+4)-cost private-coin communication protocol that computes the n-bit Equality function, to error ϵ. This improves upon the log(n/ϵ3)+O(1) upper bound implied by Newman's theorem, and matches the best known lower bound, which follows from Alon [Comb. Prob. Comput.'09], up to an additive loglog(1/ϵ)+O(1).
In the quantum model,
1) we exhibit a one-way protocol of cost log(n/ϵ)+4, that uses only pure states and computes the n-bit Equality function to error ϵ. This bound was implicitly already shown by Nayak [PhD thesis'99].
2) We show that any ϵ-error one-way protocol for n-bit Equality that uses only pure states communicates at least log(n/ϵ)−loglog(1/ϵ)−O(1) qubits.
3) We exhibit a one-way protocol of cost log(n/ϵ)+3, that uses mixed states and computes the n-bit Equality function to error ϵ. This is also tight up to an additive loglog(1/ϵ)+O(1), which follows from Alon's result.
Our upper bounds also yield upper bounds on the approximate rank and related measures of the Identity matrix. This also implies improved upper bounds on these measures for the distributed SINK function, which was recently used to refute the randomized and quantum versions of the log-rank conjecture.
The concept of $\e$-randomizing quantum channels has been introduced by Hayden, Leung, Shor and Winter in connection with approximately encrypting quantum states. They proved using a discretization argument that sets of roughly dlogd random unitary operators provide examples of such channels on $\C^d$. We show that a simple trick improves the efficiency of the argument and reduces the number of unitary operators to roughly d.
Randomization of quantum states is the quantum analogue of the classical one‐time pad. We present an improved, efficient construction of an approximately randomizing map that uses O(d/ε2) Pauli operators to map any d‐dimensional state to a state that is within trace distance ε of the completely mixed state. Our bound is a log d factor smaller than that of Hayden, Leung, Shor, and Winter, and Ambainis and Smith.Then, we show that a random sequence of essentially the same number of unitary operators, chosen from an appropriate set, with high probability form an approximately randomizing map for d‐dimensional states. Finally, we discuss the optimality of these schemes via connections to different notions of pseudorandomness, and give a new lower bound for small ε.