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arXiv:quant-ph/0505118 · Physical Review A

Continuous-variable private quantum channel

Kamil Brádler

Published Oct 10, 2005• 10 citations
arXivHTMLPDF

Abstract

In this paper we introduce the concept of quantum private channel within the continuous variables framework (CVPQC) and investigate its properties. In terms of CVPQC we naturally define a ``maximally'' mixed state in phase space together with its explicit construction and show that for increasing number of encryption operations (which sets the length of a shared key between Alice and Bob) the encrypted state is arbitrarily close to the maximally mixed state in the sense of the Hilbert-Schmidt distance. We bring the exact solution for the distance dependence and give also a rough estimate of the necessary number of bits of the shared secret key (i.e., how much classical resources are needed for an approximate encryption of a generally unknown continuous-variable state). The definition of the CVPQC is analyzed from the Holevo bound point of view which determines an upper bound of information about an incoming state an eavesdropper is able to get from his optimal measurement.

DOI
10.1103/physreva.72.042313

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In this paper we introduce the concept of quantum private channel within the continuous variables framework (CVPQC) and investigate its properties. In terms of CVPQC we naturally define a ``maximally'' mixed state in phase space together with its explicit construction and show that for increasing number of encryption operations (which sets the length of a shared key between Alice and Bob) the encrypted state is arbitrarily close to the maximally mixed state in the sense of the Hilbert-Schmidt distance. We bring the exact solution for the distance dependence and give also a rough estimate of the necessary number of bits of the shared secret key (i.e., how much classical resources are needed for an approximate encryption of a generally unknown continuous-variable state). The definition of the CVPQC is analyzed from the Holevo bound point of view which determines an upper bound of information about an incoming state an eavesdropper is able to get from his optimal measurement.
While the objective of conventional quantum key distribution (QKD) is to secretly generate and share the classical bits concealed in the form of maximally mixed quantum states, that of private quantum channel (PQC) is to secretly transmit individual quantum states concealed in the form of maximally mixed states using shared one-time pad and it is called Gaussian private quantum channel (GPQC) when the scheme is in the regime of continuous variables. We propose a GPQC enhanced with squeezed coherent states (GPQCwSC), which is a generalization of GPQC with coherent states only (GPQCo) [Phys. Rev. A 72, 042313 (2005)]. We show that GPQCwSC beats the GPQCo for the upper bound on accessible information. As a subsidiary example, it is shown that the squeezed states take an advantage over the coherent states against a beam splitting attack in a continuous variable QKD. It is also shown that a squeezing operation can be approximated as a superposition of two different displacement operations in the small squeezing regime.
We consider the private classical capacity of a quantum wiretap channel, where the users (sender Alice, receiver Bob, and eavesdropper Eve) have access to the resource of a shared quantum state, additionally to their channel inputs and outputs. An extreme case is maximal entanglement or a secret key between Alice and Bob, both of which would allow for one-time padding the message. But here both the wiretap channel and the shared state are general. In the other extreme case that the state is trivial, we recover the wiretap channel and its private capacity [N. Cai, A. Winter and R. W. Yeung, Probl. Inform. Transm. 40(4):318-336, 2004]. We show how to use the given resource state to build a code for secret classical communication. Our main result is a lower bound on the assisted private capacity, which asymptotically meets the multi-letter converse and which encompasses all sorts of previous results as special cases.
We study private quantum channels on a single qubit, which encrypt given set of plaintext states PPP. Specifically, we determine all achievable states ρ(0)ρ^{(0)}ρ(0) (average output of encryption) and for each particular set PPP we determine the entropy of the key necessary and sufficient to encrypt this set. It turns out that single bit of key is sufficient when the set PPP is two dimensional. However, the necessary and sufficient entropy of the key in case of three dimensional PPP varies continuously between 1 and 2 bits depending on the state ρ(0)ρ^{(0)}ρ(0). Finally, we derive private quantum channels achieving these bounds. We show that the impossibility of universal NOT operation on qubit can be derived from the fact that one bit of key is not sufficient to encrypt qubit.
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 prove a new version of the Holevo bound employing the Hilbert-Schmidt norm instead of the Kullback-Leibler divergence. Suppose Alice is sending classical information to Bob by using a quantum channel while Bob is performing some projective measurements. We bound the classical mutual information in terms of the Hilbert-Schmidt norm by its quantum Hilbert-Schmidt counterpart. This constitutes a Holevo-type upper bound on the classical information transmission rate via a quantum channel. The resulting inequality is rather natural and intuitive relating classical and quantum expressions using the same measure.
The study of properties of randomly chosen quantum states has in recent years led to many insights into quantum entanglement. In this work, we study private quantum states from this point of view. Private quantum states are bipartite quantum states characterised by the property that carrying out simple local measurements yields a secret bit. This feature is shared by the maximally entangled pair of quantum bits, yet private quantum states are more general and can in their most extreme form be almost bound entangled. In this work, we study the entanglement properties of random private quantum states and show that they are hardly distinguishable from separable states and thus have low repeatable key, despite containing one bit of key. The technical tools we develop are centred around the concept of locally restricted measurements and include a new operator ordering, bounds on norms under tensoring with entangled states and a continuity bound for a relative entropy measure.