Quantum Enigma Machines and the Locking Capacity of a Quantum Channel

Abstract

The locking effect is a phenomenon that is unique to quantum information theory and represents one of the strongest separations between the classical and quantum theories of information. The Fawzi-HaydenSen locking protocol harnesses this effect in a cryptographic context, whereby one party can encode n bits into n qubits while using only a constant-size secret key. The encoded message is then secure against any measurement that an eavesdropper could perform in an attempt to recover the message, but the protocol does not necessarily meet the composability requirements needed in quantum key distribution applications. In any case, the locking effect represents an extreme violation of Shannon's classical theorem, which states that information-theoretic security holds in the classical case if and only if the secret key is the same size as the message. Given this intriguing phenomenon, it is of practical interest to study the effect in the presence of noise, which can occur in the systems of both the legitimate receiver and the eavesdropper. This paper formally defines the locking capacity of a quantum channel as the maximum amount of locked information that can be reliably transmitted to a legitimate receiver by exploiting many independent uses of a quantum channel and an amount of secret key sublinear in the number of channel uses. We provide general operational bounds on the locking capacity in terms of other well-known capacities from quantum Shannon theory. We also study the important case of bosonic channels, finding limitations on these channels' locking capacity when coherent-state encodings are employed and particular locking protocols for these channels that might be physically implementable.

Robust quantum data locking from phase modulation

“…Although the phenomenon of QDL has been known for about ten years now [1], only recently has it been reconsidered in the context of noisy quantum channels [9,10]. In particular, the locking capacity of a (noisy) quantum channel has been defined in [10] as the maximum rate at which classical information can be locked through N instances of the channel with the assistance of a secret key shared by Alice and Bob which grows sub-linearly in N . One can indeed define a weak and a strong notion of locking capacity [10].…”

“…In particular, the locking capacity of a (noisy) quantum channel has been defined in [10] as the maximum rate at which classical information can be locked through N instances of the channel with the assistance of a secret key shared by Alice and Bob which grows sub-linearly in N . One can indeed define a weak and a strong notion of locking capacity [10]. In the weak case one assumes that Eve has access to the channel environment (the output of the complementary channel), while in the strong case one gives her access to the quantum system being input to the noisy channel.…”

“…It hence remains to show how and at which rate Bob can decode reliably. One way to do that is by concatenating the QDL protocol with an error correcting code [5,10,14]. Consider N 1 uses of a quantum channel.…”

“…Such additional assumptions make the accessible information criterion weaker than the commonly accepted security criterion for quantum private communication [8], which is instead related to the Holevo information. Interestingly enough, a large gap exists between these two security criteria that may allow for high rate QDL through quantum channels with poor privacy [9,10]. As a matter of fact, explicit examples of channels with low or even zero privacy that allow for QDL at high rates have been recently provided [11,12].…”

“…It is worth noticing that, compared to previously known QDL protocols that require the preparation of uniform (Haar distributed) basis vectors, our scheme greatly simplifies the structure of codewords for QDL and represents a major simplification for optical experimental implementations. Moreover, the expansion of the set of QDL codewords given in this paper paves the way to the application of random coding techniques to lock classical information into noisy quantum channels [9,10], which requires the codewords used to hide information to coincide with the codewords used to protect information from noise. Codewords randomly selected from an ensemble that attains the coherent information bound suffice to protect information from noise [20][21][22][23][24][25].…”