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IEEE International Symposium on Information Theory (ISIT)

On the Relation between Quantum Data Hiding and Quantum Key Distribution

Matthias Christandl, Mads Friis Frand-Madsen

Published Jun 25, 2023• 0 citations
Publisher

Abstract

In 2004, Horodecki et al., showed that the amount of secret key that can be extracted from noisy bipartite quantum states may exceed the amount of distillable maximally entangled quantum bits. Their work was based on the intuition of quantum data hiding, but did not offer a quantitative relationship to this phenomenon.In this work, which is set in a one-way classical communication scenario, we define a rate of quantum data hiding and provide entropic bounds on it. We then relate the quantum data hiding rate to the gap between the rates at which secret key and entanglement can be extracted, improving on earlier work on this topic.

DOI
10.1109/isit54713.2023.10206873

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In 2004, Horodecki et al., showed that the amount of secret key that can be extracted from noisy bipartite quantum states may exceed the amount of distillable maximally entangled quantum bits. Their work was based on the intuition of quantum data hiding, but did not offer a quantitative relationship to this phenomenon.In this work, which is set in a one-way classical communication scenario, we define a rate of quantum data hiding and provide entropic bounds on it. We then relate the quantum data hiding rate to the gap between the rates at which secret key and entanglement can be extracted, improving on earlier work on this topic.
An important contribution to the understanding of quantum key distribution has been the discovery of entangled states from which secret bits, but no maximally entangled states, can be extracted [Horodecki et al., Phys. Rev. Lett. 94, 200501 (2005)PRLTAO0031-900710.1103/PhysRevLett.94.200501]. The construction of those states was based on an intuition that the quantum mechanical phenomena of data hiding and privacy might be related. In this Letter we firmly connect these two phenomena and highlight three aspects of this result. First, we simplify the definition of the secret key rate. Second, we give a formula for the one-way distillable entanglement of certain private states. Third, we consider the problem of extending the distance of quantum key distribution with help of intermediate stations, a setting called the quantum key repeater. We show that for protocols that first distill private states, it is essentially optimal to use the standard quantum repeater protocol based on entanglement distillation and entanglement swapping.
We present protocols for multiparty data hiding of quantum information that implement all possible threshold access structures. Closely related to secret sharing, data hiding has a more demanding security requirement: that the data remain secure against unrestricted attacks via local operation and classical communication. In the limit of hiding a large amount of data, our protocols achieve an asymptotic rate of one hidden qubit per local physical qubit. That is, each party holds a share that is the same size as the hidden state to leading order, with accuracy and security parameters incurring an overhead that is asymptotically negligible. The data-hiding states have very unusual entanglement properties, which we briefly discuss.
When classical or quantum information is broadcast to separate receivers, there exist codes that encrypt the encoded data, such that the receivers cannot recover it when performing local operations and classical communication, but they can reliably decode if they bring their systems together and perform a collective measurement. This phenomenon is known as quantum data hiding and hitherto has been studied under the assumption that noise does not affect the encoded systems. With the aim of applying the quantum data hiding effect in practical scenarios, here, we define the data-hiding capacity for hiding classical information using a quantum channel. Using this notion, we establish a regularized upper bound on the data hiding capacity of any quantum broadcast channel, and we prove that coherent-state encodings have a strong limitation on their data hiding rates. We, then, prove a lower bound on the data hiding capacity of channels that map the maximally mixed state to the maximally mixed state (we call these channels mictodiactic-they can be seen as a generalization of unital channels when the input and output spaces are not necessarily isomorphic) and argue how to extend this bound to generic channels and to more than two receivers.
When classical or quantum information is broadcast to separate receivers, there exist codes that encrypt the encoded data, such that the receivers cannot recover it when performing local operations and classical communication, but they can reliably decode if they bring their systems together and perform a collective measurement. This phenomenon is known as quantum data hiding and hitherto has been studied under the assumption that noise does not affect the encoded systems. With the aim of applying the quantum data hiding effect in practical scenarios, here, we define the data-hiding capacity for hiding classical information using a quantum channel. Using this notion, we establish a regularized upper bound on the data hiding capacity of any quantum broadcast channel, and we prove that coherent-state encodings have a strong limitation on their data hiding rates. We, then, prove a lower bound on the data hiding capacity of channels that map the maximally mixed state to the maximally mixed state (we call these channels mictodiactic-they can be seen as a generalization of unital channels when the input and output spaces are not necessarily isomorphic) and argue how to extend this bound to generic channels and to more than two receivers.
Recent work has shown how to use the laws of quantum mechanics to keep classical and quantum bits secret in a number of different circumstances. Among the examples are private quantum channels, quantum secret sharing and quantum data hiding. In this paper we show that a method for keeping two classical bits hidden in any such scenario can be used to construct a method for keeping one quantum bit hidden, and vice--versa. In the realm of quantum data hiding, this allows us to construct bipartite and multipartite hiding schemes for qubits from the previously known constructions for hiding bits. Our method also gives a simple proof that two bits of shared randomness are required to construct a private quantum channel hiding one qubit.
Abstract — We expand on our work on Quantum Data Hiding [1] – hiding classical data among parties who are restricted to performing only local quantum operations and classical communication (LOCC). We review our scheme that hides one bit between two parties using Bell states, and we derive upper and lower bounds on the secrecy of the hiding scheme. We provide an explicit bound showing that multiple bits can be hidden bitwise with our scheme. We give a preparation of the hiding states as an efficient quantum computation that uses at most one ebit of entanglement. A candidate data hiding scheme that does not use entanglement is presented. We show how our scheme for quantum data hiding can be used in a conditionally secure quantum bit commitment scheme.