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arXiv:1608.01695 · Physical Review Letters

Masking Quantum Information is Impossible

Kavan Kishore Modi, Arun Kumar Pati, Aditi Sen, Ujjwal Sen

Published Jun 5, 2018• 86 citations
arXivHTMLPDF

Abstract

Classical information encoded in composite quantum states can be completely hidden from the reduced subsystems and may be found only in the correlations. Can the same be true for quantum information? If quantum information is hidden from subsystems and spread over quantum correlation, we call it masking of quantum information. We show that while this may still be true for some restricted sets of nonorthogonal quantum states, it is not possible for arbitrary quantum states. This result suggests that quantum qubit commitment-a stronger version of the quantum bit commitment-is not possible in general. Our findings may have potential applications in secret sharing and future quantum communication protocols.

DOI
10.1103/physrevlett.120.230501

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Classical information encoded in composite quantum states can be completely hidden from the reduced subsystems and may be found only in the correlations. Can the same be true for quantum information? If quantum information is hidden from subsystems and spread over quantum correlation, we call it masking of quantum information. We show that while this may still be true for some restricted sets of nonorthogonal quantum states, it is not possible for arbitrary quantum states. This result suggests that quantum qubit commitment-a stronger version of the quantum bit commitment-is not possible in general. Our findings may have potential applications in secret sharing and future quantum communication protocols.
To solve the problem of hiding quantum information in simplified subsystems, Modi et al. [1] introduced the concept of quantum masking. Quantum masking is the encoding of quantum information by composite quantum states in such a way that the quantum information is hidden to the subsystem and spreads to the correlation of the composite systems. The concept of quantum masking was developed along with a new quantum impossibility theorem, the quantum no-masking theorem. The question of whether a quantum state can be masked has been studied by many people from the perspective of the types of quantum states, the number of masking participants, and error correction codes. Others have studied the relationships between maskable quantum states, the deterministic and probabilistic masking of quantum states, and the problem of probabilistic masking. Quantum masking techniques have been shown to outperform previous strategies in quantum bit commitment, quantum multi-party secret sharing, and so on.
The no-masking theorem for quantum information proves that it is impossible to encode an arbitrary input state into a larger bipartite entangled state such that the full information is stored in the correlation but the individual subsystems have no information about the input state. Here, we ask the question: Is it possible to mask an observable such that the information about the observable is available in the joint system, but individual subsystems reveal nothing about the imprints of the observable? This generalizes the notion of masking to observables. We show that a universal unitary that can mask an arbitrary observable in any dimension does not exist. For a qubit system, we show that the masking operation for a given observable is locally unitarily connected to the swap operation. This suggests a conservation law for information content of observables that goes beyond the conservation laws under symmetry operations. Furthermore, we prove that the unconditional no-bit commitment result follows from the no-masking theorem for observables. Our results can have important applications in quantum information and quantum communication where we encode information not in states but in observables.
Masking of quantum information means that information is hidden from a subsystem and spread over a composite system. Modi et al. proved in [Phys. Rev. Lett. 120, 230501 (2018)] that this is true for some restricted sets of nonorthogonal quantum states and it is not possible for arbitrary quantum states. In this paper, we discuss the problem of masking quantum information encoded in pure and mixed states, respectively. Based on an established necessary and sufficient condition for a set of pure states to be masked by an operator, we find that there exists a set of four states that can not be masked, which implies that to mask unknown pure states is impossible. We construct a masker S♯S^\sharpS♯ and obtain its maximal maskable set, leading to an affirmative answer to a conjecture proposed in Modi's paper mentioned above. We also prove that an orthogonal (resp. linearly independent) subset of pure states can be masked by an isometry (resp. injection). Generalizing the case of pure states, we introduce the maskability of a set of mixed states and prove that a commuting subset of mixed states can be masked by an isometry S⋄S^{\diamond}S⋄ while it is impossible to mask all of mixed states by any operator. We also find the maximal maskable sets of mixed states of the isometries S♯{S^{\sharp}}S♯ and S⋄{S^{\diamond}}S⋄, respectively.
Abstract Quantum information masking is a protocol that hides the original quantum information from subsystems and spreads it over quantum correlation, which the original information of input states is completely unknown to local subsystems. In this paper, we explore the possibility of partial masking of quantum information in multipartite systems through using the generator matrices and stabilizer codes, as this approach is more suitable for constructing secret sharing protocols. Furthermore, if the condition for partial masking of quantum information is strengthened, it can be shown that the quantum state α | 0 ⟩ + β | 1 ⟩ can be masked in multipartite systems. Among them, partial masking of quantum information has a wider scope of application because it has fewer restrictions. To some extent, the masking of quantum information can be seen as a special case of partial masking. At last, we provide an example of how masked quantum information can be restored to its original state.
Classical information can be completely hidden in the correlations of bipartite quantum systems. However, it is impossible to hide or mask all quantum information according to the no-hiding and no-masking theorems derived recently. Here we show that any set of informationally complete quantum states is neither hidable nor maskable, thereby strengthening the no-hiding and no-masking theorems known before. Then, by virtue of Hurwitz-Radon matrices (representations of the Clifford algebra), we show that information about real quantum states can be completely hidden in the correlations, although the minimum dimension of the composite Hilbert space required increases exponentially with the dimension of the original Hilbert space. Moreover, the set of real quantum states is a maximal maskable set within quantum theory and has a surprising connection with maximally entangled states. These results offer valuable insight on the potential and limit of hiding and masking quantum information, which are of intrinsic interest to a number of active research areas.