arXiv:quant-ph/0003059 · Physical Review A

Optimal encryption of quantum bits

P. Oscar Boykin, Vwani Roychowdhury

Published Apr 22, 2003• 339 citations

Abstract

We show that $2n$ random classical bits are both necessary and sufficient for encrypting any unknown state of n quantum bits in an informationally secure manner. We also characterize the complete set of optimal protocols in terms of a set of unitary operations that comprise an orthonormal basis in a canonical inner product space. Moreover, a connection is made between quantum encryption and quantum teleportation that allows for a different proof of optimality of teleportation.

DOI: 10.1103/physreva.67.042317

Related papers

On the optimality of quantum encryption schemes

Daniel Nagaj, Iordanis Kerenidis

arXiv:quant-ph/0509169Journal of Mathematical PhysicsSep 1, 200616 citations

Optimality of private quantum channels

Jan Bouda, Mário Ziman

arXiv:0710.2004Journal of Physics A Mathematical and TheoreticalApr 30, 200710 citations

Comment on “Optimal encryption of quantum bits”

Xiangfu Zou, Daowen Qiu

Boykin and Roychowdhury [Phys. Rev. A 67, 042317 (2003)] discussed the optimal encryption of quantum bits, gave a necessary and sufficient condition (NASC) for the security of quantum encryption, and suggested an optimal quantum one-time pad encryption scheme. In this Comment, we point out that the proof of the NASC for the security of quantum encryption is not correct. Moreover, we reprove the NASC for the security of quantum encryption.

Physical Review AAug 27, 20131 citation

Private Quantum Channels and the Cost of Randomizing Quantum Information

Michele Mosca, Alain Tapp, Ronald de Wolf

arXiv:quant-ph/0003101Mar 22, 200026 citations

A simple unconditionally secure quantum bit commitment protocol via quantum teleportation

Horace P. Yuen

arXiv:quant-ph/0305142May 23, 20033 citations

Quantum teleportation transfers state, not matter, via 2 classical bits — E8 Intelligence Research

Andrew Stewart Caldin

Oct 31, 20260 citations

arXiv:quant-ph/0003059 · Physical Review A

Optimal encryption of quantum bits

P. Oscar Boykin, Vwani Roychowdhury

Published Apr 22, 2003• 339 citations

arXiv HTML PDF

Abstract

We show that

2n

random classical bits are both necessary and sufficient for encrypting any unknown state of n quantum bits in an informationally secure manner. We also characterize the complete set of optimal protocols in terms of a set of unitary operations that comprise an orthonormal basis in a canonical inner product space. Moreover, a connection is made between quantum encryption and quantum teleportation that allows for a different proof of optimality of teleportation.

DOI: 10.1103/physreva.67.042317

Related papers

On the optimality of quantum encryption schemes

Daniel Nagaj, Iordanis Kerenidis

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.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.

arXiv:quant-ph/0509169Journal of Mathematical PhysicsSep 1, 200616 citations

Optimality of private quantum channels

Jan Bouda, Mário Ziman

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 $\cal E$ and von Neumann entropy of the ciphertext state $\varrho^{(0)}$ . 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.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 $\cal E$ and von Neumann entropy of the ciphertext state $\varrho^{(0)}$. 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.

arXiv:0710.2004Journal of Physics A Mathematical and TheoreticalApr 30, 200710 citations

Comment on “Optimal encryption of quantum bits”

Xiangfu Zou, Daowen Qiu

Physical Review AAug 27, 20131 citation

Private Quantum Channels and the Cost of Randomizing Quantum Information

Michele Mosca, Alain Tapp, Ronald de Wolf

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 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.

arXiv:quant-ph/0003101Mar 22, 200026 citations

A simple unconditionally secure quantum bit commitment protocol via quantum teleportation

Horace P. Yuen

By using local quantum teleportation of a fixed state to one qubit of an entangled pair sent from the other party, it is shown how one party can commit a bit with only classical information as evidence that results in an unconditionally secure protocol. The well-known ``impossibility proof'' does not cover such protocols due to its different commitment and opening prescriptions, which necessitate actual quantum measurements among different possible systems that cannot be entangled as a consequence.By using local quantum teleportation of a fixed state to one qubit of an entangled pair sent from the other party, it is shown how one party can commit a bit with only classical information as evidence that results in an unconditionally secure protocol. The well-known ``impossibility proof'' does not cover such protocols due to its different commitment and opening prescriptions, which necessitate actual quantum measurements among different possible systems that cannot be entangled as a consequence.

arXiv:quant-ph/0305142May 23, 20033 citations

Quantum teleportation transfers state, not matter, via 2 classical bits — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum teleportation transfers quantum *state* (information), not matter — no particle traverses the channel; only a classical 2-bit message plus a pre-shared entangled pair is required. | MATH: Protocol (Bennett et al. 1993): Alice holds unknown state |ψ⟩ = α|0⟩ + β|1⟩ (|α|²+|β|²=1) and one qubit of a Bell pair |Φ⁺⟩ = (|00⟩+|11⟩)/√2. Joint Bell-basis measurement yields one of 4 outcomes {|Φ±⟩,|Ψ±⟩}, each probability 1/4. Alice sends 2 classical bits (log₂4 = 2). Bob applies one of {I, X, Z, XZ} to recover |ψ⟩ exactly. Fidelity F=1; no-cloning preserved (state destroyed at Alice). | CONNECTION: The 4 Bell states form an orthonormal basis of the 2-qubit Hilbert space ℂ²⊗ℂ² ≅ ℂ⁴ — a complete measurement basis. The 1/4 uniform outcome probability is the maximally-mixed (maximally-entangled) reference state ρ = I/4. This is the qubit analogue of a 4-fold symmetric point group (C₂ᵥ-like), and the Pauli group {I,X,Y,Z} is the order-4 operator set closing under multiplication. The √ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Quantum teleportation transfers quantum *state* (information), not matter — no particle traverses the channel; only a classical 2-bit message plus a pre-shared entangled pair is required. | MATH: Protocol (Bennett et al. 1993): Alice holds unknown state |ψ⟩ = α|0⟩ + β|1⟩ (|α|²+|β|²=1) and one qubit of a Bell pair |Φ⁺⟩ = (|00⟩+|11⟩)/√2. Joint Bell-basis measurement yields one of 4 outcomes {|Φ±⟩,|Ψ±⟩}, each probability 1/4. Alice sends 2 classical bits (log₂4 = 2). Bob applies one of {I, X, Z, XZ} to recover |ψ⟩ exactly. Fidelity F=1; no-cloning preserved (state destroyed at Alice). | CONNECTION: The 4 Bell states form an orthonormal basis of the 2-qubit Hilbert space ℂ²⊗ℂ² ≅ ℂ⁴ — a complete measurement basis. The 1/4 uniform outcome probability is the maximally-mixed (maximally-entangled) reference state ρ = I/4. This is the qubit analogue of a 4-fold symmetric point group (C₂ᵥ-like), and the Pauli group {I,X,Y,Z} is the order-4 operator set closing under multiplication. The √ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 31, 20260 citations

We show that

2n

\cal E

and von Neumann entropy of the ciphertext state

\varrho^{(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.