Quantum Codewords Research Articles

Tamper Detection against Unitary Operators

Security of a storage device against a tampering adversary has been a well-studied topic in classical cryptography. Such models give black-box access to an adversary, and the aim is to protect the stored message or abort the protocol if there is any tampering.In this work, we extend the scope of the theory of tamper detection codes against an adversary with quantum capabilities. We consider encoding and decoding schemes that are used to encode a k-qubit quantum message |m⟩ to obtain an n-qubit quantum codeword |ψm⟩. A quantum codeword |ψm⟩ can be adversarially tampered via a unitary U from some known tampering unitary family UAdv (acting on C2n).Firstly, we initiate the general study of quantum tamper detection codes, which detect if there is any tampering caused by the action of a unitary operator. In case there was no tampering, we would like to output the original message. We show that quantum tamper detection codes exist for any family of unitary operators UAdv, such that |UAdv|<22αn for some constant α∈(0,1/6); provided that unitary operators are not too close to the identity operator. Quantum tamper detection codes that we construct can be considered to be quantum variants of classical tamper detection codes studied by Jafargholi and Wichs ['15], which are also known to exist under similar restrictions.Additionally, we show that when the message set M is classical, such a construction can be realized as a non-malleable code against any UAdv of size up to 22αn.

Lossless quantum data compression with exponential penalization: an operational interpretation of the quantum R\xe9nyi entropy

Based on the problem of quantum data compression in a lossless way, we present here an operational interpretation for the family of quantum Rényi entropies. In order to do this, we appeal to a very general quantum encoding scheme that satisfies a quantum version of the Kraft-McMillan inequality. Then, in the standard situation, where one is intended to minimize the usual average length of the quantum codewords, we recover the known results, namely that the von Neumann entropy of the source bounds the average length of the optimal codes. Otherwise, we show that by invoking an exponential average length, related to an exponential penalization over large codewords, the quantum Rényi entropies arise as the natural quantities relating the optimal encoding schemes with the source description, playing an analogous role to that of von Neumann entropy.

Structured Optical Receivers to Attain Superadditive Capacity and the Holevo Limit

Attaining the ultimate (Holevo) limit to the classical capacity of a quantum channel requires the receiver to make joint measurements over long code-word blocks. For a pure-state channel, we show that the Holevo limit can be attained by a receiver that uses a multisymbol unitary transformation on the quantum code word followed by separable projective measurements. We show a concatenated coding and joint-detection architecture to approach the Holevo limit. We then construct some of the first concrete examples of codes and structured joint-detection receivers for the lossy bosonic channel, which can achieve fundamentally higher (superadditive) capacity than conventional receivers that detect each modulation symbol individually. We thereby pave the way for research into codes and structured receivers for reliable communication data rates approaching the Holevo limit.

Generating continuous variable quantum codewords in the near-field atomic lithography

Recently, Gottesman et al (2001 Phys. Rev. A 64, 012310) showed how to encode a qubit into a continuous variable quantum system. This encoding was realized by using non-normalizable quantum codewords, which therefore can only be approximated in any real physical set-up. Here we show how a neutral atom, falling through an optical cavity and interacting with a single mode of the intracavity electromagnetic field, can be used to safely encode a qubit into its external degrees of freedom. In fact, the localization induced by a homodyne detection of the cavity field is able to project the near-field atomic motional state into an approximate quantum codeword. The performance of this encoding process is then analysed by evaluating the intrinsic errors induced in the recovery process by the approximated form of the generated codeword.

Continuous variable encoding by ponderomotive interaction

Recently it has been proposed to construct quantum error-correcting codes that embed a finite-dimensional Hilbert space in the infinite-dimensional Hilbert space of a system described by continuous quantum variables [D. Gottesman et al., Phys. Rev. A 64, 012310 (2001)]. The main difficulty of this continuous variable encoding relies on the physical generation of the quantum codewords. We show that ponderomotive interaction suffices to this end. As a matter of fact, this kind of interaction between a system and a meter causes a frequency change on the meter proportional to the position quadrature of the system. Then, a phase measurement of the meter leaves the system in an eigenstate of the stabilizer generators, provided that system and meter's initial states are suitably prepared. Here we show how to implement this interaction using trapped ions, and how the encoding can be performed on their motional degrees of freedom. The robustness of the codewords with respect to the various experimental imperfections is then analyzed.

ERROR SYMMETRIZATION IN QUANTUM COMPUTERS

Errors in quantum computers are of two kinds:sudden perturbations to isolated qubits, and slow,random drifts of all the qubits. Isolated errors can becorrected by using quantum codewords that represent a logical qubit in a redundant way, by severalphysical qubits. On the other hand, slow drifts can bereduced, but not completely eliminated, by means ofsymmetrization, namely by using many replicas of the computer, and forcing their joint quantumstate to be completely symmetric. Several symmetrizationstrategies are examined and analyzed.

Correcting the effects of spontaneous emission on cold-trapped ions

We propose two quantum error correction schemes which increase the maximum storage time for qubits in a system of cold trapped ions, using a minimal number of ancillary qubits. Both schemes consider only the errors introduced by the decoherence due to spontaneous emission from the upper levels of the ions. Continuous monitoring of the ion fluorescence is used in conjunction with selective coherent feedback to eliminate these errors immediately following spontaneous emission events, and the conditional time evolution between quantum jumps is removed by symmetrizing the quantum codewords.

Quantum code words contradict local realism

Quantum code words are highly entangled combinations of two-state systems. The standard assumptions of local realism lead to logical contradictions similar to those found by Bell, Kochen, and Specker, Greenberger, Horne and Zeilinger, and Mermin. The new contradictions have some noteworthy features that did not appear in the older ones.