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arXiv:quant-ph/0601010 · IEEE Transactions on Information Theory

Optimal Encoding of Classical Information in a Quantum Medium

Noam Elron, Yonina C. Eldar

Published May 1, 2007• 20 citations
arXivHTMLPDF

Abstract

We investigate optimal encoding and retrieval of digital data, when the storage/communication medium is described by quantum mechanics. We assume an m-ary alphabet with arbitrary prior distribution, and an n-dimensional quantum system. Under these constraints, we seek an encoding-retrieval setup, comprised of code-states and a quantum measurement, which maximizes the probability of correct detection. In our development, we consider two cases. In the first, the measurement is predefined and we seek the optimal code-states. In the second, optimization is performed on both the code-states and the measurement. We show that one cannot outperform "pseudo-classical transmission," in which we transmit n symbols with orthogonal code-states, and discard the remaining symbols. However, such pseudo-classical transmission is not the only optimum. We fully characterize the collection of optimal setups, and briefly discuss the links between our findings and applications such as quantum key distribution and quantum computing

DOI
10.1109/tit.2006.894686

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Journal of Modern Optics•Nov 1, 1996•13 citations
We investigate optimal encoding and retrieval of digital data, when the storage/communication medium is described by quantum mechanics. We assume an m-ary alphabet with arbitrary prior distribution, and an n-dimensional quantum system. Under these constraints, we seek an encoding-retrieval setup, comprised of code-states and a quantum measurement, which maximizes the probability of correct detection. In our development, we consider two cases. In the first, the measurement is predefined and we seek the optimal code-states. In the second, optimization is performed on both the code-states and the measurement. We show that one cannot outperform "pseudo-classical transmission," in which we transmit n symbols with orthogonal code-states, and discard the remaining symbols. However, such pseudo-classical transmission is not the only optimum. We fully characterize the collection of optimal setups, and briefly discuss the links between our findings and applications such as quantum key distribution and quantum computing
Random access codes are a type of communication task that is widely used in quantum information science. The optimal average success probability that can be achieved through classical strategies is known for any random access code. However, only a few cases are solved exactly for quantum random access codes. In this paper, we provide bounds for the fully general setting of n independent variables, each selected from a d-dimensional classical alphabet and encoded in a D -dimensional quantum system subject to an arbitrary quantum measurement. The bound recovers the exactly known special cases, and we demonstrate numerically that even though the bound is not tight overall, it can still yield a good approximation.
Random access codes are a type of communication task that is widely used in quantum information science. The optimal average success probability that can be achieved through classical strategies is known for any random access code. However, only a few cases are solved exactly for quantum random access codes. In this paper, we provide bounds for the fully general setting of n independent variables, each selected from a d-dimensional classical alphabet and encoded in a D-dimensional quantum system subject to an arbitrary quantum measurement. The bound recovers the exactly known special cases, and we demonstrate numerically that even though the bound is not tight overall, it can still yield a good approximation.
Quantum computation and information has become a central research area in theoretical computer science. It studies how information is encoded in nature according to the laws of quantum mechanics and what this means for its computational power. On one hand, the ability of a quantum system to be in a superposition of states provides a way to encode an exponential amount of information in a quantum state. However, this information is accessible only indirectly via measurements. The goal of this thesis is precisely to investigate the fundamental question of how information is encoded and processed in quantum mechanical systems. (1) We investigate their power relative to classical encodings in the model of one-way communication complexity and show that they can be exponentially more efficient, answering a long standing open question in the area of quantum communication complexity. (2) We show how the theory developed for the study of quantum information can be employed in order to answer questions about classical coding theory and complexity. We resolve a main open question about the efficiency of Locally Decodable Codes by reducing the problem to one about quantum codes and solving the latter using tools from quantum information theory. This is the first example where quantum techniques are essential in resolving a purely classical question. (3) We investigate the power of quantum encodings in the area of cryptography. We study certain cryptographic primitives, such as Private Information Retrieval, Symmetrically-Private Information Retrieval and Coin Flipping.
We consider the rate-limited quantum-to-classical optimal transport in terms of output-constrained rate-distortion coding for both finite-dimensional and continuous-variable quantum-to-classical systems with limited classical common randomness. The main coding theorem provides a single-letter characterization of the achievable rate region of a lossy quantum measurement source coding for an exact construction of the destination distribution (or the equivalent quantum state) while maintaining a threshold of distortion from the source state according to a generally defined distortion observable. The constraint on the output space fixes the output distribution to an IID predefined probability mass function. Therefore, this problem can also be viewed as information-constrained optimal transport which finds the optimal cost of transporting the source quantum state to the destination classical distribution via a quantum measurement with limited communication rate and common randomness. We develop a coding framework for continuous-variable quantum systems by employing a clipping projection and a dequantization block and using our finite-dimensional coding theorem. Moreover, for the Gaussian quantum systems, we derive an analytical solution for rate-limited Wasserstein distance of order 2, along with a Gaussian optimality theorem, showing that Gaussian measurement optimizes the rate in a system with Gaussian quantum source and Gaussian destination distribution. The results further show that in contrast to the classical Wasserstein distance of Gaussian distributions, which corresponds to an infinite transmission rate, in the Quantum Gaussian measurement system, the optimal transport is achieved with a finite transmission rate due to the inherent noise of the quantum measurement imposed by Heisenberg’s uncertainty principle.
We consider the characterization as well as the construction of quantum codes that allow to transmit both quantum and classical information, which we refer to as `hybrid codes'. We construct hybrid codes [![n,k:m,d]!]q[\\![n,k{: }m,d]\\!]_q[![n,k:m,d]!]q​ with length nnn and distance ddd, that simultaneously transmit kkk qudits and mmm symbols from a classical alphabet of size qqq. Many good codes such as [![7,1:1,3]!]2[\\![7,1{: }1,3]\\!]_2[![7,1:1,3]!]2​, [![9,2:2,3]!]2[\\![9,2{: }2,3]\\!]_2[![9,2:2,3]!]2​, [![10,3:2,3]!]2[\\![10,3{: }2,3]\\!]_2[![10,3:2,3]!]2​, [![11,4:2,3]!]2[\\![11,4{: }2,3]\\!]_2[![11,4:2,3]!]2​, [![11,1:2,4]!]2[\\![11,1{: }2,4]\\!]_2[![11,1:2,4]!]2​, [![13,1:4,4]!]2[\\![13,1{: }4,4]\\!]_2[![13,1:4,4]!]2​, [![13,1:1,5]!]2[\\![13,1{: }1,5]\\!]_2[![13,1:1,5]!]2​, [![14,1:2,5]!]2[\\![14,1{: }2,5]\\!]_2[![14,1:2,5]!]2​, [![15,1:3,5]!]2[\\![15,1{: }3,5]\\!]_2[![15,1:3,5]!]2​, [![19,9:1,4]!]2[\\![19,9{: }1,4]\\!]_2[![19,9:1,4]!]2​, [![20,9:2,4]!]2[\\![20,9{: }2,4]\\!]_2[![20,9:2,4]!]2​, [![21,9:3,4]!]2[\\![21,9{: }3,4]\\!]_2[![21,9:3,4]!]2​, [![22,9:4,4]!]2[\\![22,9{: }4,4]\\!]_2[![22,9:4,4]!]2​ have been found. All these codes have better parameters than hybrid codes obtained from the best known stabilizer quantum codes.
The calculation method proposed here obtains an optimum detection process for an M-ary signal whose quantum states are linearly independent and nonorthogonal. Optimizations are carried out by minimizing the average probability of error or by maximizing the mutual information in the signal detection process. Quantum detection operators, in terms of which the signal detection process is mathematically described, are assumed to be projection operators defined on a Hilbert space H s spanned by the quantum states of the signal; that is, the best projection operators that describe the signal detection process are obtained. The optimization procedure is equivalent to finding an appropriate unitary transformation in the Hilbert space H s. For a binary quantum state signal, the optimum detection processes in which the average probability of error is minimized and in which the mutual information is maximized are obtained. It is found that the optimum detection process minimizing the average probability of error is the same as that maximizing the mutual information.