dorsal/arxiv
View SchemaTrading quantum for classical resources in quantum data compression
| Authors | Patrick Hayden, Richard Jozsa, Andreas Winter |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0204038 |
| URL | https://arxiv.org/abs/quant-ph/0204038 |
| DOI | 10.1063/1.1497184 |
| Journal | J. Math. Phys. Vol. 43 No. 9 pp. 4404-4444 (2002) |
Abstract
We study the visible compression of a source E of pure quantum signal states, or, more formally, the minimal resources per signal required to represent arbitrarily long strings of signals with arbitrarily high fidelity, when the compressor is given the identity of the input state sequence as classical information. According to the quantum source coding theorem, the optimal quantum rate is the von Neumann entropy S(E) qubits per signal. We develop a refinement of this theorem in order to analyze the situation in which the states are coded into classical and quantum bits that are quantified separately. This leads to a trade--off curve Q(R), where Q(R) qubits per signal is the optimal quantum rate for a given classical rate of R bits per signal. Our main result is an explicit characterization of this trade--off function by a simple formula in terms of only single signal, perfect fidelity encodings of the source. We give a thorough discussion of many further mathematical properties of our formula, including an analysis of its behavior for group covariant sources and a generalization to sources with continuously parameterized states. We also show that our result leads to a number of corollaries characterizing the trade--off between information gain and state disturbance for quantum sources. In addition, we indicate how our techniques also provide a solution to the so--called remote state preparation problem. Finally, we develop a probability--free version of our main result which may be interpreted as an answer to the question: ``How many classical bits does a qubit cost?'' This theorem provides a type of dual to Holevo's theorem, insofar as the latter characterizes the cost of coding classical bits into qubits.
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"abstract": "We study the visible compression of a source E of pure quantum signal states,\nor, more formally, the minimal resources per signal required to represent\narbitrarily long strings of signals with arbitrarily high fidelity, when the\ncompressor is given the identity of the input state sequence as classical\ninformation. According to the quantum source coding theorem, the optimal\nquantum rate is the von Neumann entropy S(E) qubits per signal.\n We develop a refinement of this theorem in order to analyze the situation in\nwhich the states are coded into classical and quantum bits that are quantified\nseparately. This leads to a trade--off curve Q(R), where Q(R) qubits per signal\nis the optimal quantum rate for a given classical rate of R bits per signal.\n Our main result is an explicit characterization of this trade--off function\nby a simple formula in terms of only single signal, perfect fidelity encodings\nof the source. We give a thorough discussion of many further mathematical\nproperties of our formula, including an analysis of its behavior for group\ncovariant sources and a generalization to sources with continuously\nparameterized states. We also show that our result leads to a number of\ncorollaries characterizing the trade--off between information gain and state\ndisturbance for quantum sources. In addition, we indicate how our techniques\nalso provide a solution to the so--called remote state preparation problem.\nFinally, we develop a probability--free version of our main result which may be\ninterpreted as an answer to the question: ``How many classical bits does a\nqubit cost?\u0027\u0027 This theorem provides a type of dual to Holevo\u0027s theorem, insofar\nas the latter characterizes the cost of coding classical bits into qubits.",
"arxiv_id": "quant-ph/0204038",
"authors": [
"Patrick Hayden",
"Richard Jozsa",
"Andreas Winter"
],
"categories": [
"quant-ph"
],
"doi": "10.1063/1.1497184",
"journal_ref": "J. Math. Phys. Vol. 43 No. 9 pp. 4404-4444 (2002)",
"title": "Trading quantum for classical resources in quantum data compression",
"url": "https://arxiv.org/abs/quant-ph/0204038"
},
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