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arXiv:1701.03372 · IEEE Transactions on Information Theory

Compression for Quantum Population Coding

Yuxiang Yang, Ge Bai, Giulio Chiribella, Masahito Hayashi

Published Jan 10, 2018• 14 citations
arXivHTMLPDF

Abstract

We study the compression of n quantum systems, each prepared in the same state belonging to a given parametric family of quantum states. For a family of states with f independent parameters, we devise an asymptotically faithful protocol that requires a hybrid memory of size (f/2)\log n, including both quantum and classical bits. Our construction uses a quantum version of local asymptotic normality and, as an intermediate step, solves the problem of compressing displaced thermal states of n identically prepared modes. In both cases, we show that (f/2)\log n is the minimum amount of memory needed to achieve asymptotic faithfulness. In addition, we analyze how much of the memory needs to be quantum. We find that the ratio between quantum and classical bits can be made arbitrarily small, but cannot reach zero: unless all the quantum states in the family commute, no protocol using only classical bits can be faithful, even if it uses an arbitrarily large number of classical bits.

DOI
10.1109/tit.2017.2788407

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We study the compression of n quantum systems, each prepared in the same state belonging to a given parametric family of quantum states. For a family of states with f independent parameters, we devise an asymptotically faithful protocol that requires a hybrid memory of size (f/2)\log n, including both quantum and classical bits. Our construction uses a quantum version of local asymptotic normality and, as an intermediate step, solves the problem of compressing displaced thermal states of n identically prepared modes. In both cases, we show that (f/2)\log n is the minimum amount of memory needed to achieve asymptotic faithfulness. In addition, we analyze how much of the memory needs to be quantum. We find that the ratio between quantum and classical bits can be made arbitrarily small, but cannot reach zero: unless all the quantum states in the family commute, no protocol using only classical bits can be faithful, even if it uses an arbitrarily large number of classical bits.
We study the compression of arbitrary parametric families of n identically prepared finite-dimensional quantum states, in a setting that can be regarded as a quantum analogue of population coding. For a family with f free parameters, we propose an asymptotically faithful protocol that requires a memory of overall size (f/2) log n. Our construction uses a quantum version of local asymptotic normality and, as an intermediate step, solves the problem of the optimal compression of n identically prepared displaced thermal states. Our protocol achieves the ultimate bound predicted by quantum Shannon theory. In addition, we explore the minimum requirement for quantum memory: On the one hand, the amount of quantum memory used by our protocol can be made arbitrarily small compared to the overall memory cost; on the other hand, any protocol using only classical memory cannot be faithful.
We consider the problem of compression of the quantum information carried by ensemble of mixed states. We prove that for arbitrary coding schemes the least number of quantum bits (qubits) per message needed to convey the signal states asymptotically faithfully is bounded from below by the Holevo function $S(\ensuremath{\varrho})\ensuremath{-}{\ensuremath{\sum}}_{i}{p}_{i}S({\ensuremath{\varrho}}_{i})$. We also show that a compression protocol can be composed with another one, provided that the latter offers perfect transmission. Such a compound protocol is applied to the case of binary source. It is conjectured to reach the obtained bound. Finally, we point out that in the case of mixed signal states there could be a difference between the maximal compression rates at the coding schemes that are ``blind'' to the signal and the ones that assume the knowledge about the identities of the signal states.
We show that the quantum Fisher information about any phase parameter encoded in a family of pure quantum states can be faithfully compressed into a single qubit, accompanied by a logarithmic amount of classical bits. When the phase is encoded into many identical copies of a qubit state on the equator of the Bloch sphere, we show that the compression can be implemented sequentially, by iteratively compressing pairs of qubits into a single qubit. We experimentally demonstrate this building block in a photonic setup, developing two alternative compression strategies, based on Type-I fusion gate and a postselected implementation of the CNOT gate.
Two-Ievel (qubit) clock systems are often used to perform precise measurement of time. In this work, we propose a compression protocol for n identically prepared states of qubit clocks. The protocol faithfully encodes the states into (1/2) logn qubits and (1/2) logn classical bits and works even in the presence of noise. If the purity of the clock states is fixed, (1/2) logn qubits are sufficient. We also prove that this protocol requires the minimum amount of total memory among all protocols with vanishing error in the large n limit.
Abstract We design quantum compression algorithms for parametric families of tensor network states. We first establish an upper bound on the amount of memory needed to store an arbitrary state from a given state family. The bound is determined by the minimum cut of a suitable flow network, and is related to the flow of information from the manifold of parameters that specify the states to the physical systems in which the states are embodied. For given network topology and given edge dimensions, our upper bound is tight when all edge dimensions are powers of the same integer. When this condition is not met, the bound is optimal up to a multiplicative factor smaller than 1.585. We then provide a compression algorithm for general state families, and show that the algorithm runs in polynomial time for matrix product states.
Free classical communication can yield an unbounded single-shot saving in quantum compression, even though it leaves the asymptotic rate of an i.i.d. source unchanged. At every fixed error in (0,1)(0,1)(0,1), we exhibit sources whose unassisted compression cost is arbitrarily large, yet a single qubit suffices with classical communication and no preshared entanglement. We establish this separation by determining the exact optimal source fidelity for compression through an mmm-dimensional noiseless quantum channel, with and without free classical communication, while preserving correlations with an inaccessible reference. Without classical assistance, the optimal fidelity is exactly the Ky Fan norm ∥ρ∥(m)\|ρ\|_{(m)}∥ρ∥(m)​, the sum of the mmm largest eigenvalues of the source state. With classical assistance, we obtain an explicit spectral threshold formula and a protocol attaining it. Classical communication strictly improves fidelity whenever mmm is smaller than the source rank. The corresponding compression costs are two smoothed versions of the log-rank entropy, also called max-entropy: trace-distance smoothing under local operations and quadratic smoothing under LOCC. For i.i.d. copies of a fixed source with positive entropy variance, however, the two costs have identical first-, second-, and third-order asymptotic terms. For nonlattice spectra, we determine the constant-order advantage: classical communication saves log⁡2(e)−1\log_2(e)-1log2​(e)−1 bits per block asymptotically, and the ratio of assisted to unassisted minimum communication dimensions converges to 2/e2/e2/e.