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arXiv:1506.03542 · Physical Review Letters

Efficient Quantum Compression for Ensembles of Identically Prepared Mixed States

Yuxiang Yang, Giulio Chiribella, Daniel Ebler

Published Feb 23, 2016• 28 citations
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Abstract

We present one-shot compression protocols that optimally encode ensembles of N identically prepared mixed states into O(logN) qubits. In contrast to the case of pure-state ensembles, we find that the number of encoding qubits drops down discontinuously as soon as a nonzero error is tolerated and the spectrum of the states is known with sufficient precision. For qubit ensembles, this feature leads to a 25% saving of memory space. Our compression protocols can be implemented efficiently on a quantum computer.

DOI
10.1103/physrevlett.116.080501

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We present one-shot compression protocols that optimally encode ensembles of N identically prepared mixed states into O(logN) qubits. In contrast to the case of pure-state ensembles, we find that the number of encoding qubits drops down discontinuously as soon as a nonzero error is tolerated and the spectrum of the states is known with sufficient precision. For qubit ensembles, this feature leads to a 25% saving of memory space. Our compression protocols can be implemented efficiently on a quantum computer.
We present an universal protocol, capable of compressing NNN copies of a completely unknown mixed state into O(log⁡N)O(\log N)O(logN) qubits in a lossless fashion. Our result demonstrates that the exponential compression of identically prepared pure states, recently implemented in a prototype experiment by Rozema et. al. [Phys. Rev. Lett. 113, 160504 (2014)], can be extended to arbitrary mixed states, despite the fact that the latter cover an exponentially large space. Furthermore, we highlight a striking difference between the pure and the mixed state case, showing that, if the spectrum is known with sufficient precision, the number of qubits needed for compression of a mixed state will drop down discontinuously as soon as a small recovery error is allowed.
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.
We propose a scheme for an exact efficient transformation of a tensor product state of many identically prepared qubits into a state of a logarithmically small number of qubits. Using a quadratic number of elementary quantum gates we transform NNN identically prepared qubits into a state, which is nontrivial only on the first $\ensuremath{\lceil}\mathrm{log}{}_{2}(N+1)\ensuremath{\rceil}$ qubits. This procedure might be useful for quantum memories, as only a small portion of the original qubits has to be stored. Another possible application is in communicating a direction encoded in a set of quantum states, as the compressed state provides a high-effective method for such an encoding.
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 revisit the task of visible compression of an ensemble of quantum states with entanglement assistance in the one-shot setting. The protocols achieving the best compression use many more qubits of shared entanglement than the number of qubits in the states in the ensemble. Other compression protocols, with potentially larger communication cost, have entanglement cost bounded by the number of qubits in the given states. This motivates the question as to whether entanglement is truly necessary for compression, and if so, how much of it is needed. Motivated by questions in communication complexity, we lift certain restrictions that are placed on compression protocols in tasks such as state-splitting and channel simulation. We show that an ensemble of the form designed by Jain, Radhakrishnan, and Sen (ICALP'03) saturates the known bounds on the sum of communication and entanglement costs, even with the relaxed compression protocols we study. The ensemble and the associated one-way communication protocol have several remarkable properties. The ensemble is incompressible by more than a constant number of qubits without shared entanglement, even when constant error is allowed. Moreover, in the presence of shared entanglement, the communication cost of compression can be arbitrarily smaller than the entanglement cost. The quantum information cost of the protocol can thus be arbitrarily smaller than the cost of compression without shared entanglement. The ensemble can also be used to show the impossibility of reducing, via compression, the shared entanglement used in two-party protocols for computing Boolean functions.
Quantum data compression is of great significance to both quantum computation and quantum communication due to the limited resources for quantum storage. Here in this study, we present an efficient multi-qubit quantum data compression protocol in which N identically prepared qubits can be perfectly compressed into qubits. Moreover, the feasible implementation of the quantum data compression is also designed based on the IBM Quantum Experience. The experiment reveals efficient information compression performance using the current scheme. We believe this method might be useful in quantum information which one could make the encoding of information on the qubits more efficiently, and also reducing the resources consumption for quantum storage.