Randomizing Quantum States: Constructions and Applications

By Patrick M. Hayden, Debbie W. Leung, Peter W. Shor, Andreas J. Winter. 2003

359 citations recorded by the source corpus.

The construction of a perfectly secure private quantum channel in dimension d is known to require 2 log d shared random key bits between the sender and receiver. We show that if only near-perfect security is required, the size of the key can be reduced by a factor of two. More specifically, we show that there exists a set of roughly d log d unitary operators whose average effect on every input pure state is almost perfectly randomizing, as compared to the d2 operators required to randomize perfectly. Aside from the private quantum channel, variations of this construction can be applied to many other tasks in quantum information processing. We show, for instance, that it can be used to construct LOCC data hiding schemes for bits and qubits that are much more efficient than any others known, allowing roughly log d qubits to be hidden in 2 log d qubits. The method can also be used to exhibit the existence of quantum states with locked classical correlations, an arbitrarily large amplification of the correlation being accomplished by sending a negligibly small classical key. Our construction also provides the basic building block for a method of remotely preparing arbitrary d-dimensional pure quantum states using approximately log d bits of communication and log d ebits of entanglement.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/95adbc1cb9c884b85da7c92aa74b3632a98aa128

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Citations in the corpus

Random matrix techniques in quantum information theory (2015, background): “eview { did not have an obvious non-random answer, therefore it became not only natural, but also important, to consider how ‘typical’ quantum objects behave. This was arguably initiated by the paper [HLSW04]. The results obtained in the rst papers were of striking importance in quantum information ”
Aspects of Generic Entanglement (2004, methodology): “Recall next from the proof of Lemma II.3 in [ 22 ] that the inequalities (2) and (3) for s = 1 themselves come from exploiting the moment generating function, in particular, applying the general upper bound”
Entropic uncertainty relations—a survey (2009, methodology): “It was shown in (Hayden et al. , 2004) that L = (log d ) 4 unitaries U j chosen from the Haar measure randomly and independently obey with high probability, and for sufﬁciently large dimen-sion d .”
Quantum, classical, and total amount of correlations in a quantum state (2004, methodology): “By elementary linear algebra, e Now we show, using the “operator Chernoﬀ bound”, lemma A.3 in appendix A, that we can select a small subensemble of these unitaries doing the same job to suf-ﬁcient approximation (this is an argument like those used in [13]).”
Advances in quantum cryptography (2019, neutral): “[682] showed that a random choice of k random observables (distributed according to the unitary invariant measure) satisfies a maximally strong EUR with probability arbitrary close to 1, provided that d is large enough and k grows at least logarithmically in d (see also Refs.”
Quantum computation vs. firewalls (2013, background): “Such a set is called an -net, and it is not too hard to get an estimate of how many elements it must have [33].”
Local random quantum circuits are approximate polynomial-designs: numerical results (2012, methodology): “Examples of the use of random unitaries, drawn from the Haar measure on the unitary group, include the encoding for almost every known protocol for sending information down a quantum channel [2], approximate encryption of quantum information [46], quantum data-hiding [46], information locking [46], ”