For example, using Fano’s inequality, we obtain H( R| B)≤ h( p R)+ p Rlog 2( d−1), where p R is the probability that the outcomes of R and R′ are not equal and h is the binary entropy function. The probability with which the measurements on A and B disagree can be directly used to upper bound the entropies H( R| B) and H( S| B). Furthermore, a second measurement (of R′ or S′) should be applied to B trying to reproduce the outcome of the first. Analogously to the uncertainty game, we measure A with one of two observables, R or S. In the following, we will explain how it can be applied to the task of witnessing entanglement and to construct security proofs in quantum cryptography.įor the application to witnessing entanglement, consider a source that emits a two-particle state ρ A B. It is therefore qualitatively different from existing classical bounds.Īside from its fundamental significance, our result has an impact on the development of future quantum technologies. As a negative conditional entropy H( A| B) is a signature of entanglement 8, the uncertainty relation takes into account the entanglement between the particle and the memory. Fourth, in terms of new applications, the most interesting case is when A and B are entangled, but not maximally so. However, if the particle, A, is in a mixed state then H( A)>0 and the resulting bound is stronger than equation (1) even when there is no quantum memory. If the state of the particle, A, is pure, then H( A)=0 and we again recover the bound of Maassen and Uffink, equation (1). Third, in the absence of the quantum memory, B, we can reduce the bound equation (2) to H( R)+ H( S)≥log 21/ c+ H( A). As H( R| B)≤ H( R) and H( S| B)≤ H( S) for all states, we recover Maassen and Uffink’s bound, equation (1). Second, if A and B are not entangled (that is, their state is a convex combination of product states) then H( A| B)≥0. As discussed above, Bob can guess both R and S perfectly with such a strategy. As log 21/ c cannot exceed log 2 d, the bound in equation (2) reduces to H( R| B)+ H( S| B)≥0, which is trivial, because the conditional entropy of a system after measurement given the quantum memory cannot be negative. First, if the particle, A, and memory, B, are maximally entangled, then H( A| B)=−log 2 d, where d is the dimension of the particle sent to Alice. We continue by discussing some instructive examples. We sketch the proof of this relation in the Methods section and defer the full proof to the Supplementary Information. The extra term H( A| B) appearing on the right-hand side quantifies the amount of entanglement between the particle and the memory. The uncertainty about the outcome of measurement R given information stored in a quantum memory, B, is denoted by the conditional von Neumann entropy, H( R| B). It provides a bound on the uncertainties of the measurement outcomes that depends on the amount of entanglement between the measured particle, A, and the quantum memory, B. We proceed by stating our uncertainty relation, which applies in the presence of a quantum memory. Hence, the uncertainties about both observables, R and S, vanish, which shows that if one tries to generalize equation (1) by replacing the measure of uncertainty about R and S used there (the Shannon entropy) by the entropy conditioned on the information in Bob’s quantum memory, the resulting relation no longer holds. Then, for any measurement she chooses, there is a measurement on Bob’s memory that gives the same outcome as Alice obtains. To do so, he should maximally entangle his quantum memory with the particle he sends to Alice. However, with access to a quantum memory, Bob can beat this bound. We detail the application of our result to witnessing entanglement and to quantum key distribution.Įquation (1) bounds Bob’s uncertainty in the case that he has no quantum memory-all information Bob holds about the particle is classical, for example, a description of its density matrix. Here, we extend the uncertainty principle to incorporate this case, providing a lower bound on the uncertainties, which depends on the amount of entanglement between the particle and the quantum memory. However, if the particle is prepared entangled with a quantum memory, a device that might be available in the not-too-distant future 2, it is possible to predict the outcomes for both measurement choices precisely. It implies that one cannot predict the outcomes for both possible choices of measurement to arbitrary precision, even if information about the preparation of the particle is available in a classical memory. The principle bounds the uncertainties about the outcomes of two incompatible measurements, such as position and momentum, on a particle. The uncertainty principle, originally formulated by Heisenberg 1, clearly illustrates the difference between classical and quantum mechanics.
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