Improved bounds in the entropic uncertainty and certainty relations for complementary observables

Sánchez-Ruiz, Jorge

doi:10.1016/0375-9601(95)00219-s

Cited by 180 publications

(157 citation statements)

References 8 publications

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“…But, on the other hand, we also get relations that upper bound the uncertainties of incompatible observables. In the literature, these are known as certainty relations [35,36], and here we give such relations that allow for quantum side information. For example, we get, in terms of the conditional min-entropy (again up to a small error term > 0),…”

Section: B Uncertainty and Certainty Relationsmentioning

confidence: 99%

Entanglement-assisted guessing of complementary measurement outcomes

2014

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Heisenberg's uncertainty principle implies that if one party (Alice) prepares a system and randomly measures one of two incompatible observables, then another party (Bob) cannot perfectly predict the measurement outcomes. This implication assumes that Bob does not possess an additional system that is entangled to the measured one; indeed, the seminal paper of Einstein, Podolsky, and Rosen (EPR) showed that maximal entanglement allows Bob to perfectly win this guessing game. Although not in contradiction, the observations made by EPR and Heisenberg illustrate two extreme cases of the interplay between entanglement and uncertainty. On the one hand, no entanglement means that Bob's predictions must display some uncertainty. Yet on the other hand, maximal entanglement means that there is no more uncertainty at all. Here we follow an operational approach and give an exact relation-an equality-between the amount of uncertainty as measured by the guessing probability and the amount of entanglement as measured by the recoverable entanglement fidelity. From this equality, we deduce a simple criterion for witnessing bipartite entanglement and an entanglement monogamy equality. I. UNCERTAINTY RELATIONSHeisenberg's uncertainty principle forms one of the fundamental elements of quantum mechanics. Originally proven for measurements of position and momentum, it is one of the most striking examples of the difference between a quantum and a classical world [1]. Uncertainty relations today are probably best known in the form given by Robertson [2], who extended Heisenberg's result to two arbitrary observables X and Z. More precisely, Robertson's relation states that when measuring the state |ψ using either X or Z, one findswhere Y = ψ|Y 2 |ψ − ψ|Y |ψ 2 for Y ∈ {X,Z} is the standard deviation resulting from measuring |ψ with observable Y . In the modern-day literature, uncertainty is usually measured in terms of entropies (starting with [3][4][5]; see [6] for a survey). One of the reasons this is desirable is that Eq. (1) makes no statement if |ψ happens to give zero expectation on [X,Z] [7]. To see how uncertainty can be quantified in terms of entropies, let us start with a simple example. Throughout, we let Alice (A) denote the system to be measured. For now, let us consider measuring a single qubit in the state ρ A using two incompatible measurements given by the Pauli σ x or σ z eigenbases, and let K be the random variable associated with the measurement outcome. We have from [8] that for any state ρ A ,where H (K| = θ ) = − k p k| =θ log p k| =θ is the Shannon entropy (all logarithms are base 2 in this article) of the probability distribution over measurement outcomes k ∈ {0,1} when we perform the measurement labeled θ on the state ρ A , and each measurement is chosen with probability p θ = 1/2. To see that this is an uncertainty relation, note that if one of the two entropies is zero, then (2) tells us that the other is necessarily nonzero, i.e., there is at least some amount of uncertainty. If we measure a d A -dimensional...

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Section: B Uncertainty and Certainty Relationsmentioning

confidence: 99%

Entanglement-assisted guessing of complementary measurement outcomes

2014

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“…In fact, analytical solutions have been found so far only for a few two-dimensional (qubit) cases, where the Bloch vectors of POVM elements constitute an n-gon [6,52,103], a tetrahedron [90] or an octahedron [32,98]. All these POVMs are symmetric (group covariant), but, as we shall see, symmetry alone is not enough to solve the problem analytically.…”

Section: Introductionmentioning

confidence: 98%

Highly symmetric POVMs and their informational power

Słomczyński

Szymusiak

2015

Quantum Inf Process

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We discuss the dependence of the Shannon entropy of normalized finite rank-1 POVMs on the choice of the input state, looking for the states that minimize this quantity. To distinguish the class of measurements where the problem can be solved analytically, we introduce the notion of highly symmetric POVMs and classify them in dimension 2 (for qubits). In this case, we prove that the entropy is minimal, and hence, the relative entropy (informational power) is maximal, if and only if the input state is orthogonal to one of the states constituting a POVM. The method used in the proof, employing the Michel theory of critical points for group action, the Hermite interpolation, and the structure of invariant polynomials for unitary-antiunitary groups, can also be applied in higher dimensions and for other entropy-like functions. The links between entropy minimization and entropic uncertainty relations, the Wehrl entropy, and the quantum dynamical entropy are described.

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“…However, to apply this to our cipher W n , we would like to look at a set of only 2 n bases and we want a bound on the sum of the entropies H[|u , B i ] and not the sum of the collision probabilities. This can be solved following a line of arguments from Sánchez-Ruiz [7]. Using Jensen's inequality, we can compute as follows:…”

Section: Theoremmentioning

confidence: 99%

“…Shannon entropy can be combined with known results on so called entropic uncertainty relations [5,3,7] and mutually unbiased bases [8]. We note that somewhat related techniques are used in concurrent independent work by DiVincenzo et al [2] to handle a different, non-cryptographic scenario.…”

Section: Introductionmentioning

confidence: 99%

On the Key-Uncertainty of Quantum Ciphers and the Computational Security of One-Way Quantum Transmission

Damgård¹,

Pedersen²,

Salvail³

2004

Advances in Cryptology - EUROCRYPT 2004

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Abstract. We consider the scenario where Alice wants to send a secret (classical) n-bit message to Bob using a classical key, and where only one-way transmission from Alice to Bob is possible. In this case, quantum communication cannot help to obtain perfect secrecy with key length smaller then n. We study the question of whether there might still be fundamental differences between the case where quantum as opposed to classical communication is used. In this direction, we show that there exist ciphers with perfect security producing quantum ciphertext where, even if an adversary knows the plaintext and applies an optimal measurement on the ciphertext, his Shannon uncertainty about the key used is almost maximal. This is in contrast to the classical case where the adversary always learns n bits of information on the key in a known plaintext attack. We also show that there is a limit to how different the classical and quantum cases can be: the most probable key, given matching plain-and ciphertexts, has the same probability in both the quantum and the classical cases. We suggest an application of our results in the case where only a short secret key is available and the message is much longer. Namely, one can use a pseudorandom generator to produce from the short key a stream of keys for a quantum cipher, using each of them to encrypt an n-bit block of the message. Our results suggest that an adversary with bounded resources in a known plaintext attack may potentially be in a much harder situation against quantum stream-ciphers than against any classical stream-cipher with the same parameters.

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Improved bounds in the entropic uncertainty and certainty relations for complementary observables

Cited by 180 publications

References 8 publications

Entanglement-assisted guessing of complementary measurement outcomes

Entanglement-assisted guessing of complementary measurement outcomes

Highly symmetric POVMs and their informational power

On the Key-Uncertainty of Quantum Ciphers and the Computational Security of One-Way Quantum Transmission

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