Conditional entropic uncertainty relations for Tsallis entropies
The entropic uncertainty relations are a very active field of scientific inquiry. Their applications include quantum cryptography and studies of quantum phenomena such as correlations and non-locality. In this work we find state-independent entropic uncertainty relations in terms of the Tsallis entropies for states with a fixed amount of entanglement. Our main result is stated as Theorem 1. Taking the special case of von Neumann entropy and utilizing the concavity of conditional von Neumann entropies, we extend our result to mixed states. Finally we provide a lower bound on the amount of extractable key in a quantum cryptographic scenario.
Keywords:Conditional uncertainty relations Tsallis entropies Quantum cryptography
Formulated by Heisenberg heisenberg1927anschaulichen (), the uncertainty relation gives insight into differences between classical and quantum mechanics. According to the relation, simultaneous measurements of some non-commuting observables of a particle cannot be predicted with arbitrary precision.
Numerous studies over the uncertainty relations led to entropic formulation by Białynicki-Birula and Mycielski bialynicki1975uncertainty (), as a sum of two continuous Shannon entropies, for probability distributions of position and momentum. As our goal is to consider general observables, let us choose two Hermitian non-commuting operators and . The first uncertainty relation that holds for a pair of arbitrary observables was derived by Deutsch deutsch1983uncertainty ()
where and denote the Shannon entropies of the probability distributions obtained during measurements of and respectively. If , are the eigenvectors of and , then . Maassen and Uffink maassen1988generalized () obtained a stronger result
where , and are the same as in relation proposed by Deutsch.
The entropic uncertainty relations are a very active field of scientific inquiry wehner2010entropic (); coles2017entropic (). One of the reasons are the applications in quantum cryptography koashi2005simple (); divincenzo2004locking (); damgaard2008cryptography (). Another area where entropic uncertainty relations are widely used are studies of quantum phenomena such as correlations and non-locality guhne2004characterizing (); oppenheim2010uncertainty (); rastegin2016separability (). Some results were generalized, hence entropic formulations of the uncertainty relation in terms of Rényi entropies is included in zozor2013generalized (). Uncertainty relations for mutually unbiased bases and symmetric informationally complete measurements in terms of generalized entropies of Rényi and Tsallis can be found rastegin2013uncertainty ().
In it kaniewski2014entropic () was shown that entropic uncertainty relations can be derived for binary observables from effective anti-commutation, which can be important in device-independent cryptography. This result was generalized in xiao2017uncertainty () for entropic uncertainty relations in the presence of quantum memory.
The majorization-based bounds of uncertainty relation were first introduced by Partovi in partovi2011majorization (), which was generalized in puchala2013majorization (); friedland2013universal (). In puchala2013majorization (), majorization techniques were applied to obtain lower bound of the uncertainty relation, which can give the bound stronger than the well know result of Massen and Uffink. The formulation of strong majorization uncertainty relation presented in rudnicki2014strong () is involved, but in the case of qubits it can be expressed as
The asymptotic analysis of entropic uncertainty relations for random measurements has been provided in adamczak2016asymptotic () with the use majorization bounds. Some interesting results along these lines are included in rudnicki2015majorization (); rastegin2016majorization (); puchala2015certainty ().
In berta2010uncertainty (), Berta et al. considered the uncertainty relation for a system with the presence of a quantum memory. In this setup, the system is described by a bipartite density matrix . Quantum conditional entropy can be defined as
where denotes the von Neumann entropy of the state . Eq. (4) is also known as the chain rule. We also introduce the states and as
which are post-measurement states, when the measurements were performed on the part . Berta et al. berta2010uncertainty () showed that a bound on the uncertainties of the measurement outcomes depends on the amount of entanglement between measured particle and the quantum memory. As a consequence, they formulated a conditional uncertainty relation given as
Entropy quantifies the amount of entanglement between the particle and the memory. The bound of Berta et al. berta2010uncertainty () was improved by Coles and Piani in coles2014improved () through replacing the state-dependent value with larger parameter. The result of Coles and Piani was improved in xiao2016improved (). This relation was also generalized for Rényi entropies and several important result can be found in coles2012uncertainty (); muller2013quantum (); tomamichel2014relating (). The uncertainty relation is also considered in the context of quantum-to-classical randomness extractors (QC-extractors) berta2014quantum (). It is proved that QC-extractors gives rise to uncertainty relation with the presence of a quantum memory.
In the absence of the quantum memory the bound (6) reduces to (2) for pure . The results by Berta et al. berta2010uncertainty () can be applied to the problem of entanglement detection li2011experimental () or quantum cryptography coles2017entropic (). The bound quantified by Berta et al. berta2010uncertainty () was experimentally validated prevedel2011experimental ().
In this paper we aim at finding state-independent entropic uncertainty relations in terms of von Neumann and Tsallis entropies. Our results apply to states with fixed amount of entanglement. The Tsallis entropy tsallis1988possible () is a non-additive generalization of von Neumann entropy and for a state it is defined as
where are the eigenvalues of and . Tsallis entropy is identical to the Havrda-Charvát structural -entropy havrda1967quantification () in information theory. Note that when we have . The chain rule applies to Tsallis entropies, hence
We will use the following notation for Tsallis point entropy
In the limit we recover
2 Qubit conditional uncertainty relations
Without a loss of generality let us assume that we start with an entangled state , where . In this case, the parameter describes the entanglement between the parties and . We chose the eigenvectors of and as and , where
is a real rotation matrix. Hence, instead of optimizing the uncertainty relation over all possible states , we will instead optimize over . Hereafter we assume . In this case we have
It is important to notice, that we can restrict our attention to real rotation matrices. This follows from the fact, that any unitary matrix is similar to real rotation matrix. Matrices are similar, , if for some permutation matrices and diagonal unitary matrices , we have puchala2013majorization (). Next we note that the eigenvalues of states are invariant with respect to the equivalence relation.
We should also note here, that the two qubit scenario, simple as it is, may be easily generalized to an arbitrary dimension of system .
As we are interested in binary measurements, the states and are rank-2 operators. The non-zero eigenvalues of can be easily obtained as
To obtain the eigenvalues of we need to replace with .
2.1 Analytical minima
Using eigenvalues of and , we arrive at
Let us perform detailed analysis on the case when , ie. the von Neumann entropy case. We get
In order to obtain an uncertainty relation, we need to minimize this quantity over the parameter . This is a complicated task even in the case and has been studied earlier bosyk2011comment (). We guess that is an extremal point of (14). Unfortunately, this point is the global minimum only when
A numerical solution of this inequality is shown in Fig. 1. When this condition is satisfied, the uncertainty relation is
When the condition in Eq. (16) is not satisfied, our guessed extreme point becomes a maximum and two minima emerge, symmetrically to . The reasoning can be generalized to in a straightforward, yet cumbersome way. We will omit the details here. The solutions of an inequality similar to (16) for various values of are shown in Fig.1
2.2 Bounding the relative entropies
In order to study the case of general Tsallis entropies , we introduce the following proposition
Let and , then
In the cases and we have an equality.
Next we note, that . We will show, that has no other zeros on interval . We calculate
which is positive for and . Therefore we obtain, that is strictly convex on and .
Now let us assume, that for we have . Then by Rolle‘s theorem, there exist points such that . Together with fact that we obtain a contradiction with the convexity of on .
Last thing to show is that for some we have . To show it we write
Now we note, that is positive for and . This follows form convexity of on these sets and the fact, that it has a minimum, . From this fact there exist such that .
The equalities in the case follow from a direct inspection.
Now we are ready to state and prove the main result of this work
Let , where . Let us choose two observables and with eigenvectors and , where is as in Eq. (11). Then, the Tsallis entropic conditional uncertainty relation is
In the limit we get the following uncertainty relation for Shannon entropies
Using the concavity of the conditional von Neumann entropy, we may generalize bound (24) to mixed states . We get
The state dependent entropic uncertainty relation for reads
A comparison with the known entropic uncertainty relations for and is shown in Fig. 2. As can be seen, our result gives a tighter bound than the one obtained by Massen and Uffink for all values of . The bound is also tighter than when is in the neighborhood of .
3 Security of quantum key distribution protocols
One of the possible application of the uncertainty relation is quantum cryptography, where the relation allows us to bound of the amount of key the parties are able to extract per state.
Assume that an eavesdropper creates a quantum system . Next, parts and are distributed to Alice and Bob. The generation of a secret key is based on measurements and performed by Alice and Bob, respectively. Subsequently, Alice and Bob inform each other of their choices of measurements. The security of the key depends on the correlation between the measurement outcomes.
According to the investigations of Devetak and Winter devetak2005distillation (), the amount of extractable key is quantified as . Using our bound we are able to bound the amount of extractable key in the terms of von Neumann entropies by
In the above is the conditional entropy of the state shared by Alice and Bob, when both parties execute the measurement schemes respectively. This relates our result to shor2000simple (). In our case Alice and Bob need to upper bound entropies and . The former entropies can be bounded by quantities such as frequency of the agreement of the outcomes.
In this paper, we have derived new state-independent uncertainty relations in terms of von Neumann and Tsallis entropies for qubits and binary observables with respect to quantum side information. Our bounds were compared with well-know bounds derived by Massen and Uffink maassen1988generalized (), Rudnicki et al. rudnicki2014strong () and Berta et al. berta2010uncertainty (). This paper can be also treated as a generalization of results included in bosyk2011comment ().
Presented results are expected to have application to witnessing entanglement or in quantum cryptography as a measure of information in quantum key distribution protocols. Verification of our results in potential applications seems to be interesting task.
Acknowledgements.The authors acknowledge the support by the Polish National Science Center under the Project Numbers 2013/11/N/ST6/03090 (D. K.), 2015/17/B/ST6/01872 (Ł. P.) and 2016/22/E/ST6/00062 (Z. P.).
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