Entropic uncertainty relations and their applications

Heisenberg’s uncertainty principle forms a fundamental element of quantum mechanics. Uncertainty relations in terms of entropies were initially proposed to deal with conceptual shortcomings in the original formulation of the uncertainty principle and, hence, play an important role in quantum foundations. More recently, entropic uncertainty relations have emerged as the central ingredient in the security analysis of almost all quantum cryptographic protocols, such as quantum key distribution and two-party quantum cryptography. This review surveys entropic uncertainty relations that capture Heisenberg’s idea that the results of incompatible measurements are impossible to predict, covering both finite- and infinite-dimensional measurements. These ideas are then extended to incorporate quantum correlations between the observed object and its environment, allowing for a variety of recent, more general formulations of the uncertainty principle. Finally, various applications are discussed, ranging from entanglement witnessing to wave-particle duality to quantum cryptography.

Figure 1

Physical scenario relevant to preparation uncertainty relations. Each incoming particle is measured using either measurement $P$ or measurement $Q$, where the choice of the measurement is random. An uncertainty relation says we cannot predict the outcomes of both $P$ and $Q$. If we can predict the outcome of $P$ well, then we are necessarily uncertain about the outcome of measurement $Q$, and vice versa.

Figure 2

Diagram showing a guessing game with players Alice and Bob. First, Bob prepares $A$ in state ${\rho }_A$ and sends it to Alice. Second, Alice measures either $\mathbb{X}$ or $\mathbb{Z}$ with equal probability and stores the measurement choice in the bit $\mathrm{\Theta }$. Third, Alice stores the measurement outcome in bit $K$ and reveals the measurement choice $\mathrm{\Theta }$ to Bob. Bob’s task is to guess $K$ (given $\mathrm{\Theta }$). Entropic uncertainty relations like the Maassen-Uffink relation (5) can be understood as fundamental constraints on the optimal guessing probability.

Figure 3

Illustration for example t3, where a particle is initially confined to the two small boxes at the end and excluded from the long middle box. Then the particle is allowed to go free into the middle box.

Figure 4

Two probability distributions with the same standard deviation but different entropy, as explained in example t5.

Figure 5

Rényi entropies of $X$ with probability distribution as in example t6 with $|X|=65$ compared to a uniform random variable $U$ on 4 bits.

Figure 6

Various literature bounds on entropic uncertainty for qubit orthonormal bases as a function of the maximum overlap $c$. The region above $q_{\mathrm{opt}}$ contains pairs $(c,q)$ that can be achieved by quantum mechanics.

Figure 7

Comparison of classical and quantum memory.

Figure 8

The guessing game in the presence of a quantum memory system. First, Bob prepares $AB$ in state ${\rho }_{AB}$ and then sends system $A$ to Alice. Second, Alice performs either the $\mathbb{X}$ or $\mathbb{Z}$ measurement on $A$ and then announces the measurement choice $\mathrm{\Theta }$ to Bob. Bob’s task is to correctly guess $K$. The question is thus: how uncertain is Bob about Alice’s measurement outcome $K$, given that he has access to $B$ and $\mathrm{\Theta }$?

Figure 9

For $d=5$ and a various number of MUBs $n\le d+1=6$ the lower bounds (lb) are plotted on the entropic uncertainty $H_{\mathrm{coll}}(K|\mathrm{\Theta })$ from Eq. (194) as a function of $H_{\mathrm{coll}}(A|B)$. Moreover, for $n<6$ we have only the trivial upper bound (ub) on $H_{\mathrm{coll}}(K|B\mathrm{\Theta })$, whereas for $n=6$ the lower and upper bounds coincide as in Eq. (185).

Figure 10

Diagram of the tripartite quantum memory setup. First, a source prepares $ABC$ in state ${\rho }_{ABC}$, and sends $A$ to Alice, $B$ to Bob, and $C$ to Charlie. Second, Alice measures either $\mathbb{X}$ or $\mathbb{Z}$ on $A$ and asks how uncertain is Bob about her $X$ outcome, given $B$, and how uncertain is Charlie about her $Z$ outcome, given $C$? As shown in Eq. (206) there is a trade-off that is quantified by the complementarity of the measurements $X$ and $Z$. W interpret this scenario as a guessing game, also called a monogamy game. In this game, Bob and Charlie play against Alice. They prepare ${\rho }_{ABC}$ where they send $A$ to Alice, Bob keeps $B$, and Charlie keeps $C$. Alice then randomly chooses a measurement obtaining the measurement outcome $K$. Afterward, she sends her choice of basis to Bob and Charlie. They win the game if and only if both output $K$. This game measures the same kind of uncertainty as Eq. (206), explicitly exploiting the monogamy of entanglement: if Bob produces $K=X$ correctly in case Alice measured $\mathbb{X}$, then this is a certificate that Charlie cannot produce a good guess of $K=Z$ in case Alice measured $\mathbb{Z}$.

Figure 11

How to convert the dynamical evolution of a system into (a) a bipartite mixed state or (b) a tripartite pure state.

Figure 12

Gaussian wave packet in position space with ${\mathrm{\Gamma }}_Q(q)$ as in Eq. (256), as well as the finite resolution discretization from Eq. (260) in intervals of size $\delta$.

Figure 13

Comparison of the lower bounds in Eqs. (269) and (270) on the uncertainty generated by the finite-spacing position and momentum measurements $Q_{\delta }$, $P_{\delta }$ as in Eq. (260). Note that the latter bound becomes negative and hence trivial for larger spacings ${\delta }_q{\delta }_p\gtrsim 8.5$.

Figure 14

The uncertainty $h(Q|B)+h(P)$ of the EPR state from example t33 in terms of the squeezing strength $r$.

Figure 15

Preparation and measurement phase of the QKD protocol described in Sec. 6b1.

Figure 16

The noisy-storage model: [252] and [144] assumed that during waiting times $\mathrm{\Delta }t$ in the protocol, the adversary can keep only quantum information in an imperfect and limited storage device described by a quantum channel $\mathcal{F}$. This is the only restriction and the adversary is otherwise arbitrarily powerful. In particular, he can first store all incoming qubits and has a quantum computer to encode them into an arbitrary quantum error-correcting code to protect them against the noise of the channel $\mathcal{F}$. He can also keep an unlimited amount of classical memory and perform any operation instantaneously.

Figure 17

Entanglement witnessing for a bipartite two-qubit state using mutually unbiased observables. Suppose Alice and Bob observe $\mathrm{Pr}[X_A=X_B=0]=\mathrm{Pr}[X_A=X_B=1]=(1-e_X)/2$ and $\mathrm{Pr}[X_A=0,X_B=1]=\mathrm{Pr}[X_A=1,X_B=0]=e_X/2$, and analogously for $Z$ and $e_Z$. The region below the curve indicates the region for which one can guarantee entanglement for the respective witnesses.

Figure 18

Mach-Zehnder interferometer for single photons. A photon impinges on a beam splitter, after which we label the two possible paths by the $\mathbb{Z}$ basis states $|0⟩,|1⟩$. The photon may interact with some environment $E$ inside the interferometer. Then a phase $\varphi$ is applied to the lower path, and the two paths are recombined on a second beam splitter. Finally the photon is detected at either $D_0$ or $D_1$.

Figure 19

Two scenarios considered by [36], which capture (a) the noise of an attempted $\mathbb{X}$ measurement, and (b) the disturbance of the $\mathbb{Z}$ observable.