Entropic uncertainty relations for quantum information scrambling

How violently do two quantum operators disagree? Different fields of physics feature different measures of incompatibility: (i) In quantum information theory, entropic uncertainty relations constrain measurement outcomes. (ii) In condensed matter and high-energy physics, the out-of-time-ordered correlator (OTOC) signals scrambling, the spread of information through many-body entanglement. We unite these measures, proving entropic uncertainty relations for scrambling. The entropies are of distributions over weak and strong measurements' possible outcomes. The weak measurements ensure that the OTOC quasiprobability (a nonclassical generalization of a probability, which coarse-grains to the OTOC) governs terms in the uncertainty bound. The quasiprobability causes scrambling to strengthen the bound in numerical simulations of a spin chain. This strengthening shows that entropic uncertainty relations can reflect the type of operator disagreement behind scrambling. Generalizing beyond scrambling, we prove entropic uncertainty relations satisfied by commonly performed weak-measurement experiments. We unveil a physical significance of weak values (conditioned expectation values): as governing terms in entropic uncertainty bounds.

Entropic uncertainty relations and OTOCs occupy disparate subfields, but both quantify operator disagreement. We unite these quantifications, deriving three entropic uncertainty relations for QI scrambling (Theorem 1). Each relation contains a bound dependent on the quasiprobabilities in terms of which the OTOC decomposes [35,36]. Quasiprobabilities resemble probabilities but can behave nonclassically, assuming negative and nonreal values. Quasiprobabilities describe weak measurements, which barely disturb the measured quantum system [37]. Our uncertainty relations generalize: Quasiprobabilities and weak values (conditioned expectation values) bound entropic uncertainty relations that govern weak measurements (Theorem 2). Before detailing our results, we review entropic uncertainty relations, OTOCs, and quasiprobabilities.
The Maassen-Uffink relation exemplifies entropic un-certainty relations [7]: is small, the bound is tight. c is smallest when the eigenbases are mutually unbiased bases (MUBs): | a , λ a |b m , α bm | = 1 √ d , wherein d := dim(H) denotes the Hilbert space's dimensionality. For example, the eigenbases of the Pauli operatorsσ x andσ z form MUBs. Henceσ x andσ z are said to fail maximally to commute.
Entropic uncertainty relations have been generalized in three relevant ways [15]: (i) Measurements of observableŝ A andB have been extended to measurements of positive operator-valued measures (POVMs, or generalized measurements [41]) [14,43]. (ii) The Shannon entropy H has been generalized to Rényi entropies H α [7]. (iii) The entropies have been smoothed with an error-tolerance parameter ε [44].
The OTOC captures a similar divergence. Let us construct two protocols that differ largely by an initial perturbation. The system could consist of an N -site chain of spin- 1 2 degrees of freedom, or qubits. Suppose, for simplicity, thatρ = |ψ ψ| is pure. Protocol I consists of (i) preparing the system in |ψ , (ii) perturbing the system with a localV (as by operating on spin 1 withσ z 1 ), (iii) evolving the system under theÛ generated by a nonin-tegrableĤ, (iv) perturbing with a localŴ (such as the final spin'sσ z N ), and (v) evolving the system backward underÛ † . This protocol prepares |ψ I :=Ŵ (t)V |ψ .
How much does the initialV perturbation affect the system's final state? The answer manifests in the overlap The final expression characterizes nonlocally coupled systems, such as the Sachdev-Ye-Kitaev (SYK) model [19,28,47,48]. The Lyapunov-type exponent λ controls the exponential decay. [In local systems, F (t) decays polynomially. Generally, F (t) decays during a time window around the scrambling time, t * .] Hence F (t) reflects a Lyapunov-type divergence reminiscent of chaotic sensitivity to initial perturbations. Smallness of F (t) reflects highly nonlocal entanglement: After t * , no local probeŴ can recover information about the earlierV . This many-body nonlocality is called scrambling [26,49].
Quasiprobabilities: Quasiprobability distributions represent quantum states as probability distributions µ(q, p) represent classical statistical-mechanical states. The phase-space distribution µ(q, p) is a function of variables q and p, such as position and momentum. Can we represent a quantum stateσ with an analogousμσ(q, p)? We must relax one or more axioms of probability theory [50]. Relaxing different axioms leads to different quasiprobability representationsμσ.
Outline:Ãρ governs terms in the entropic uncertainty relations for scrambling. We present and analyze these relations in Sec. I. Numerical simulations of a spin chain show that scrambling tightens the uncertainty bounds. We also generalize beyond scrambling, unveiling a physical significance of general KD quasiprobabilities and weak values (conditioned expectation values of observables [61,62]): KD quasiprobabilities and weak values govern terms in entropic uncertainty bounds obeyed by weak measurements. We generalize also from the fourpoint F (t) to 2K -point OTOCs, forK = 1, 2, 3, 4, . . .. Such high-order OTOCs reflect later, subtler stages of scrambling and equilibration [36,60,[63][64][65][66]. This work's significance is expounded upon in the Sec. II.

I. RESULTS
Uncertainty relations and OTOCs, reflecting quantum operator disagreement in different subfields, cry out for unification. But how can one form an uncertainty relation for scrambling? One might try substitutingV and W (t) into Ineq. (2). But the resulting bound would bear no signature of scrambling. Moreover, simulations imply, simple choices ofV andŴ (t) eigenbases fail to become MUBs after t * [36].
We must find POVMs such that the uncertainty bound reflects scrambling. Three hints suggest that the reflection should manifest inÃρ. First, entropic uncertainty relations followed from "stripping off" eigenvalues a , leaving probabilities. The quasiprobabilityÃρ follows from "stripping the eigenvalues v and w m off" the OTOC [Eq. (14)].
Second, entropic uncertainty relations depend on the maximum overlap (4). c equals a product of inner products. So doesÃρ [Eq. (12)], whose sum equalsÃρ. . Indeed, c consists of two inner products, whereasÃ1 contains four. (The quasiprobability is evaluated, here, on the identity operator1, rather than on a normalized quantum stateρ. The elimination of state dependence fromÃ, we will see, follows from the state independence of entropic uncertainty bounds.) Having motivated expectations about entropic uncertainty relations for scrambling, we now fulfill those expectations. We first choose uncertainty-relation POVMs suited to scrambling (Sec. I A). The POVMs' possible outcomes obey probability distributions whose entropies we define (Sec. I B). We then present (Sec. I C) and analyze (Sec. I D) the entropic uncertainty relations for scrambling. Numerical simulations of a spin chain illustrate the results (Sec. I E). We generalize to higher-point OTOCs (Sec. I F), then to weak values and quasiprobabilities beyond scrambling (Sec. I G).

I A. Choosing POVMs
We continue to focus on a quantum many-body system illustrated with a chain of N qubits. To simplify notation, we now omit hats from operators. Manybody quantities are defined as in the introduction: the Hilbert space H, its dimensionality d, the arbitrary state ρ ∈ D(H), the Hamiltonian H, the time-evolution unitary U , the local operators V and W (illustrated with σ z 1 and σ z N ), the Heisenberg-picture W (t), the projectors Π V v and Π wm , the eigenvalues v and w m , the lists λ v and α wm of degeneracy parameters, the eigenstates |v , λ v and |w m , α wm , the OTOC F (t), and the finegrained and coarse-grained OTOC quasiprobabilitiesÃ ρ andÃ ρ .
The Hilbert space H is assumed to be discrete, in accordance with [14,43]. (Continuous-variable systems are addressed in Sec. II.) We emphasize nonintegrable, nonlocal Hamiltonians. We assume that V and W are Hermitian, for simplicity, but the results generalize: Each of V and W can be Hermitian and/or unitary [35,36]. (If V is unitary but not Hermitian, for example, measurements of V are replaced with measurements of the Hermitian operator that generates V .) We adapt the formalism used by Tomamichel in [14]. He invokes POVMs X := M X x (e.g., a measurement of an observable X) and Y := M Y y (e.g., a measurement of an observable Y ). Which POVMs would seed c withÃ 1 , uniting entropic uncertainty relations with scrambling? Equation (7) offers guidance. The OTOC emerges from a forward protocol, in which V precedes W (t), and a reverse protocol. Let us conjecture that X should consist of a V measurement followed by a W (t) measurement, while Y should consist of the reverse. 2 Two reasons suggest that the V measurements should be weak: (i) Equation (7) links the OTOC to the classical butterfly effect. The butterfly effect encapsulates sensitivity to tiny initial perturbations. The V measurement, if weak, is perturbative in the coupling strength.
(ii) Weak measurements of V can be used to measure the OTOC and its quasiprobabilities experimentally [35,36].
Having motivated our choices of POVMs, we introduce definitions. X consists of a weak measurement of V , followed by a projective measurement of W (t). We use the term "weak measurement of V " as in [36]: A projector Π V v1 is effectively measured weakly. One can effectively measure a qubit system's Π V v1 by, e.g., coupling the detector to V and calibrating the detector appropriately. The experimenter chooses the value of v 1 ; the choice directs the calibration. See Sec. I E 1 and [36, Sec. I D 4] for example implementations. This composite POVM, which we call "the forward measurement," replaces Tomamichel's X . The reverse process replaces Tomamichel's Y, for a definition of "reverse" formalized after the weak measurement.
To measure Π V v weakly, one prepares a detector D in a state |φ . The system's Π V v is coupled weakly to a detector observable, via an interaction unitary V int . A detector observable is measured projectively, yielding an outcome j .
The weak measurement induces dynamics modeled with Kraus operators [41,67,68]. Kraus operators represent the system-of-interest evolution effected by a coupling to another system (which can be regarded as measuring the first system): The operators satisfy the completeness relation Let σ temporarily denote the state occupied by the system before the coupling.
The detector has a probability Tr of registering the outcome j . The outcome-dependent g V j ∈ C quantifies the interaction strength. The experimenter can tune g V j , whose smallness reflects the measurement's weakness: g V j 1. We refer to various g V j 's collectively as g's.
Imagine strongly measuring the detector observable without having coupled D to the system. The outcome j has a probability p V j of obtaining. We invoke Kraus operators' unitary equivalence [68] to ensure that p V j ∈ R. The forward POVM M F v1,j1,w2 is defined through the composite Kraus operators Ending the protocol with a weak measurement might disconcert measurement theorists. But this reverse protocol captures the forward-and-reverse OTOC spirit in Eq. (7), as explained earlier. To round out the reversal, we not only swap the V measurement with the W (t), but also Hermitian-conjugate. Conjugation negates imaginary numbers, representing, e.g., the time-reversal of magnetic fields. Let us clarify which variables are chosen and which vary randomly. w 2 is a random outcome whose value varies from realization to realization of the forward POVM. w 3 is a random outcome whose value varies from realization to realization of the reverse POVM. The experimentalist chooses the values of v 1 and v 2 . Though a forward trial's v 1 and w 2 can differ from a reverse trial's v 2 and w 3 , both protocols' measurements [of V and of W (t)] are essentially the same.

I B. Entropies
Consider preparing the system in the state ρ, then measuring the forward POVM, M F v1,j1,w2 . One prepares a detector D F in some fiducial state. Some detector observable is effectively coupled to the system's Π V v1 . Then, some D F observable couples to a classical 3 registerD V F . D V F records the outcome j 1 . Next, the system's W (t) couples to a classical registerD . This register records the outcome w 2 .
The two-register system ends in the state The eigenvalues, Tr , form a probability distribution over the possible pairs (j 1 , w 2 ) of measurement outcomes. Entropies of the distribution equal entropies of ρ F . In defining the entropies, we follow Tomamichel's conventions [14]. However, he considers subnormalized quantum states σ, whose traces Tr(σ) ≤ 1. We focus on normalized states, Tr(σ) = 1, except when otherwise specified.
The order-α Rényi entropy of a quantum state σ is We choose for all logarithms to be base-2, following [14]. The von Neumann entropy is The min entropy is defined as p max denotes the greatest eigenvalue of σ. The max entropy is The Schatten 1-norm is denoted by ||.|| 1 . The general Schatten p-norm of a Hermitian operator σ = j s j |s j s j | is for p ≥ 1 [69]. H max reflects the discrepancy between σ and the maximally mixed state [14, p. 60]: The fidelity between normalized states σ and γ is 4 F (σ, γ) := || √ σ √ γ|| 1 . H max depends on the fidelity between σ and We notate the detector state's Rényi entropies as following [14]. We have now introduced the forward-POVM entropies. The two-detector state ρ R , and the entropy H α (W (t)V ), are defined analogously. H max and H min , like H vN , quantify rates at which information-processing and thermodynamic tasks can be performed. Applications include quantum key distribution, randomness extraction, erasure, work extraction, and work expenditure (e.g., [14,44,[70][71][72][73][74][75]). Quantum states desired for such tasks cannot be prepared exactly.
The ε-smooth min entropy follows from maximizing H min over the ε-ball: The ε-smooth max entropy follows from minimizing H max over the ε-ball [14, p. 84]: Setting ε to zero, we recover the min and max entropies:

I C. Entropic uncertainty relations for QI scrambling
We can now reconcile the two subfields' notions of quantum operator disagreement. Theorem 1. The forward and reverse POVMs satisfy the entropic uncertainty relations for ε ≥ 0 and 1 α + 1 β = 2. The bound depends on the coarse-grained OTOC quasiprobability: wherein δ ab denotes the Kronecker delta.
The uncertainty relations are proved in Suppl. Mat. A. They follow from three general uncertainty relations: Result 7 in [14], Corollary 2.6 in [43], and Ineq. (13) in [76]. The OTOC POVMs (16) and (17) are substituted into the general uncertainty relations. The POVMs' maximum overlap, c, cannot obviously be inferred from parameters chosen, or from measurements taken, in an OTOC-inference experiment. We therefore bound c, us-ingÃ ρ and the Schatten p-norm's monotonicity in p: We substitute in for the K's from Eq. (15), then multiply out. In each of several terms, two K's contribute Π V v 's, while two K's contribute 1's. These terms contain quasiprobaiblity valuesÃ 1 . We isolate the terms by Taylor-expanding the logarithm in the g's.

I D. Analysis
Four points merit analysis: the form of f (v 1 , v 2 ), implications for the butterfly effect, simple limits, and conditions that render the bound nontrivial.
Form of f (v 1 , v 2 ): The bound contains three quasiprobability-dependent terms, in line (34). The other terms are "background terms": They contain classical probabilities, accessible without weak measurements.
The first term, in line (32) lacks g's. Unsuppressed by weak-coupling constants, this log dominates f (v 1 , v 2 ). The two terms in (33) depend on g's linearly. Only through sums Tr Π of classical probabilities do these terms depend on projectors Π. This dependence characterizes also the g 2 terms in line (35).
Three g 2 terms depend on quasiprobability valuesÃ 1 . The quasiprobability's gracing higher-order terms accords with intuition: Scrambling is a subtle feature of quantum equilibration, detectable in just (≥ 2)-point correlators. Likewise, the OTOC quasiprobability governs second-order terms in the uncertainty bound.
The quasiprobability's being evaluated on 1, rather than on ρ, should not surprise us. The need for stateindependent bounds partially motivated the recasting of uncertainty relations in terms of entropies [15]: Stateindependent bounds reflect the operators' disagreement, unpolluted by any particular state ρ.
The first term's dominance, the δ w2w3 , and the min ensure that w 2 = w 3 throughout the min's argument. The finalÃ 1 has four arguments, (v 1 , w 2 , v 2 , w 3 ), constrained only by the δ w2w3 . In each otherÃ 1 , the first argument must equal the third, even before the minimization is imposed. For example, the first quasiprobability value has the formÃ 1 (v 1 , w 2 , v 1 , w 3 ). The V eigenvalues equal each other, due to Ineq. (37). One v 1 comes from the K V,v1 j1 † ; and one, from the K V,v1 j1 . Implications for the butterfly effect: The V measurements' weakness might surprise OTOC connoisseurs. But the weakness bolsters an analogy between the OTOC and the butterfly effect characteristic of classical chaos [20,29,30,77]. The classical butterfly effect manifests when a tiny perturbation snowballs into a drastic change. This perturbation has been likened to operation by a unitary V . But the trace norm of V grows exponentially with the system size N in the spin-chain example: ||V || 1 = 2 N . Ambiguity therefore obscures how V can be regarded as a tiny perturbation. V should be associated with a weak measurement, Theorem 1 clarifies. The measurement is perturbative in g V j . Nontriviality conditions: The Rényi entropies are nonnegative, H α (σ) ≥ 0, so the bound is nontrivial when positive: f (v 1 , v 2 ) > 0. The bound is positive when its first term is positive, when the coupling is weak. The first term simplifies to . The trace is large in the system size, equaling 2 N −1 in the spin-chain example. One might worry that this trace swells the log, drawing the bound far below zero. The probabilities p V j can offset the enormity. Let us focus on the spin-chain example and approximate Weak measurements as in [61] satisfy this requirement. Let each detector manifest as a particle, e.g., in a potential that defines a dial. Let O F/R denote the strongly measured detector observable (e.g., the positionx). Let O F/R denote the conjugate observable (e.g., the momen- a Gaussian state that peaks sharply at someÕ F/R eigenvalue (e.g., a sharp momentum-space wave packet). The probabilities p V j can be small enough that f (v 1 , v 2 ) > 0. We present an example in Sec. I E.
The g-free log encodes randomness in a measurement of a detector that has never coupled to the system. Hence the log fails to reflect disagreement between V and W (t). The disagreement manifests in the g-dependent terms.
Simple limits: Three simple limits illuminate the bound's behavior: early times (t ≈ 0), late times (t ≥ t * ), and the weak limit (g → 0). We focus on a chaotic spin chain, for concreteness. Numerical simulations (Sec. I E) support these arguments.
Early times (t ≈ 0): V and W (t) ≈ W nontrivially transform primarily far-apart subsystems. Hence Tr Π traces are large, dragging the ∼ g terms in Eq. (32), and the four terms ∼ −g 2 , below zero. The g's mitigate the dragging's magnitude. Still, the bound is expected to be relatively loose before t * .
Late times (t ≥ t * ): V can fail to commute with . . will shrink: Consider a one-qubit system, as a simple illustration. Suppose that The traces' smallness tightens the uncertainty bound, as expected when the system is scrambled (as explained in the introduction). 5 The bound likely does not remain at its maximum possible value at all t > t * , however. As W (t) evolves, the bound should fluctuate around a relatively large value. 5 This expectation is borne out when v 1 = v 2 , as implied by (i) Suppl. Mat. B and (ii) reasoning, similar to that in the Suppl. Mat., about the first two terms in line (34). Supplementary Material B also shows why the quasiprobability tightens the bound when (i) v 1 = −v 2 and (ii) g V j 1 g V j 2 approximately equals a negative real number.
Weak limit (g → 0): The system fails to couple to the detectors D F and D R .
Let us analyze the uncertainty relations' right-hand sides (RHSs) first.

I E. Numerical simulations of a spin chain
We illustrate Theorem 1 with an interacting spin chain. The set-up and weak-measurement implementation are described in Sec. I E 1. The detector probabilities p V j , the weak-measurement Kraus operators K V,v j , the couplings g V j , and the entropies H α are calculated in Sec. I E 2. We present and analyze results in Sec. I E 3. The code used to calculate the entropies, simulate the weak measurements, etc. is available on Github, at https://github.com/abartolo-tb/Entropic-Unc-and-Quasiprobs.

I E 1. Spin-chain set-up
Consider a one-dimensional (1D) chain of N = 8 qubits. The OTOC operators manifest as single-qubit Pauli operators: V = σ z 1 , and W = σ z N . The operators' precise forms do not impact our chaotic-system results, however.
Model: The chain evolves under the power-law quantum Ising Hamiltonian [78] (see [79] for a similar model). Spin j interacts with each spin that lies within a distance 0 , for every j = 1, 2, . . . , N . The interaction strength declines with distance as a power law controlled by ζ. We choose J = 1, ζ = 6, and 0 = 5, as in [78]. Planck's constant is set to one: = 1. We set the transverse field h x to s1.05. The longitudinal field h z j = 0.375(−1) j flips from site to site. The transverse-field Ising model with a longitudinal field reproduces our results' qualitative features. But the power-law quantum Ising model mimics all-to-all interactions, such as in the SYK model [19,28,47,48]. Around t = t * , therefore, the OTOC decays almost exponentially. Exponential decay evokes classical chaos, as discussed in the introduction.
Weak-measurement implementation: Section I D guides our implementation, which parallels [61]. We illustrate with the forward-protocol weak measurement, temporarily dropping F subscripts. We also reinstate operators' hats.
The detector D consists of a particle that scatters off the system. The detector could manifest as a photon, as in circuit QED [80] and in purely photonic experiments [81]. Letŷ denote the longitudinal direction, which points from the detector's initial position to the system.
Letx denote a transversal direction; and |φ , thex component of the detector's initial state. |φ consists of a Gaussian, centered on the transverse-momentum eigenvalue p ≡ p x = 0. ∆ denotes the Gaussian's standard deviation. The displaced detector positionx − x 01 couples 6 to the system'sΠV v1 . [The displacement prevents the minimization in (32) from choosing the detector-measurement outcome x = 0. This choice would set g V x1 to g V x0 = 0, eliminating the weak measurement.] The interaction unitary has the form The interaction strengthg governs the outcomedependent coupling gV j1 . Numerical experiments show thatg = 0.02 and x 0 = 10 keep perturbatively small while strengthening the bound. The detector'sx is measured strongly. Let L > 0 denote the measurement's precision. Positions x 1 and x 2 can be distinguished if they lie a distance |x 2 − x 1 | ≥ L apart. Hence the classical registerD has a discrete spectrum {x 1 }. We simulated a register whose L = 0.1.
Detector probability pV j1 = pV x1 : Consider preparing the detector in |φ , then measuringx. The measurement 6ΠV v 1 can effectively be measured weakly via coupling of D to V =σ z 1 . The interaction unitary will have the form exp − i g x ⊗σ z 1 . The Pauli operator decomposes asσ z 1 = ± 2ΠV ± −1 . Hence the interaction unitary has the form . The Kraus operator becomes x 1 | exp ± i g x ⊗1 exp ∓ 2i g x ⊗ΠV ± |φ . The lefthand exponential can be absorbed into the strong measurement of D: Consider wishing to measureΠV + weakly. Instead of measuring the detector's {|x 1 } strongly, one measures has a probability pV x1 L = | x 1 |φ | 2 L of yielding a position within L of x 1 . By Eq. (39), Equation (42) determines the condition under which the uncertainty bound is nontrivial.
Entropies H α : Let us reinstate F subscripts and remove operators' hats. We illustrate the entropies' analytical forms with The measurement operators have the form by Eq. (16). We substitute in from Eq. (15), multiply out, and substitute into Eq. (50): .
The H ε min of the quantum state ρ F has reduced to the H ε min of a probability distribution. The other entropies have analogous forms.
We will focus on H min , H max , and H vN , for simplicity. Explorations of ε > 0 are discussed in Sections I E 3 and II. The bound grows at t = t * , confirming expectations: At the scrambling time, the OTOC drops. A decayed OTOC reflects noncommutation of V and W (t). The worse two operators commute, the stronger their entropic uncertainty relations; the stronger the uncertainty bound f (v 1 , v 2 ). Hence Theorem 1 unites information scrambling and OTOCs with entropic uncertainty relations, as claimed. Figure 2 shows the quasiprobability's contribution to the uncertainty bound (32). Figure 3 shows the LHSs of Ineqs. (29) and (30) (H ε min + H ε max and H vN + H vN ) at ε = 0, with the shared RHS f (v 1 , v 2 ). This figure is more zoomed-out than Fig. 2; hence the tightening is too small to detect. This reduced visibility is expected: Scrambling is a subtle, high-order stage of quantum equilibration. It manifests in the g 2 terms of f (v 1 , v 2 ), just asÃ ρ can be inferred from high-order terms in weak-measurement experiments [35,36].

I E 3. Spin-chain results
The LHSs lie ∼ 10 bits above the bound. The gap stems from the Tr Π W (t) w3 = 2 N −1 in Eq. (32). This gap bodes ill for the large-system limit, N → ∞, of interest in holography. But the gap scales only linearly, not exponentially, with N . Furthermore, many of today's experiments (e.g., [85]) will correspond to reasonably small gaps. Additionally, Sec. I G presents weak-measurement entropic uncertainty relations independent of scrambling. Those uncertainty relations need not have such a gap. We will illustrate with a qubit example whose bound is tight at zeroth order in g, in Sec. I G. Figure 4 illustrates how tight the bound can grow in an exceptional parameter regime. The top curves represent H min + H max and H vN + H vN . These curves dip at t ≈ t * because (i) ρ is a W (t ≈ t * ) eigenstate and (ii) the POVMs' W (t) measurements are fine-grained. That is, recall the forward and reverse POVMs (16) and (17). The W (t) measurements are replaced with measurements of U † |w , α w U . The POVM outcomes become highly predictable around t * , so the bound grows tight to within 0.53 bits. 7 7 In addition to choosing ρ and to fine-graining, we raised the interaction strength tog = 0.16. The outcome-dependent coupling strengths g V x are comparable to the detector probabilities: x . This comparability invalidates the Taylor expansion that leads to Eq. (32). Equation (A15) in Suppl. Mat. A gives the pre-Taylor-expansion bound. This bound appears as the solid, green, bottom curve in Fig. 4. The bound would rise more than in the earlier figures, if the POVMs' W (t) measure-  Fig. 1, with three exceptions. First, the initial state ρ is a W (t) eigenstate, wherein the time t is evaluated at the scrambling time t * . Second, the W (t) measurements in the positive operator-valued measures (16) and (17)  H PQIM , which is nonintegrable. Integrable Hamiltonians' OTOCs revive and decay repeatedly, as information recollects from across the system and spreads again. The revivals and decays lift and suppress f (v 1 , v 2 ), we have confirmed using a transverse-field Ising model. The relevant plots are omitted but can be found on the Github repository.
Higher-point OTOCs F (K ) (t) equilibrate at later times t and can be inferred from sequences of 2K − 1 weak measurements. , followed by more weak measurements, until a weak measurement of Π The weak measurement of an observable Θ = B(t 2 ), C(t 3 ), . . . is represented by a Kraus operator The j α denotes the weak measurement's outcome, p Θ jα denotes the detector probability, and g Θ jα denotes the outcome-dependent weak-coupling strength.
The von Neumann uncertainty relation has the form The term contains the quasiprobability behind theK -fold OTOC. 8 Hence our entropic-uncertainty-relation results extend to arbitrary-point OTOCs.
I G. Beyond scrambling: Entropic uncertainty relations for weak values and Kirkwood-Dirac quasiprobabilities Weak values, like OTOCs, encode time reversals and measurement sequences [61,62]. Consider preparing a quantum system in a state |i at a time t = 0, evolving the system for a time t under a unitary U t , measuring a nondegenerate observable F = f f |f f |, and obtaining the outcome f . Let A = a a|a a| denote a nondegenerate observable that fails to commute with F .
Which value can most reasonably be attributed, retrodictively, to the A at a time t ∈ (0, t ), given that |i was prepared and that the measurement yielded f ? The weak value wherein |f := U t −t |f and |i := U t |i denote timeevolved states. A wk is an expectation value of A conditioned on the state preparation and the postselection. Consider eigendecomposing A, then factoring out the sum and eigenvalues. We multiply the numerator and denominator by i |f , then invoke Eq. (8): wherein p(f |i) = | f |i | 2 . (We have absorbed the U t 's into theÃ.) The KD quasiprobability governs the conditional quasiprobabilityÃ (1) |i i| (a, f )/p(f |i) that, if |i is 8 Entropic uncertainty relations for ≥ 3 measurements have been derived [16]. Could such relations containK -fold OTOCs? The match appears unnatural, for two reasons. First, consider the minimal generalization of F (t), in which every observable equals W (t) or V : W (t)V . . . W (t)V . Each POVM involves only two observables, W (t) and V , not three observables. Second, suppose that (i) A, . . . , R are unitary, as well as Hermitian and (ii) ρ is pure. |F (K ) | equals an overlap | ψ II |ψ I |, as F (t) was shown to in the introduction. Implementing A(t 1 ), then B(t 2 ), etc., then E(tK ) prepares |ψ II . Implementing an analogous sequence prepares |ψ II . The overlap |F (K ) | compares one sequence to the other, rather than comparing all the observables that define the sequences. An entropic uncertainty relation, in contrast, reflects all the observables' disagreements with each other. prepared and the F measurement yields f , a is the value most reasonably attributable to A retrodictively.
A wk generalizes to arbitrary initial states ρ and to degenerate observables A = a a Π A a and F = f f Π F f : wherein p(f |ρ) = Tr Π ρ . One can infer A wk experimentally by preparing ρ, evolving the system under U t , measuring A weakly, evolving the system under U t −t , and measuring F strongly. One performs this protocol in many trials. A wk is inferred from the measurement statistics.
We introduce another physical significance in this section: Weak values and KD quasiprobabilities govern terms in entropic uncertainty relations for POVMs that involve weak measurements. We present the results (Sec. I G 1), then illustrate with a qubit example (Sec. I G 2).

I G 1. Entropic uncertainty relations for A wk andÃ
(1) ρ Consider a quantum system associated with a Hilbert space H. Let ρ ∈ D(H) denote any state of the system. Let A = a a Π A a , F = f f Π F f , and I = i λ i Π I i be eigenvalue decompositions of observables. [The index i should not be confused with √ −1. The index serves similarly to the i that labels the initial state |i in Eq. (60).] The uncertainty relation forÃ (1) features a POVM that we label I. One measures Π A a weakly, then F strongly: The weak- Here, O g 2 signifies terms of second order in the Hamiltonian's coupling parameter (e.g., theg in the spin-chain example of Sec. I E).
We define as POVM II a strong measurement of I: M II i := Π I i . POVM II prepares a maximally mixed state over an I eigenspace.
Define the entropies H α (AF ) ρ , H α Ā F ρ , and H α (I) ρ via analogy with the QI-scrambling entropies (Sec. I B). One can infer the weak value 9 upon preparing the state Π I i /Tr Π I i , measuring A weakly, and postselecting a strong F measurement on f .
Theorem 2. POVMs I and II obey three entropic uncertainty relations dependent on a KD quasiprobability for σ := Π I i /Tr Π I i : wherein POVMsĪ and II obey three entropic uncertainty relations dependent on the weak value A wk (i, f ): wherein The smoothing parameter ε ≥ 0, the Rényi order parameters α and β satisfy 1 α + 1 β = 2, and ρ denotes an arbitrary state.
The proof is analogous to the proof of Theorem 1. The forward and reverse POVMs are replaced with POVMs I and II (orĪ and II). 9 We have tweaked our notation for A wk . The first argument equals, rather than the state ρ, the label i of the eigenvalue λ i that specifies ρ = Π I i /Tr Π I i .

I G 2. Qubit example
Let us illustrate the uncertainty relation (65) for A wk . We consider a qubit system, denoted with a subscript s, and a qubit detector, denoted with a subscript d. Let I = σ z s , A = σ y s , and F = σ x s . A can be weakly measured as follows. The detector is prepared in the state |x+ . A z-controlled y conditions a rotation of the detector's state on the system's state. The interaction Hamiltonian H int =g (σ y d ⊗ σ z s ) generates the unitary The detector's σ y d is measured strongly, yielding the outcome j = y d = ±1.
We can now assemble the ingredients in Ineq. (65). The coupling-free probabilities Mat. C for a derivation); the outcome-dependent weak couplings, ; and the weak values, A wk (z s , x s ) = x s |σ y s |z s / x s |z s = x s z s i. The weak values' nonreality is nonclassical: A has only real eigenvalues a, but the conditioned average A wk is imaginary [94].
Let us calculate the bound (68).
When this factor equals its maximum value of |g| Having evaluated the RHS of Ineq. (65), we turn to the LHS. We calculate the POVM probabilities, then their entropies. ρ denotes an arbitrary system state, exemplified by |z+ .
POVM II consists of a strong I = σ z s measurement. The possible outcomes z s have probabilities q II zs = z s |ρ|z s of obtaining. If ρ = |z+ z+|, then q II zs=1 = 1, and q II zs=−1 = 0. The max entropy is H max q II zs = log 1 = 0. Smoothing cannot change this value.
POVMĪ consists of a weak A = σ y s measurement followed by a strong F = σ x s measurement. The possible outcome tuples (y d , x s ) correspond to the probabilities qĪ y d ,xs = x s |K Y y d ρ K Y y d † |x s . We substitute in, then multiply out: If ρ = |z+ z+|, the distribution is uniform: qĪ y d ,xs = When ρ = |z+ z+|, therefore, the uncertainty relation reads, 2.00 ≥ 2.00 − 2 ln 2 |g| + O g 2 . Ifg = 2.00 × 10 −2 , as in Sec. I E, the relation approximates to 2.00 ≥ 1.94. The bound is satisfied and is tight at order g 0 .

II. DISCUSSION
We have reconciled two measures of disagreement between quantum operators: entropic uncertainty relations and out-of-time-ordered correlators (OTOCs). The reconciliation unites several subfields of physics: (i) quasiprobabilities and weak measurements tie (ii) quantum information theory to (iii) condensed matter and (iv) high-energy physics. Information theory and complexity theory have begun intersecting with condensed matter and high-energy physics recently, shedding light on black holes, information propagation, and space-time (e.g., [26,[99][100][101][102][103][104]). This paper broadens the intersection into quasiprobability and quantum-measurement theory and farther into quantum information theory.
This broadening has two more important significances: one for OTOC theory and one for weak-measurement theory. First, the extension reconciles the OTOC's V with the tiny perturbation that triggers violent consequences in the classical butterfly effect: V can naturally be regarded, our uncertainty relations show, as being measured weakly. The weak measurement is perturbative literally, in the coupling strength g.
Within measurement theory, second, we have uncovered a physical significance of weak values A wk and Kirkwood-Dirac quasiprobabilitiesÃ (1) 1 : These quantities govern terms in entropic uncertainty relations obeyed by weak measurements. Quantum information theory therefore sheds light on mathematical objects whose interpretations have been debated in quantum optics, quantum foundations, and quantum computation.
Testing Theorem 2 experimentally requires even fewer resources: Interacting many-body systems are unnecessary, and one weak measurement per trial suffices. Tantalizingly, though, two [130][131][132] and three [125] sequential weak measurements have been realized recently. They can be applied to (i) characterize higher-order terms in Eqs. (65) and (68) and (ii) test entropic uncertainty relations for POVMs of many-weak-measurement sequences, in the spirit of Sec. I F.
Finally, nonclassicality ofÃ 1 ,Ã 1 , and A wk might strengthen the uncertainty bounds. The quasiprobabilities behave nonclassically by acquiring negative real and nonzero imaginary components. A wk behaves nonclassically by lying outside the spectrum of A. Such nonclassical mathematical behavior can signal nonclassical physics [52][53][54][55][56][57]. Nonclassicality's potential to tighten uncertainty bounds, as discussed in Sec. I E 3, merits study.  [133]). We use Tomamichel's notation, for concreteness. But the three uncertainty relations have the same RHSs. Hence our use of [14,Result 7] translates directly into uses of the other two bounds.
Tomamichel presents in [14] (see also [134][135][136]). B and C denote conditioned-on systems, information accessible to an agent performing an information-processing task. We trivialize the conditioning, setting the states of B and C proportional to 1.
The POVM overlap c is defined as The outer square-root equals, by Eqs. (16) and (17), The two central projectors have collapsed into one: Π w2 . The operator O is Hermitian and so eigendecomposes. The eigenvalues are real and nonnegative, being the squares of the singular values of M F v1,j1,w2 M R v2,j2,w3 . Also a physical argument implies the eigenvalues' reality and nonnegativity: O is proportional to a quantum state: Π represents the state that is maximally mixed over the eigenvalue-w 3 eigenspace of W (t). Imagine preparing Π , subjecting the state to the quantum channel defined by the operation elements [41] K V,v2 j2 † j2 , 10 subjecting the state to the channel defined by , and then measuring W (t) projectively. The resultant state, σ f , is proportional to O. The proportionality 10 We must prove that defines a quantum chan- does by definition, so each K V,v 2 j 2 maps the input Hilbert space to the output Hilbert space, and differs from factor equals Tr(O), the joint probability that (i) this realization of the initial channel's action is labeled by j 2 , (ii) this realization of the second channel's action is labeled by j 1 , and (iii) the W (t) measurement yields outcome w 3 . has two distinct eigenvalues η: η = 0, of degeneracy , and η = 1, of degeneracy Tr Π . Let Λ r η denote the r th O eigenvalue associated with any eigenvector in the η eigenspace of Π w3 . If d η denotes the degeneracy of Λ r η , r = 1, 2, . . . d η . (We have omitted the η dependence from the symbol r for notational simplicity.) Every eigenvalue-0 eigenvector of Π is an eigenvalue-0 eigenvector of O: Λ r We use this eigenvalue decomposition to evaluate the RHS of Eq. (A4), working from inside to outside. The outer square-root has the form The projectors project onto orthogonal subspaces, so We take the trace, Tr . The limit as α → ∞ gives the RHS of Eq. (A4): Only the greatest eigenvalue to survives: is neither a parameter chosen by the experimentalist nor obviously experimentally measurable. Hence bounding the entropies with Λ max 1 is useless.
Probabilities and quasiprobabilities are measurable. Tr(O) equals a combination of probabilities and quasiprobabilities. We therefore seek to shift the Tr of Eq. (A8) inside the [.] α and the √ . . Equivalently, we seek to shift the of Eq. (A9) inside the (.) α/2 . We do so at the cost of introducing an inequality: for all α/2 ≥ 1. This inequality follows from the Schatten p-norm's monotonicity. The Schatten p-norm of an operator σ is defined as ||σ|| p := Tr √ σ † σ p 1/p , for p ∈ [1, ∞). As p increases, the Schatten norm decreases monotonically: only by complex conjugation of the coupling g V j 2 ∈ C.
Hence j 2 K V,v 2 j 2 K V,v 2 j 2 † = 1, as required of Kraus operators. This mathematical result complements physical intuition: Suppose that the detector manifests as a qubit. A common interaction rotates the detector's state conditionally on the system's state [36,60,137]. Let K V,v 2 j 2 j 2 follow from a rotation in some fiducial direction.
follows from a rotation in the opposite direction. Now, suppose that the detector manifests as a particle in some potential. A common interaction conditionally kicks the detector. If K V,v 2 j 2 j 2 follows from a kick in one direction, follows from a kick in the opposite.
We have invoked the trace's cyclicality and Π We substitute into the trace from Eqs. (16) and (17): Tr Π W (t) Multiplying out yields Six of the traces are instances ofÃ 1 .
Only the first term is constant in g. If g is small, therefore, the maximum obtains where the first term maximizes, where w 2 = w 3 . Hence every RHS term is implicitly evaluated at w 2 = w 3 .
We Next, we factor out the p V j1 p V j2 Tr Π W (t) w3 and invoke the log law for multiplication: + log (1 + [terms small in g]) . We then Taylor-expand in the g's. of V and W (t). Let us set w 2 = w 3 and replace ρ with 1. We recall that w , v m = ±1, that the Pauli operators' traces vanish, and that the Pauli operators square to 1. The expression simplifies: Let us analyze the expression piecemeal. First, the V W (t) ≈ 0 at early times, because the influence from V has not reached W (t). Random-matrix-theory cancellations suppress V W (t) at late times. Second, the OTOC begins at F (t ≈ 0) ≈ 1 and drops to F (t ≥ t * ) ≈ 0. Third, suppose that v 1 = −v 2 .
In summary, the finalÃ 1 value in (35) points to v 1 = −v 2 as a condition under which the uncertainty bound is relatively tight. We arbitrarily chose v 1 = 1.