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Initial entanglement between detectors allows violating Heisenberg’s uncertainty relations

. Physics textbooks introduce an inequality derived by Kennard that concerns the impossibility of preparing a quantum state with well deﬁned momentum and position. However, Heisenberg had formulated two diﬀerent inequalities: one stating the impossibility of a simultaneous detection of position and momentum with arbitrary precision, and another one imposing a tradeoﬀ between the precision on the measurement of a variable and the backaction on a subsequent measurement of a conjugated variable. Here, we explore the connections among these three inequalities, and we show that the latter can be violated, if the detectors are initially entangled. The results, besides being of fundamental interest, can be useful for building up an ideal momentum, or position, detector (i.e. one that introduces no noise in the measurement, besides the intrinsic statistical noise of the input state).


Introduction
Recently, there have been several experiments claiming to have violated Heisenberg's uncertainty relation [1][2][3][4]. At first sight, this may seem as revolutionary, since the uncertainty principle is one of the pillars of quantum mechanics. However, there are two important considerations to make: (1) Heisenberg [5] formulated an inequality about the uncertainty of a position measurement and the disturbance in a subsequent, or simultaneous, momentum measurement. The inequality demonstrated in quantum mechanics textbooks is due to Kennard [6], who proved it shortly after Heisenberg's semi-qualitative formulation, and it concerns the impossibility of preparing an ensemble of systems such that a measurement of momentum in a subensemble yields a variance σ 2 P and a measurement of position in another subensemble, at the same time, yields a variance σ 2 Q < 2 /4σ 2 P . (2) The inequality violated in the experiments concerns finite-dimensional systems, and was not formulated by Heisenberg. Instead, this inequality was retrofitted, following the reasoning: Heisenberg formulated an uncertainty relation for uncertainty in position and disturbance in a later measurement of momentum and another relation for the uncertainties in a joint measurement of momentum and position ε Q η P |Q , for sequential measurements, (1a) ε Q|P ε P |Q , for joint measurements; Kennard proved a rigorous inequality for the preparation variances therefore, Heisenberg would have endorsed the corresponding inequality for uncertainty in a variable and disturbance in the other variable, at least according to Ozawa [9] ε A η B|A However, a simple inspection reveals that equations (1e) and (1f) cannot hold. Indeed, for a finite-dimensional system, both the uncertainty ε and the disturbance η, whatever its definition is, must be finite. On the other hand, the precision of a measurement can be made, in principle, arbitrary small. Hence the products ε A η B|A and ε A|B ε B|A cannot have a lower bound. This consideration is to be contrasted with the original inequalities formulated in (1a) and (1b), for an infinite-dimensional Hilbert space. In this case, indeed, one can imagine that the disturbance η P |Q or the uncertainty ε P |Q diverge if the uncertainty ε Q|P tends to zero, so that the inequalities make sense, and cannot be discarded a priori.
The goal of the present manuscript, therefore, is to verify whether the inequalities (1a) and (1b) can be violated. In order to do this, it is necessary to define the uncertainty of a measurement, and the disturbance of a subsequent or simultaneous measurement. It turns out that the latter definition is a very controversial subject [10][11][12][13]. Therefore, the violation of the Heisenberg inequality in the uncertainty-disturbance formulation depends fatally on one's definition of disturbance. Here, we shall argue that our definition translates the intuitive notion of disturbance of a measurement. Furthermore, independently of the definition, we shall prove a somewhat surprising corollary: it is possible to make a very precise measurement of momentum by preceding it with an imprecise measurement of position. This results is uncontroversial, and it opens up a new possibility for realizing a precise measurement apparatus. So far, a precise apparatus should be prepared in a very sharp state of the pointer variable J, i.e. a state very close to the ideal state |J = 0 . The result of the present manuscript, however, allows to make a very precise measurement, say, of momentum, even though the detector is not in such an initial state, provided that its output variable is (almost) perfectly anticorrelated to the input variable of the first measurement. In quantum mechanics, perfect anticorrelations can be achieved only by means of entanglement. Hence, our main result can be summarized as in the title of this contribution: Initial entanglement between detectors allows violating Heisenberg's uncertainty relations.
A short account of the results presented here was given in Ref. [12]. Bullock and Busch have expanded the subject further [14]. Here, we shall present a more detailed derivation, in particular we shall justify the additional term due to the finite resolution of the readout that contributes to the uncertainty of the measurement. By making only the hypothesis of a uniform resolution of the detectors, we prove that the finite resolution contributes with additive terms to the cumulants of the observable outputs. We also include an appendix on the connection with the POVM approach that so far was published only in the pre-print version of [12].

Definitions
In the following, we define what we mean by 'uncertainty' and 'disturbance' in operational terms. Our definitions do not rely on any specific theory, so that they can be translated into different mathematical formulas depending on the underlying theory. In the next section, we shall provide the translation of these definitions in quantum mechanical terms.

Uncertainty
The uncertainty in a measurement is not in any way connected to the sample variance. A simple illustration is the following: suppose you are measuring the height of a large group of people. Say that the histogram is well approximated by a Gaussian with averageh = 175.cm and a variance σ h = 10.cm. Let us consider a single person, whose height is reported by our apparatus as h = 169.3cm. What is the uncertainty in the height of this person? Certainly not 10.cm! The uncertainty on an individual measurement depends on the resolution of the meter, and on how the meter interacted with the measured system. The uncertainty cannot be lower than the resolution, which for a typical tape is 1mm (if we exclude interpolation due to the human eye being a meter per se). For instance, was the 0 of the tape placed at the feet of the person, or by mistake a non-zero value was used? Was the tape held straight? And so on. So, how do we estimate the uncertainty in an individual measurement? Usually, we do by calibration. We where δ is in principle the best precision that we can achieve in the preparation, and we observe the outputs A of the apparatus. Unless our measuring apparatus is better than our preparation apparatus, we shall observe a distribution centered in A with a spread ∆ > δ. The quadratic difference ε 2 = ∆ 2 − δ 2 is the statistical uncertainty of our apparatus. Usually, δ is negligible compared to ∆. We note that in principle the uncertainty varies with the output h.
We also introduce the systematic uncertainty (error, in the common terminology): it is the difference between the average value A in the limit of a large number of observations N → ∞, and the 'true' value A 0 , This values is also known a priori, in principle.

Disturbance
The disturbance is the most controversial quantity to be defined. Since in quantum mechanics it is problematic to assume that a system possesses a property, unless it is prepared in an eigenstate of the corresponding observable, we avoid interpreting the disturbance as being a perturbation in an observable, but we interpret this concept as a disturbance of a measurement of an observable, i.e. we maintain an operational stance, in line with the Bohr's approach to quantum mechanics, and we choose to speak only of results of observations.
We say that a measurement of a quantity B at a time t B is disturbed by a measurement of a quantity A at time t A if the variance increases or the average changes with respect to the case when the measurement of A is not performed. Let B and ∆ 2 B the average value and the variance observed when the measurement of A is not performed, and B A and ∆ 2 B|A the corresponding values when A is measured. 1 We shall also define the same values for the case when the measurement A is performed and the corresponding output is A 0 , B A 0 and ∆ 2 B|A 0 . We define the average statistical disturbance and the average systematic disturbance as We define also the statistical disturbance and the systematic disturbance as η 2 It is worth noting that Busch, Lahti and Werner [11] propose a different definition of disturbance. The statistics of the B measurement when the A measurement is performed is compared to the statistics of an ideal B measurement when A is not performed, . This formulation is in line with the interpretation that what is disturbed is the observable, not the measurement of the observable. Besides this difference in the definition, our results agree with those of [11].
Our definition, on the other hand, should be taken with a grain of salt. Let us imagine a broken detector, which always yields the same output B 0 . This measurement would not be disturbed by any measurement performed before it, trivially, and the Heisenberg relation (1a) would be violated. Our definition should be applied only to a 'faithful' measurement. By this term, we mean a measurement that yields different average outputs for different inputs, provided that the inputs are sufficiently far from each other (if their difference is below the resolving power of the measuring apparatus, of course the output will be the same).

Methods
In the theory of sequential or joint measurements it is usually assumed that the initial state of the system and of the two probes factorizes as 1 Clearly, causality implies that B = B A and ∆ 2 B = ∆ 2 B|A if the measurement of A is performed after B (or better, if the measurement B is either in the past light cone or outside the light cone of the measurement of A). Our definition, however, holds also if causality were violated. The main idea of the present work is to question the preceding assumption. While it is uncontroversial that the initial state of the system must be uncorrelated to that of the detectors (otherwise, we would have a measurement before the interaction started), there is no compelling reason to assume that the two detectors are initially uncorrelated. Therefore, we shall write, in full generality, We assume a von Neumann interaction, leading to the following time-evolution of the system and the meters withΦ α representing the input variable of the detector α. We consider the readout states of the detectorsF A (µ), described in the appendix, as Gaussian. It is convenient to work with the characteristic function Z of the readouts, i.e. the Fourier transform of the joint probability P(µ A , µ B ).
For later convenience, we introduce the column vectors (indicated by horizontal lists within The components of the vectors are labeled by an index α taking values in the set {P, Q}, so that s P = q and s Q = p. We also introduce the vector S = [P, Q]. With these definitions, e.g. the term pQ + qP can be written as S · s. For a 1-dimensional system, the Moyal quasi-characteristic function M (q, p) [15] is defined as the Fourier transform of the Wigner quasiprobability W (P, Q) [16].
We recall that M (q, 0) = Z(q) is the characteristic function for the probability distribution in K, i.e. M (q, 0) = dP exp(iP q) P |ρ|P , and M (0, q) is the characteristic function for the probability in Q. Let M sys (s) the initial Moyal function of the system, and M det (φ, j) that of the probes. If the probes are initially uncorrelated, then M det (φ, j) = M Q (φ Q , j Q )M P (φ P , j P ). However, we shall consider the general case when the probes are initially correlated, so that . The variances and covariances of the probes can be expressed as logarithmic derivatives of their Moyal function: We defined asδ 2 α the initial variance in the input variable Φ α and as δ 2 α the initial variance in the output variable J α . By Kennard relation, applied to the probes, We shall use the relation between the final Moyal function for system and probes in terms of the initial ones [17,18] M (s; φ, j) = M sys (s + φ)M det (φ, j + 2α 0 s + α τ φ) , The derivation of this equation is provided in the Supplemental Material of Ref. [18]. The symbol τ ∈ {+, −, 0} specifies the order of the measurements: τ = 0 means joint measurements, τ = + means a position measurement followed by a momentum measurement, and τ = − means the vice versa. We recall that the Moyal function in Eq. (9) provides a complete description of the system+probes, and it is equivalent to knowing the density matrix. The characteristic function of the readout probability Z(φ) = dµ exp [iJ · φ]P (J) is obtained by tracing out the system (s = 0) and by putting j = 0 in Eq. (9). We note that Z can be inferred directly from the experimental data.
References [17,18], in deriving (9), assumed as readout states the projectorsF det (µ) = |J = µ J = µ|. In a realistic measurement with finite resolution, however,F det (µ) = dJ p(µ|J)Π(J), where p(µ|J) is the probability of observing an output µ given that the value of the pointer is J. We recall that for a uniform resolution this latter probability depends only on the difference of its arguments, p(µ|J) = f (µ − J). Here, we make only this hypothesis on the resolution probability, we do not assume e.g., that it is a Gaussian. The results of Refs. [17,18] are then easily generalized thanks to the convolution theorem: the characteristic function, i.e. the Fourier transform of the joint probability of observing µ = (µ P , µ Q ) as the output, is obtained by introducing a factor z(φ) = dµ exp[iµ · φ]f (µ), For definiteness, we consider τ = +, i.e. a measurement of position is made first, and a measurement of momentum follows. Then, Z(φ) = z(φ)M sys (φ)M det (φ, [0, φ P ]). Notice the appearance of the term [0, φ P ] as the second argument of M det . By taking the second derivatives of ln (Z), we obtain the statistical variances of the readout Since the finite resolution enters with a multiplicative term, we find that its contribution to the cumulants is simply of adding a term. The last two terms in Eq. (14) arise because of the derivation rule for a function g(u) = f (u, u) If the Q measurement were not performed, the statistical variance in the P measurement would be ∆ 2 P = σ 2 P + δ 2 P + δ 2 P .
Therefore the statistical disturbance is Here, κ is the initial covariance between the input variable Φ Q of the first probe and the output variable J P of the second probe. The statistical noise for the position measurement is while for the momentum measurement We can make the systematic error D Q = J Q and disturbance D P |Q = J P + Φ Q vanish, by applying two appropriate translations to the initial state of the probe (since we are looking for minimal disturbance and error, it is sensible to assume that the probes are not biased).

Results and applications
Inspection of (17) reveals that the square of the disturbance can not only be arbitrary small, but even negative, This means that the first measurement actually did not disturb the second one, but rather it helped make it sharper. At the same time, while the disturbance reaches zero, the error (or uncertainty) in the first measurement does not diverge. Therefore, the errordisturbance relation can be violated. In order for η 2 P |Q to become negative, it is necessary that the correlator κ defined in (7d) becomes negative. The lowest disturbance, and therefore the lowest error in the second measurement, is achieved when the input variable Φ Q of the first probe is perfectly anti-correlated to the output variable J P of the second probe.
On the other hand, multiplying the errors in (18) and (19) yields because of Kennard relation applied to the variableĴ Q andĴ P +Φ Q , which are canonically conjugated.
In summary, we found that the error-disturbance relation can be violated, with our definition of disturbance, if the detectors have initial correlations, while the error-error relation cannot be violated. We note that what we define as the error is equivalent to the disturbance of [11,14], and therefore, using this alternative definition of disturbance, one would conclude that the error-disturbance relation cannot be violated. Irrespectively of the definition, there is a useful consequence of our result: it is possible to make a measurement of momentum infinitely sharp by preceding it with a position measurement. This is because the former measurement perturbs the momentum by a quantity Φ Q , the write-in variable of the position probe. If this variable is perfectly anti-correlated to the readout variable J P of the second probe, the outcome of the latter will be automatically corrected. Now, in quantum mechanics, two variables can be perfectly anti-correlated only if the two probes are in a maximally entangled state. Specifically, they need to be in an Einstein-Podolsky-Rosen state [19]. 2 Appendix A. Description of the measurement In a measurement, a meter interacts with a quantum system. What sets apart the meter from the system is its ability to cause a sensation in a human being, if not directly, through a chain of amplifications and further interactions with the visible electromagnetic field. Here, the effect of this von Neumann chain [21] is accounted for by considering the readout state of the probe to be a mixed state.
We can justify this assumption in the following way: There are many microscopic states that correspond to the same perceived readout J of a detector, let them |J, m , with m a collective quantum number. The probability of observing an output J is therefore where U is the time-evolution operator, andΠ det (J) is the projection operator on the subspace spanned by |J, m when m varies for fixed J. Next, we consider that there is always some finite resolution in the observation and that the observation may fail, both these processes being characterized by a conditional probability p(µ|J) of observing a value µ when the actual value is J. Strictly speaking, the observed µ should take discrete values, however, since we are more familiar with integrals than with sums, it is often treated as a continuous variable.
The probability of observing µ is therefore whereF det (µ) = dJp(µ|J)Π det (J) is a family of positive operators describing the readout states. These must satisfy the normalization dµF det (µ) = 1, with dµ a Lebesgues-Stieltjes measure, corresponding to a discrete or continuous distribution of outputs µ. The readout statesF det (µ) are not normalized, but in general Tr[F det (µ)] = pr(µ) = 1. The quantities pr(µ) are the priors [22], and they are not necessarily one, unlessF det (µ) is a rank-1 projector. Instead, they represent the unnormalized probability of obtaining an outcome µ in absence of any other information. They are not a conventional probability, since with d the dimension of the Hilbert space of the meter (which could be infinite). However, the only relevant quantity are the ratios pr(µ 1 )/pr(µ 2 ) representing relative probabilities. Furthermore, it follows from their definition that the readout states are classical, i.e. [F det (µ 1 ),F det (µ 2 )] = 0, ∀µ 1 , µ 2 . This hypothesis allows to individuate one, or more, privileged basis, |J , the one that diagonalizes at once allF det (µ). The existence of such a basis can also be justified by the decoherence approach [23,24]. TheF det (µ) are labeled by the average of the operatorĴ = dJJ |J J| and they have a spread δ (µ) in J, i.e., withρ det (µ) =F det (µ)/pr(µ) the normalized readout states. The spread δ (µ) is but the resolution of the probe, and in principle it can be different for different outputs. and let the readout µ take discrete values µ n spaced by (δ n + δ n+1 )/2 from one another (then dµ is a distribution formed by a sum of Dirac deltas). In this case the priors are pr(µ n ) = δ n . In general, we can writeF with p(µ|J) a probability distribution for µ. The priors are thus pr(µ) = dJp(µ|J). Assuming a uniform resolution δ (µ) = δ , Eq. (A.5) is satisfied for p(µ|J) = f (µ − J), where f (µ) is a probability distribution having zero average and spread δ . An important example is given by the Gauss-Laplace distribution, with µ ∈ R and uniform priors pr(µ) = 1.
The conditional state of the system for given µ iŝ Equivalent formulas arise when treating postselected weak measurement [25,26], but with the difference that U is expanded in a perturbation series and the rôles of the system and the probe are reversed. Let us introduce the preparation basis |I , the one in whichρ det = dI w(I)|I I|. We can rewrite the conditional state of the system The measurement can also be described in the language of positive-operator valued measures (POVM) [27][28][29][30][31][32]. The linear operations where D is a measurable domain in the space of possible outcomes (i.e. an element of a σ-algebra in the Kolmogorov formalism), are called instruments if they preserve the non-negativity of ρ.
In the POVM formalism, the instruments are introduced as the basic objects describing the measurement. Besides preserving the nonnegativity of the density operator, they must also satisfy [27,33]  (i) I ∅ = 0, i.e., the measurement always gives an outcome (notice, however, that the outcome is not necessarily a number, but it could consist, e.g., in the observation that the apparatus has failed, etc.). (ii) I ∪D j = j I D j for any family of disjoint elements of the σ-algebra.
The instruments defined in Eq. (A.13) automatically satisfy the three above properties. An interesting question is whether any instrument can be written as in Eq. (A.13).
The instruments describe the change of the density matrix one assigns to the system, conditioned on the information that the outcome of the measurement was in D. In other words, the new density matrix is J,I is univocally defined as a positive operator. In other words, we consider the eigenvalues of the positive operator E J,I (µ), take their arithmetic square root, and define E 1/2 J,I (µ) as the operators having the same eigenstates as E J,I (µ), and as eigenvalues the said arithmetic root. The additional operations V J,I (µ) represent an unwanted and avoidable feedback on the system. For instance, in a Stern-Gerlach apparatus, the beam exiting the magnetic field gradient could pass through a uniform magnetic field, which rotates the spin, so that a second measurement would not confirm the result of the first one. Ideally, V J,I (µ) = 1. Only by making subsequent measurements on the system, is it possible to check whether the measurement process is introducing this unwanted feedback.
We are interested in particular to the case of two sequential measurements. A probe measurinĝ A couples first to the system, then, after the first probe decouples, a second probe measuringB couples. The time-evolution operator factorizes then as U = U B U A , where U A (respectively, U B ) acts on the Hilbert space of the system and the probe A (respectively, B). As we are making two measurements, it is natural to assume that the readout states factor,F det (µ) =F A (µ A )F B (µ B ), each factor acting on a different Hilbert space. A fundamental consideration is that, if we assume as well that the initial state of the two probes factors as ρ det = ρ A ⊗ ρ B , i.e. if we assume no initial correlations between the probes, then the probability of observing the outcomes µ A , µ B can be written P(µ A , µ B ) = Tr sys {I µ B [I µ A (ρ sys )]} (A. 18) and the conditional state of the system is where the composing instruments are defined by However, if we do assume initial correlations, then the total instrument cannot be written as a composition of two instruments, contrary to what is commonly assumed (see e.g. [34]), and we can only associate one set of instrument, defined on the probability space M A × M B . Finally, we note that the marginal probabilities are Notice that the correlations are swapped: from correlations between A and B, they become correlations between B and the system, after tracing out the first probe. Applying Bayes' rule, we see that neither conditional probability unless ρ det = ρ A ⊗ ρ B . This is because, for a given output µ A (respectively, µ B ), the system becomes correlated with the probe B (respectively, A).