Are there fundamental limits for observing quantum phenomena from within quantum theory?
aa r X i v : . [ qu a n t - ph ] S e p Are there fundamental limits for observing quantum phenomena from within quantum theory?
Johannes Kofler and ˇCaslav Brukner Institut f¨ur Quantenoptik und Quanteninformation, ¨Osterreichische Akademie der Wissenschaften,Boltzmanngasse 3, 1090 Wien, Austria Fakult¨at f¨ur Physik, Universit¨at Wien, Boltzmanngasse 5, 1090 Wien, Austria
Does there exist a limit for the applicability of quantum theory for objects of large mass or size, or objectswhose states are of large complexity or dimension of the Hilbert space? The possible answers range frompractical limitations due to decoherence within quantum theory to fundamental limits due to collapse modelsthat modify quantum theory. Here, we suggest the viewpoint that there might be also fundamental limits withoutaltering the quantum laws . We first demonstrate that for two quantum spins systems of a given spin length, noviolation of local realism can be observed, if the measurements are su ffi ciently coarse-grained. Then we showthat there exists a fundamental limit for the precision of measurements due to (i) the Heisenberg uncertaintyrelation which has to be applied to the measuring apparatus, (ii) relativistic causality, and (iii) the finiteness ofresources in any laboratory including the whole universe. This suggests that there might exist a limit for the sizeof the systems (dimension of the Hilbert space) above which no violation of local realism can be seen anymore. Despite the enormous success of quantum physics and itswide range of applications, the region of the whole parame-ter space over which the validity of quantum physics has beendirectly tested is still rather modest. In Ref. [1], Leggett ar-gues that, taking for example the length scale, it is commonlyclaimed that quantum laws are valid down to the Planck scale( ∼ − m) and up to the size of the characteristic length scaleof the Universe ( ∼ + m). This results in 62 orders of mag-nitude, compared to about 25 orders of magnitude over whichthe theory has been directly tested so far. Notwithstandingrecent experimental achievements [2–4] that could demon-strate quantum interference in large systems, it remains anopen question: Are there principal limitations on observingquantum phenomena of objects of large mass or size, or ob-jects whose states are of large complexity or dimension of theHilbert space?
Here we suggest a possible a ffi rmative answer to the abovequestion when considering the dimensionality of the Hilbertspace. This explanation di ff ers conceptually from decoher-ence [5, 6] or collapse theories [7, 8]. Fully within quantumtheory, our approach puts the emphasis on the observability of quantum e ff ects and shows that the necessary measurementaccuracy to see such e ff ects in systems of su ffi ciently largeHilbert space dimension cannot be met because of the con-junction of quantum physics itself, relativity theory, and thefiniteness of resources in any laboratory .To illustrate our idea, let us start by investigating the exper-imental requirements for achieving a violation of local real-ism for systems of increasing Hilbert space dimension. Sucha violation in a Bell experiment [9] is generally accepted as agenuine quantum phenomenon. We consider 2 spin- s particlesin a generalized singlet state | ψ i = √ s + s X m = − s ( − s − m | m i A |− m i B , (1)where, | m i A ( | m i B ) denotes the eigenstates of the spin oper-ator’s z -component of Alice (Bob). Measuring spin compo-nents on either side, this state allows to violate local realismfor arbitrarily large s (dimension 2 s +
1) but it is necessary that the inaccuracy of the angle settings of Alice and Bob, ∆ θ , is atmost in the order of the inverse spin size: ∆ θ . s . This is thecase in the Clauser-Horne-Shimony-Holt-type inequality [10]used in Ref. [11], where the di ff erence between setting angleshas to be about s + as well as in Ref. [12] where the settingangle has to fulfil 0 < sin θ ≈ θ < s .Meanwhile more e ffi cient inequalities for higher-dimensional systems have been found [13, 14]. So onecould argue that for the state (1) there might exist Bellinequalities which do not require such a strict condition as ∆ θ . s . However, it has been shown by Peres that for aresolution which is much worse than the intrinsic quantumuncertainty of a spin coherent state, i.e. ∆ θ ≫ √ s , (2)all Bell inequalities will necessarily be satisfied for thestate (1), since the correlations between outcomes of inaccu-rate measurements become (classical) correlations betweenclassical spins [11]. This approach was extended to the timeevolution of quantum systems and the concept of macroscopicrealism as introduced by Leggett and Garg [15]. It was shownthat for “classical Hamiltonians” and under the restriction ofcoarse-grained measurements, an arbitrarily large quantumspin evolves as an ensemble of classical spins following a clas-sical mechanical evolution [16].We will now extend the result of Peres to arbitrary states of two spin- s systems, taking the restriction of coarse-grainedmeasurements where neighboring spin directions cannot bedistinguished. We first introduce the basic mathematical con-cepts for the further analysis. The (normalized and positive) Q -distribution [17] of a two-system state ˆ ρ AB is given by Q AB ( Ω A , Ω B ) ≡ s + π ! h Ω A , Ω B | ˆ ρ AB | Ω A , Ω B i (3)with Ω i the spin direction and | Ω i i the spin coherent [18] statesfor system i = A , B . In a coarse-grained spin measurement ofsystem i , the whole unit sphere is decomposed into a numberof mutually disjoint angular regions (“slots”) Ω ( k ) i , labeled by k . (The decompositions for A and B need not be the same.)A positive operator valued measurement (POVM) on system i has the elements [19]ˆ P ( k ) i ≡ s + π " Ω ( k ) i | Ω i ih Ω i | d Ω i (4)which correspond to these coarse-grained slots ( P k ˆ P ( k ) i = m for sys-tem A and the outcome n for system B is given by w ( mn ) AB = Tr[ ˆ ρ AB ˆ P ( m ) A ˆ P ( n ) B ] or, equivalently, just via integration over the(positive and normalized) Q -distribution: w ( mn ) AB = " Ω ( m ) A " Ω ( n ) B Q AB ( Ω A , Ω B ) d Ω A d Ω B . (5)(Please note that in general Q AB ( Ω A , Ω B ) does not factorize,i.e. it cannot be written as a product Q A ( Ω A ) Q B ( Ω B ) of two Q -functions of the individual systems.) Upon measurement, thestate ˆ ρ AB is reduced to ˆ ρ ( mn ) AB = ˆ M ( m ) A ˆ M ( n ) B ˆ ρ AB ˆ M ( m ) † A ˆ M ( n ) † B / w ( mn ) AB ,with ˆ M ( k ) i the Kraus operators obeying ˆ M ( k ) † i ˆ M ( k ) i = ˆ P ( k ) i .The corresponding Q -distribution of the reduced state is Q ( mn ) AB ( Ω A , Ω B ) = ( s + π ) h Ω A , Ω B | ˆ ρ ( mn ) AB | Ω A , Ω B i .Under the restriction of su ffi ciently coarse-grained mea-surements where the (polar and azimuthal) angular size ofthese regions, ∆Θ , has to be much larger than the inversesquare root of the spin length s , ∆Θ ≫ / √ s , the Q -distribution before measurement is very well approximated bythe (weighted) mixture of the Q -distributions of the possiblereduced states ˆ ρ ( mn ) AB [19, 20]: Q AB ( Ω A , Ω B ) ≈ X m X n w ( mn ) AB Q ( mn ) AB ( Ω A , Ω B ) . (6)Moreover, this condition holds for all possible “settingchoices” of decompositions for the angular regions of the sys-tems A and B . We could also decompose the regions intoa di ff erent set of mutually disjoint regions, denoted by ¯ Ω ( k ′ ) i (where the decompositions of A and B need not be the same).Then we would get the similar condition Q AB ( Ω A , Ω B ) ≈ P m ′ P n ′ w ( m ′ n ′ ) AB ¯ Q ( m ′ n ′ ) AB ( Ω A , Ω B ), where the ¯ Q ( m ′ n ′ ) AB are the Q -functions for the reduced states under decomposition into¯ Ω ( k ′ ) i . This means that under su ffi ciently coarse-grained mea-surements one can consider all results as stemming froman underlying probability distribution, representing a clas-sical ensemble of spins [16, 19]. In particular, there ex-ists a joint (positive and normalized) probability p ( mm ′ nn ′ ) ≡ p ( Ω ( m ) A , ¯ Ω ( m ′ ) A , Ω ( n ) B , ¯ Ω ( n ′ ) B ) for the (potential) values correspond-ing simultaneously to Ω ( m ) A and ¯ Ω ( m ′ ) A for spin A and Ω ( n ) B and¯ Ω ( n ′ ) B for spin B , which is given by the integration over theintersections of the corresponding regions: p ( mm ′ nn ′ ) = " Ω ( m ) A ∩ ¯ Ω ( m ′ ) A " Ω ( n ) B ∩ ¯ Ω ( m ′ ) B Q AB ( Ω A , Ω B ) d Ω A d Ω B . (7)All the above can of course be easily generalized to morethan two di ff erent compositions for each system and to more than two systems. Q AB can be understood as providing aprobability distribution over local hidden variables ( Ω A , Ω B ).Under su ffi ciently coarse-grained spin measurements ∆Θ ≫ / √ s , no Bell inequality can be violated, as a joint probabil-ity distribution exists. Therefore, the criterion for having achance to see deviations from a fully classical description ofthe two spins reads ∆ θ . √ s . (8)It is clear that for increasingly large s it is hard to meetthis experimental requirement and violate local realism or seenon-classical correlations. But in fact, we suggest that theremight exist even a fundamental upper limit on s —stemmingfrom the Heisenberg uncertainty principle, relativity theory,and finiteness of resources—up to which, for a given mea-surement device, one can still see non-classical correlations.The measurements are done with Stern-Gerlach magnetsor similar devices. The angle of a magnet has to be setwith an accuracy ∆ θ . The Heisenberg uncertainty implies ∆ L ∆ θ ≥ ~ /
2, where ∆ L is the intrinsic uncertainty of the an-gular momentum of the whole magnet and ~ ∼ − Js it thereduced Planck constant. Note that in general the form of theangular momentum uncertainty relation is state-dependent as θ is 2 π -periodic and its variance is naturally bounded fromabove [11, 21]. However, in our case of a well aligned mea-surement apparatus, θ is sharply peaked with a very smallwidth ∆ θ ≪ θ, ˆ L ] = i ~ . (9)In Ref. [22] it was shown that the Planck length is a deviceindependent limit which determines the inaccuracy of any dis-tance measurement. Following these thoughts, we can derivea bound on the angular inaccuracy ∆ θ from within quantumphysics. First assume that the spin enters the inhomogeneousmagnetic field of the Stern-Gerlach magnet at time t = τ . The Hamiltonian of a freely rotating magnet is ˆ H = ˆ L I , (10)where ˆ L is the angular momentum operator of the magnet and I ∼ MR its moment of inertia with M and R the mass andcharacteristic size, respectively. (We often neglect factors ofthe order of 1 throughout our derivations.) In the Heisenbergpicture, the time evolution of the polar angle is given by theHeisenberg equation of motion dˆ θ/ d t = − i [ˆ θ, ˆ H ] / ~ = ˆ L / I .Therefore, for the measurement duration τ , ˆ θ ( τ ) = ˆ θ (0) + ˆ L τ/ I ,where ˆ L is independent of time. We recall the Robertson in-equality ∆ A ∆ B ≥ |h [ ˆ A , ˆ B ] i| [23], which holds for any twoobservables ˆ A and ˆ B . Using the commutation relation (9), weobtain ∆ θ (0) ∆ θ ( τ ) ≥ ~ τ I ∼ ~ τ MR . (11)It follows that at least one of the two quantities, ˆ θ (0) and ˆ θ ( τ ),has a spread of ∆ θ & R r ~ τ M , (12)which is denoted as the standard quantum limit .Using condition (8), we obtain the constraint on the spinsize such that non-classicality can possibly be seen: s . MR ~ τ . (13)Choosing typical laboratory values R ∼ M ∼ τ ∼ s . .In order to obtain a fundamental limit on s , we followRef. [22] and impose physical constraints:By relativistic causality the operative size R of the freelymoving measurement device cannot exceed the distance thatlight can travel during the interaction time τ : R ≤ c τ , with c ∼ m / s the speed of light. Note that this e ff ective mea-surement apparatus not only contains the Stern-Gerlach mag-net but also the table on which it is mounted and possibly thewhole earth etc. Using this constraint, ineq. (12) becomes ∆ θ & r ~ cMR , (14)and ineq. (8) then reads s . cMR ~ . (15)Taking again R ∼ M ∼ s . .As a fundamental limit one can choose as size and massof the device the radius and mass of the observable universe, R U ∼ m and M U ∼ kg, respectively. This leads tothe condition s . cM U R U ~ ∼ . (16)Note that under the stronger accuracy condition ∆ θ . s forthe state (1), the spin size for which the Bell inequalities ofRefs. [11, 12] can be violated is only s . . Both limitsare exceedingly large. However, insofar as the size and massof the universe as ultimate resources are finite, there is a fun-damental limit on how large the Hilbert space of the systemscan be such that one is still able to observes genuine quan-tum features. Despite the fact that the Hilbert space for a spin10 (10 ) can be formed by only about 200 (400) qubits, thequestion whether or not these two limits are trivial depends onwhether a physical spin (measured by a Stern-Gerlach appa-ratus) of such size can in principle be formed. In order to additionally avoid gravitational collapse , thesize of the measurement apparatus must be larger than theSchwarzschild radius corresponding to its mass M : R ≥ GM / c , with G ∼ − m kg − s − the gravitational con-stant [24]. Using this constraint, ineq. (14) becomes ∆ θ & l P R , (17)and ineq. (8) then reads s . R l P , (18)where l P ≡ p ~ G / c ∼ − m is the Planck length. Thislimit can intuitively be understood since the inaccuracy in themeasurement of an angle (which is the ratio of two distances)of, say, a rod of length R is essentially given by the inaccu-racy in the position measurement of its extremal point (givenby the Planck length) divided by the length of the rod. Thelatter, of course, has an uncertainty ∆ R itself, but this leads toa negligible higher order e ff ect.For R ∼ s . . As an alternative fun-damental limit one can again take the size of the universe R U ∼ m, which leads to the condition s . R U l P ∼ . (19)The limit for s in conditions (16) and (19) being similar, re-flects the fact that our observable universe is close to be ablack hole.Several remarks have to be made at this point:(i) The fact that the standard quantum limit can be beatenby contractive states [25–27] does not change the valid-ity of inequality (12) as two subsequent measurementsare still bounded by (11) [22].(ii) The assumption of a free time evolution (10) must bejustified. One could imagine a large setup with all kindsof fields and rods and clever mechanisms which com-pensates movements of the magnet within itself. Butsuch a construction terminates at the causal radius.(iii) One might argue that only angle di ff erences are impor-tant in the Bell experiment and not the local anglesthemselves. This fact, however, cannot be exploitedas the two (space-time) measurement regions must be space-like separated and no rigid connection can exist.(iv) We did not take into account other inaccuracies , in par-ticular the ones in position and momentum of the spinparticles, in the inhomogeneous magnetic field, the statepreparation, the ones during the measurement proce-dure on the screen after the magnet, and inaccuraciesin the reference frames of Alice and Bob themselves.All these components of the experimental setup haveto obey the Heisenberg uncertainty as well and maybeimpose a much stricter limit. In this sense, we havederived a very conservative upper bound on the maxi-mal spin length ( cMR / ~ or R / l P , respectively) beyondwhich it is impossible to observe the quantum featuresof an arbitrary state.(v) We note that a violation of Bell’s inequality remainspossible for arbitrarily large spin size s , if the two par-ties Alice and Bob can perform arbitrary unitary trans-formations before their measurements even if the latterare still coarse-grained [19, 28]. Consider, for exam-ple, the macroscopically entangled state ( | s i A |− s i B + |− s i A | s i B ) / √
2. For observing a violation of Bell’sinequality, it is su ffi cient that Alice and Bob per-form coarse-grained which-hemisphere measurementson their local spin systems, but then it is necessary thatthey have the ability to produce Schr¨odinger cat-likestates of the form |± α i A = cos α | s i A ± sin α |− s i A . Sucha combination of a “non-classical” transformation and a(“classical”) coarse-grained measurement is e ff ectivelynot a coarse-grained measurement in which neighbor-ing spin directions in real configuration space arebunched together . Such measurement observables aredenoted as “unreasonable” [11]. (Reasonable coarse-grained observables correspond to measurements thatbunch together those outcomes that are neighboring inreal space.) Conclusion . We demonstrated that a violation of local real-ism cannot be seen for spins of a certain size because of theHeisenberg uncertainty relation, relativistic causality, (gravi-tational collapse,) and the finiteness of resources in any lab-oratory. The view taken by most scientists is that the con-cepts of physical theories being established due to and veri-fied by experiments are independent of the amount of physicalresources needed to carry out these experiments. In stark con-trast, Benio ff [29] and Davies [30] recently argued that physi-cal laws should not be treated as infinitely precise, immutablemathematical constructs, but must rather respect the finitenessof resources in the universe. This might impose a fundamen-tal limit on the precision of the laws and the specifiability ofphysical states. We enforce this view by proposing that quan-tum mechanics itself puts a limit on the possibility to observequantum phenomena if only a restricted amount of physicalresources is available. Acknowledgments . We thank M. Aspelmeyer, T. Paterek,and A. Zeilinger for helpful discussions. This work wassupported by the Austrian Science Foundation FWF within Project No. P19570-N16, SFB and CoQuS No. W1210-N16. [1] A. J. Leggett, J. Phys.: Cond. Mat. , R415 (2002).[2] M. Arndt, O. Nairz, J. Voss-Andreae, C- Keller, G. van derZouw, and A. Zeilinger, Nature , 680 (1999).[3] J. R. Friedman, V. Patel, W. Chen, S. K. Tolpygo, and J. E.Lukens, Nature , 43 (2000).[4] B. Julsgaard, A. Kozhekin, and E. S. Polzik, Nature , 400(2001).[5] W. H. Zurek, Phys. Today , 36 (1991).[6] W. H. Zurek, Rev. Mod. Phys. , 715 (2003).[7] G. C. Ghirardi, A. Rimini, T. Weber, Phys. Rev. D , 470(1986).[8] R. Penrose, Phil. Trans. R. Soc. Lond. A , 1927 (1998).[9] J. S. Bell, Physics , 195 (1964).[10] J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt, Phys.Rev. Lett. , 880 (1969).[11] A. Peres, Quantum Theory: Concepts and Methods (KluwerAcademic Publishers, 1995).[12] N. D. Mermin, Phys. Rev. D , 356 (1980).[13] D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu,Phys. Rev. Lett. , 040404 (2002).[14] M. Junge, C. Palazuelos, D. Perez-Garcia, I. Villanueva, M. M.Wolf, arXiv:0910.4228v1 [quant-ph].[15] A. J. Leggett and A. Garg, Phys. Rev. Lett. , 857 (1985).[16] J. Kofler and ˇC. Brukner, Phys. Rev. Lett. , 180403 (2007).[17] G. S. Agarwal, Phys. Rev. A , 2889 (1981); G. S. Agarwal,Phys. Rev. A , 4608 (1993).[18] J. M. Radcli ff e, J. Phys. A: Gen. Phys. , 313 (1971); P. W.Atkins and J. C. Dobson, Proc. R. Soc. A , 321 (1971).[19] J. Kofler and ˇC. Brukner, Phys. Rev. Lett. , 090403 (2008).[20] Independent of the concrete implementation, the POVM ele-ments ˆ P ( k ) i (and the Kraus operators ˆ M ( k ) i ) in su ffi ciently coarse-grained measrements ( ∆Θ ≫ / √ s ) behave almost as projec-tors for all coherent spin states | Ω i i except for those near a slotborder, i.e. ˆ P ( k ) i | Ω i i ≈ | Ω i i for Ω i inside Ω ( k ) i and ˆ P ( k ) i | Ω i i ≈ for Ω i outside Ω ( k ) i [19].[21] D. T. Pegg, S. M. Barnett, R. Zambrini, S. Franke-Arnold, andM. Padgett, New J. Phys. , 62 (2005).[22] X. Calmet, M. Graesser, and S. D. H. Hsu, Phys. Rev. Lett. ,211101 (2004).[23] H. P. Robertson, Phys. Rev. , 163 (1929).[24] It is of course conceivable that an observer performs experi-ments inside a black hole. Such an observer, however, cannotcommunicate his results to regions outside the black hole.[25] H. P. Yuen, Phys. Rev. Lett. , 719 (1983).[26] C. M. Caves, Phys. Rev. Lett. , 2465 (1985).[27] V. Giovannetti, S. Lloyd, and L. Maccone, Science , 1330(2004).[28] W. Son, J. Kofler, M. S. Kim, V. Vedral, and ˇC. Brukner, Phys.Rev. Lett. , 110404 (2009).[29] P. Benio ff , arXiv:quant-ph / Randomness and Complexity, from Leib-niz to Chaitin , ed. C. S. Calude (World Scientific, Singapore,2007); electronic version: arXiv:quant-ph //