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Interpretations of quantum measurement without the collapse postulate
Volume 128, number 1,2
PHYSICS LETTERS A
21 March 1988
INTERPRETATIONS OF QUANTUM MEASUREMENT WITHOUT THE COLLAPSE POSTULATE D. HOME Physics Department, Bose Institute, Calcutta 700009, India
and M.A.B. WHITAKER Department of Physics, University of Ulster, Coleraine, Northern Ireland Received 23 October 1987; revised manuscript received 18 January 1988; accepted for publication 19 January 1988 Communicated by J.P. Vigier
Interpretations ofquantum measurement without the collapse postulate are discussed. That in which the final state-function is a linear superposition of macroscopic states is analysed, and contrasted with an interpretation in which macroscopic states are represented by a mixture in the final state. Application is made to a thought-experiment previously analysed by the present authors, and by Hnizdo.
There has long been considerable discussion concerning the necessity or otherwise ofassuming wavefunction collapse at a measurement [1—3], and a number of recent papers have again addressed the question [4—10]. Home and Whitaker [9] and Hnizdo [10] have analysed a thought experiment designed to test the collapse postulate, with differing conclusions. The situation may be clarified by noting that different versions of the “no collapse” postulate may lead to different final state-functions for the combined system of measured and measuring apparatuses in a range of experiments. To take a simple example, let us suppose that the z-component ofspin is measured for a system whose initial state-function is (l/,,~/~)(X~ +~). In (i) a conventional von Neumann interpretation [1] involving collapse, the final state of spin and measuring apparatus will be a mixture written as — Bz BZ ‘1 ~ X + +~X ‘ / with equal weight factors, where B~and B~are the states ofthe apparatus corresponding to spin up and down respectively. In a necessarily irreversible process, as well as (a) a change from pure to mixed state Z
—
—
—
at the measurement, there is achieved (b) a one-toone correlation between eigenstates of measured system and measuring apparatus, the latter eigenstates being orthogonal states of a macroscopic system. Thus either (a) or (b) individually leads to the result that there can be no interference between x~-and x~-following the measurement. Since, as stated in ref. [9], the view that no collapse is necessaryis put forward chiefly by advocates of (ii) the ensemble interpretation [11], it is interesting to apply such an interpretation to the same problem, following Landsberg and Home [12]. In this case the final state-function becomes
w=
~
~
(2)
V
Though feature (a) of the von Neumann interpretation is suppressed, feature (b), the correlation of observed and observing systems, remains, and again there cannot be interference between the two terms at times after the measurement. This, we believe, corresponds to the argument of Hnizdo [10], who thus recovers most of the standard results following from the collapse postulate;
0375-9601 788/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 128, number 1,2
PHYSICS LETTERS A
one notes, for example, his reference to the “endless chain of apparatuses”. We do not feel, however, that such a function as (2) can be regarded as satisfactory. It does not recognise that the measuring system is left in a definite state rather than in a linear superposition of such states. It cannot be enough that the final state is a pure state which may behave as if it were a mixture, and thus be described as being empirically equivalent to a mixture. This point has recently been stressed by Leggett [131. The same author considers the cornmon situation where the microsystem is absorbed in the detection process (for example, a photomultiplier detector or counter), so the macroscopic states include the “measured” states. Again his conclusion is that writing a final state as a linear combination of macroscopic states cannot be a true and complete reflection of reality [14]. A final state which represents (iii) a mixture, but one in which the phase relationship between Xz+ and Xz_ is maintained (corresponding to no collapse of wave-function) is
(x~-+.~t)B~,
(X~-+xl )Bt
(3)
where the weight factors are equal. With this funclion, both (a) collapse and (b) correlation between states of measuring and measured systems, are relinquished. Unlike (i) and (ii), interference between Xz+ and xt following the measurement is not precluded. It could perhaps be asserted that postulate (iii) is • . . in pnnciple objectionable on the grounds that it contradicts the requirement that if a measurement performed upon a system yields a particular value, then any subsequent measurement of the same quantity performed immediately on the same system will yield again that value with certainty. However postulate (iii) may be defended on the following grounds: (A) The above-stated requirement refers to successive measurements on a single quantum system. Proponents of the ensemble interpretation (e.g. Ballentine [11]) insist that in quantum mechanics one is concerned only with the statements dealing with statistically meaningful measuremental results pertaming to an ensemble of identically prepared systems. Unless one shows some contradiction with postulate (iii) at the statistical level, it cannot be .
2
.
.
21 March 1988
ruled out a priori merely on the basis of the above requirement. Margenau [15] has put forward this argument in a particularly cogent manner, and Jammer [2, pp. 227—229] also gives a clear discussion. (B) The requirement is not in general operationally applicable because most of the actual measurements (e.g. using counters, photographic plates, photocells) “annihilate” the measured micro-system thereby rendering any subsequent measurement on that system impossible. The different final functions may now be applied to the thought-experiment analysed in refs. [9] and [10], in which one component of an electron beam divided by a Stern—Gerlach apparatus interacts with an atomic spin-system to produce an intermediate state-function of the form
—r= [g1
çii~=
(r)A~~± +g2(r)A-_~t] (4) ~2 where g1 and g2 are spatial wave-functions of the two components of the beam, and A~ and At atomic spin-states. A measurement is then made of the zcomponent of A. According to the three interpretalions above, the final state of the combined system ,
-
~
(1)
(ii)
—~—
...
(tii)
(g1A~~~B~ +g2At~tBt)
(g1A+~~ +g2At~t)B~
1 (~‘~xz+ +g2A~~t )Bi where B~and Bt are the states of the macroscopic system measuring A~,and the mixtures of (i) and (iii) have equal weight factors. With the form (iii) (and recombination of beams so that overlap ofg1 and g2 is non-zero), interference between ~ and At~t is possible, and this is manifested in a measurement of a, given by a~ (where r~and O~2 are the spin components of the atom and the electron in the xz-plane, as discussed in ref. [9]). With either of forms (i) or (ii), no interference is possible. They will both lead to (a>
Volume 128, number 1,2
PHYSICS LETTERS A
being equal to cos 01 cos 02; assuming the overlap between g1 and g2 is unity, form (iii) makes equal to cos (01—02). In discussing results with and without collapse, it seems that Home and Whitaker [91 used interpretation (iii) for “no collapse”, and found to be different from (i), the case with collapse, while Hnizdo [101 used (ii), and found no difference from (i). Having discussed the thought experiment, we may comment on its connection with arguments (A) and (B) above. To (A) we add the remark that in this paper it is shown that even at the statistical level, it is possible to discriminate between (i) and (iii) using the thought experiment. On the “annihilation” difficulty mentioned in (B), we remark that the difficulty is circumvented in our thought experiment. Atomic spin-system A is connected to a macroscopic detector that registers, for example, whether the atom makes any transition from the excited spin-state by detecting the emitted photons. This constitutes measurement of the z-component ofA, which may then be followed by measurement ofa. Thus we reallydeal with successive measurements, and we show that the predicted values of differ depending on whether one invokes the collapse postulate. To sum up, it is clear that merely to distinguish between “collapse” and “no collapse” [9] does not fully characterise the results of a measurement procedure. In particular, following Leggett [13,141, we do not feel that a final function of form (ii) can be regarded as satisfactory. We would like to add a comment concerning Hnizdo’s final remarks that “there should be no doubt as to the correctness and consistency of the standard measurement postulate of quantum mechanics”, and his reference to Bohr [161. It is surprising to see Bohr put forward as representing the “orthodox” position on quantum measurement, particularly if, as one assumes, the position is taken to include the collapse postulate. Stapp [17] emphasises that the Copenhagen interpretation of Bohr and Heisenberg was “diametrically opposed” to what he calls the “absolute-p interpretation” which requires the collapse idea. Indeed Teller [18] points out that Bohr only ever made one — extremely oblique comment on the collapse postulate. The actual philosophy of Bohr is of a subtlety rather beyond, and in —
21 March 1988
many ways out-of-tune with the basic ideas often put forward as “orthodox” [17—19]. We would strongly question whether there exists an “orthodox” interpretation ofquantum mechanics and quantum measurement in which such matters as those discussed in ref. [91 become unproblematic. Even leaving aside new experimental challenges from neutron interferometry [7,8] and macroscopic quantum tunnelling [13,14], the very existence of a large and growing literature on foundational issues [2,20] surely confirms that the problems associated with measurement EPR, cat, Wigner’s friend and so on, the understanding ofthe collapse mechanism, the attempts at putting forward novel interpretations, are all valid subjects for current appraisal. —
References [1] J. von Neumann, Mathematical foundations of quantum mechanics (Princeton Univ. Press, Princeton, 1955) [2] M. Jammer, The philosophy ofquantum mechanics (Wiley, New York, 1974). [3] N.F. Mott, Proc. R. Soc. A 126 (1929) 79. [4] A.B. Pippard, Eur. J. Phys. 7 (1986) 43. [5] J.S. Bell, in: Quantum gravity, Vol. 2, eds. C.J. Isham, R. Penrose and DW. Sciama (Clarendon, Oxford, 1981) p. 611. [61P.J. Bussey, Phys. Lett. A 106 (1984) 407; 120 (1987) 51. [7] C. Dewdney, Ph. Guéret, A. Kyprianidis and J.P. Vigier, Phys. Lett. A 102 (1984) 291; Found. Phys. 15(1985)1031; J.P. Vigier, in:measurement Proc. mt. Conf. on New techniques and ideas in quantum theory (New York Academy of Sciences, New York, 1986). [8] H. Rauch, H. Treimerand U. Bonse, Phys. Lett. A 47 (1974) 369. [9] D. Home and M.A.B. Whitaker, Phys. Leit. A 117 (1986) 439. [10] V. Hnizdo, Phys. Lett.A 120(1987) 263. [11] H. Margenau, Phys. Rev. 49 (1936) 240; J.L. Park, Am. J. Phys. 36 (1968) 211; L.E. Ballentine, Rev. Mod. Phys. 42 (1970) 358; Newton, Am. 48 Am. (1980) 1029.55 (1987) 226. [12] R.G. P.T. Landsberg andJ.D.Phys. Home, J. Phys. [13] A.J. Leggett, in: The lesson of quantum theory, eds. J. de Boer, E. Dal and 0. Ulfbeck (North-Holland, Amsterdam, 1986); in: Quantum implications — Essays in honour of. David Bohm, eds. B.J. Hiley and D. Peat (Routledge and Kegan Paul, London, 1987). [14] A.J. Leggeit, in: Proc. 2nd mt. Symp. on Foundations of quantum mechanics in the light of new M. Namiki, Y. Ohnuki, Y. Murayama and technology, S. Namura eds. (Physical Society of Japan, Tokyo, 1987).
3
Volume 128, number 1,2
PHYSICS LETTERS A
[15] H. Margenau, Philos. Sci. 4 (1937) 337. [16] N. Bohr, Essays 1958—1962 on atomic physics and human knowledge (Interscience, New York, 1963) p.3. [17] H.P. Stapp, Am. J. Phys. 40 (1972) 1098. [18] P. Teller, in: PSA 1980, Vol. 2, eds. PD. Asquith and R.N. Giere (Philosophy of Science Association, East Lansing, 201. Michigan, 1981) p.
4
21 March 1988
[191 H.J. Folse, The philosophy of Niels Bohr (North-Holland. Amsterdam, 1985). [20]J.A. Wheeler and W.H. Zurek, eds., Quantum theory and measurement (Princeton Univ. Press. Princeton, 1983).
PHYSICS LETTERS A
21 March 1988
INTERPRETATIONS OF QUANTUM MEASUREMENT WITHOUT THE COLLAPSE POSTULATE D. HOME Physics Department, Bose Institute, Calcutta 700009, India
and M.A.B. WHITAKER Department of Physics, University of Ulster, Coleraine, Northern Ireland Received 23 October 1987; revised manuscript received 18 January 1988; accepted for publication 19 January 1988 Communicated by J.P. Vigier
Interpretations ofquantum measurement without the collapse postulate are discussed. That in which the final state-function is a linear superposition of macroscopic states is analysed, and contrasted with an interpretation in which macroscopic states are represented by a mixture in the final state. Application is made to a thought-experiment previously analysed by the present authors, and by Hnizdo.
There has long been considerable discussion concerning the necessity or otherwise ofassuming wavefunction collapse at a measurement [1—3], and a number of recent papers have again addressed the question [4—10]. Home and Whitaker [9] and Hnizdo [10] have analysed a thought experiment designed to test the collapse postulate, with differing conclusions. The situation may be clarified by noting that different versions of the “no collapse” postulate may lead to different final state-functions for the combined system of measured and measuring apparatuses in a range of experiments. To take a simple example, let us suppose that the z-component ofspin is measured for a system whose initial state-function is (l/,,~/~)(X~ +~). In (i) a conventional von Neumann interpretation [1] involving collapse, the final state of spin and measuring apparatus will be a mixture written as — Bz BZ ‘1 ~ X + +~X ‘ / with equal weight factors, where B~and B~are the states ofthe apparatus corresponding to spin up and down respectively. In a necessarily irreversible process, as well as (a) a change from pure to mixed state Z
—
—
—
at the measurement, there is achieved (b) a one-toone correlation between eigenstates of measured system and measuring apparatus, the latter eigenstates being orthogonal states of a macroscopic system. Thus either (a) or (b) individually leads to the result that there can be no interference between x~-and x~-following the measurement. Since, as stated in ref. [9], the view that no collapse is necessaryis put forward chiefly by advocates of (ii) the ensemble interpretation [11], it is interesting to apply such an interpretation to the same problem, following Landsberg and Home [12]. In this case the final state-function becomes
w=
~
~
(2)
V
Though feature (a) of the von Neumann interpretation is suppressed, feature (b), the correlation of observed and observing systems, remains, and again there cannot be interference between the two terms at times after the measurement. This, we believe, corresponds to the argument of Hnizdo [10], who thus recovers most of the standard results following from the collapse postulate;
0375-9601 788/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 128, number 1,2
PHYSICS LETTERS A
one notes, for example, his reference to the “endless chain of apparatuses”. We do not feel, however, that such a function as (2) can be regarded as satisfactory. It does not recognise that the measuring system is left in a definite state rather than in a linear superposition of such states. It cannot be enough that the final state is a pure state which may behave as if it were a mixture, and thus be described as being empirically equivalent to a mixture. This point has recently been stressed by Leggett [131. The same author considers the cornmon situation where the microsystem is absorbed in the detection process (for example, a photomultiplier detector or counter), so the macroscopic states include the “measured” states. Again his conclusion is that writing a final state as a linear combination of macroscopic states cannot be a true and complete reflection of reality [14]. A final state which represents (iii) a mixture, but one in which the phase relationship between Xz+ and Xz_ is maintained (corresponding to no collapse of wave-function) is
(x~-+.~t)B~,
(X~-+xl )Bt
(3)
where the weight factors are equal. With this funclion, both (a) collapse and (b) correlation between states of measuring and measured systems, are relinquished. Unlike (i) and (ii), interference between Xz+ and xt following the measurement is not precluded. It could perhaps be asserted that postulate (iii) is • . . in pnnciple objectionable on the grounds that it contradicts the requirement that if a measurement performed upon a system yields a particular value, then any subsequent measurement of the same quantity performed immediately on the same system will yield again that value with certainty. However postulate (iii) may be defended on the following grounds: (A) The above-stated requirement refers to successive measurements on a single quantum system. Proponents of the ensemble interpretation (e.g. Ballentine [11]) insist that in quantum mechanics one is concerned only with the statements dealing with statistically meaningful measuremental results pertaming to an ensemble of identically prepared systems. Unless one shows some contradiction with postulate (iii) at the statistical level, it cannot be .
2
.
.
21 March 1988
ruled out a priori merely on the basis of the above requirement. Margenau [15] has put forward this argument in a particularly cogent manner, and Jammer [2, pp. 227—229] also gives a clear discussion. (B) The requirement is not in general operationally applicable because most of the actual measurements (e.g. using counters, photographic plates, photocells) “annihilate” the measured micro-system thereby rendering any subsequent measurement on that system impossible. The different final functions may now be applied to the thought-experiment analysed in refs. [9] and [10], in which one component of an electron beam divided by a Stern—Gerlach apparatus interacts with an atomic spin-system to produce an intermediate state-function of the form
—r= [g1
çii~=
(r)A~~± +g2(r)A-_~t] (4) ~2 where g1 and g2 are spatial wave-functions of the two components of the beam, and A~ and At atomic spin-states. A measurement is then made of the zcomponent of A. According to the three interpretalions above, the final state of the combined system ,
-
~
(1)
(ii)
—~—
...
(tii)
(g1A~~~B~ +g2At~tBt)
(g1A+~~ +g2At~t)B~
1 (~‘~xz+ +g2A~~t )Bi where B~and Bt are the states of the macroscopic system measuring A~,and the mixtures of (i) and (iii) have equal weight factors. With the form (iii) (and recombination of beams so that overlap ofg1 and g2 is non-zero), interference between ~ and At~t is possible, and this is manifested in a measurement of a, given by a~ (where r~and O~2 are the spin components of the atom and the electron in the xz-plane, as discussed in ref. [9]). With either of forms (i) or (ii), no interference is possible. They will both lead to (a>
Volume 128, number 1,2
PHYSICS LETTERS A
being equal to cos 01 cos 02; assuming the overlap between g1 and g2 is unity, form (iii) makes equal to cos (01—02). In discussing results with and without collapse, it seems that Home and Whitaker [91 used interpretation (iii) for “no collapse”, and found to be different from (i), the case with collapse, while Hnizdo [101 used (ii), and found no difference from (i). Having discussed the thought experiment, we may comment on its connection with arguments (A) and (B) above. To (A) we add the remark that in this paper it is shown that even at the statistical level, it is possible to discriminate between (i) and (iii) using the thought experiment. On the “annihilation” difficulty mentioned in (B), we remark that the difficulty is circumvented in our thought experiment. Atomic spin-system A is connected to a macroscopic detector that registers, for example, whether the atom makes any transition from the excited spin-state by detecting the emitted photons. This constitutes measurement of the z-component ofA, which may then be followed by measurement ofa. Thus we reallydeal with successive measurements, and we show that the predicted values of differ depending on whether one invokes the collapse postulate. To sum up, it is clear that merely to distinguish between “collapse” and “no collapse” [9] does not fully characterise the results of a measurement procedure. In particular, following Leggett [13,141, we do not feel that a final function of form (ii) can be regarded as satisfactory. We would like to add a comment concerning Hnizdo’s final remarks that “there should be no doubt as to the correctness and consistency of the standard measurement postulate of quantum mechanics”, and his reference to Bohr [161. It is surprising to see Bohr put forward as representing the “orthodox” position on quantum measurement, particularly if, as one assumes, the position is taken to include the collapse postulate. Stapp [17] emphasises that the Copenhagen interpretation of Bohr and Heisenberg was “diametrically opposed” to what he calls the “absolute-p interpretation” which requires the collapse idea. Indeed Teller [18] points out that Bohr only ever made one — extremely oblique comment on the collapse postulate. The actual philosophy of Bohr is of a subtlety rather beyond, and in —
21 March 1988
many ways out-of-tune with the basic ideas often put forward as “orthodox” [17—19]. We would strongly question whether there exists an “orthodox” interpretation ofquantum mechanics and quantum measurement in which such matters as those discussed in ref. [91 become unproblematic. Even leaving aside new experimental challenges from neutron interferometry [7,8] and macroscopic quantum tunnelling [13,14], the very existence of a large and growing literature on foundational issues [2,20] surely confirms that the problems associated with measurement EPR, cat, Wigner’s friend and so on, the understanding ofthe collapse mechanism, the attempts at putting forward novel interpretations, are all valid subjects for current appraisal. —
References [1] J. von Neumann, Mathematical foundations of quantum mechanics (Princeton Univ. Press, Princeton, 1955) [2] M. Jammer, The philosophy ofquantum mechanics (Wiley, New York, 1974). [3] N.F. Mott, Proc. R. Soc. A 126 (1929) 79. [4] A.B. Pippard, Eur. J. Phys. 7 (1986) 43. [5] J.S. Bell, in: Quantum gravity, Vol. 2, eds. C.J. Isham, R. Penrose and DW. Sciama (Clarendon, Oxford, 1981) p. 611. [61P.J. Bussey, Phys. Lett. A 106 (1984) 407; 120 (1987) 51. [7] C. Dewdney, Ph. Guéret, A. Kyprianidis and J.P. Vigier, Phys. Lett. A 102 (1984) 291; Found. Phys. 15(1985)1031; J.P. Vigier, in:measurement Proc. mt. Conf. on New techniques and ideas in quantum theory (New York Academy of Sciences, New York, 1986). [8] H. Rauch, H. Treimerand U. Bonse, Phys. Lett. A 47 (1974) 369. [9] D. Home and M.A.B. Whitaker, Phys. Leit. A 117 (1986) 439. [10] V. Hnizdo, Phys. Lett.A 120(1987) 263. [11] H. Margenau, Phys. Rev. 49 (1936) 240; J.L. Park, Am. J. Phys. 36 (1968) 211; L.E. Ballentine, Rev. Mod. Phys. 42 (1970) 358; Newton, Am. 48 Am. (1980) 1029.55 (1987) 226. [12] R.G. P.T. Landsberg andJ.D.Phys. Home, J. Phys. [13] A.J. Leggett, in: The lesson of quantum theory, eds. J. de Boer, E. Dal and 0. Ulfbeck (North-Holland, Amsterdam, 1986); in: Quantum implications — Essays in honour of. David Bohm, eds. B.J. Hiley and D. Peat (Routledge and Kegan Paul, London, 1987). [14] A.J. Leggeit, in: Proc. 2nd mt. Symp. on Foundations of quantum mechanics in the light of new M. Namiki, Y. Ohnuki, Y. Murayama and technology, S. Namura eds. (Physical Society of Japan, Tokyo, 1987).
3
Volume 128, number 1,2
PHYSICS LETTERS A
[15] H. Margenau, Philos. Sci. 4 (1937) 337. [16] N. Bohr, Essays 1958—1962 on atomic physics and human knowledge (Interscience, New York, 1963) p.3. [17] H.P. Stapp, Am. J. Phys. 40 (1972) 1098. [18] P. Teller, in: PSA 1980, Vol. 2, eds. PD. Asquith and R.N. Giere (Philosophy of Science Association, East Lansing, 201. Michigan, 1981) p.
4
21 March 1988
[191 H.J. Folse, The philosophy of Niels Bohr (North-Holland. Amsterdam, 1985). [20]J.A. Wheeler and W.H. Zurek, eds., Quantum theory and measurement (Princeton Univ. Press. Princeton, 1983).