ARTICLE IN PRESS Physica E 42 (2010) 327–329
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Entangled states in quantum mechanics Janis Ruzˇa ¯ Riga Technical University, Institute Technical Physics, Riga, Latvia
a r t i c l e in fo
Available online 24 June 2009
In some circles of quantum physicists, a view is maintained that the nonseparability of quantum systems—i.e., the entanglement—is a characteristic feature of quantum mechanics. According to this view, the entanglement plays a crucial role in the solution of quantum measurement problem, the origin of the ‘‘classicality’’ from the quantum physics, the explanation of the EPR paradox by a nonlocal character of the quantum world. Besides, the entanglement is regarded as a cornerstone of such modern disciplines as quantum computation, quantum cryptography, quantum information, etc. At the same time, entangled states are well known and widely used in various physics areas. In particular, this notion is widely used in nuclear, atomic, molecular, solid state physics, in scattering and decay theories as well as in other disciplines, where one has to deal with many-body quantum systems. One of the methods, how to construct the basis states of a composite many-body quantum system, is the so-called genealogical decomposition method. Genealogical decomposition allows one to construct recurrently by particle number the basis states of a composite quantum system from the basis states of its forming subsystems. These coupled states have a structure typical for entangled states. If a composite system is stable, the internal structure of its forming basis states does not manifest itself in measurements. However, if a composite system is unstable and decays onto its forming subsystems, then the measurables are the quantum numbers, associated with these subsystems. In such a case, the entangled state has a dynamical origin, determined by the Hamiltonian of the corresponding decay process. Possible correlations between the quantum numbers of resulting subsystems are determined by the symmetries—conservation laws of corresponding dynamical variables, and not by the quantum entanglement feature. & 2009 Elsevier B.V. All rights reserved.
Keywords: Coupled states Entanglement Correlations
1. Introduction ¨ The notion of ‘‘entangled’’ states was introduced by Schrodinger  in connection with a quantum measurement problem and EPR paradox . According to the adopted deﬁnition, an entangled state is a quantum state of a composite system, which cannot be expressed via a product of quantum states of its forming subsystems. After works of Bell , the entangled states acquired a status of fundamental signiﬁcance, since it was assumed that these states are responsible for a nonlocal character of the quantum world. At the same time, one should recognize that these states, although, under other names, such as—coupled, bounded states, have already been widely used in atomic, molecular, nuclear, solid state physics, in scattering and reaction theories, i.e., anywhere, where one studies many-body quantum systems. It is signiﬁcant, that physicists, working in these areas, have not seen any speciﬁc in the entangled states, related with the exceptional role they might play. All the observed phenomena were described and explained remaining within the orthodox framework of quantum theory without referring to the nonlocality
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notion. Therefore, it would be important to establish the reason, why in the quantum methodology, unlike to the regular domains of quantum theory, the entangled states started to play such a seemingly fundamental role. Looking retrospectively, one could extract the following moments, which lead to this state of affairs. The ﬁrst one is related with the rather superﬁcial attitude to the construction of coupled—entangled—states of composite systems. It has mainly a technical character and is associated with the ignorance of relations between quantum numbers of a composite system and its forming subsystems. Due to this reason, for example, a partial trace of the density matrix of a composite system is mistakenly interpreted as the physical density matrix of the associated subsystem. The second one is related with the incorrect physical interpretation of entangled states. In the following exposition we shall consider these two moments in more detail.
2. Entangled states and their interpretation In order to obtain the state vectors for an arbitrary quantum system, employing the consequent approach, it is necessary to
ARTICLE IN PRESS J. Ruzˇa / Physica E 42 (2010) 327–329
¨ solve the corresponding Schrodinger equation. This task becomes extremely complicated in the case of composite or many-body quantum system, when a direct solution of this equation is practically impossible. However, sometimes one can obtain the ¨ necessary information about the Schrodinger equation’s Hilbert space—its structure and basis states, without solving it. One of the basic approaches to the construction of the basis states of a composite system is the so-called genealogical decomposition method, introduced by Goudsmit and Bacher  and developed by Racah . It allows one to construct recurrently by particle number the basis state vectors of a composite system from the basis states of its forming free subsystems. In order to illustrate the essence of this method, let us consider the most general case, when we have the n body system with basis state vector jnGS, and its two ﬁxed subsystems, consisting of n1 and n2 (n1 þ n2 ¼ n) particles with basis vectors jn1 G1 S and jn2 G2 S, correspondingly. Next, let us consider the decomposition: X jn1 G1 Sjn2 G2 SAG1 G2 ;G ; ð1Þ jnGS ¼ G1 G2
where AG1 G2 ;G is the decomposition coefﬁcients, called the genealogical coefﬁcients. In this construction, it is assumed that all state vectors, entering in Eq. (1), have a common structure, that is G, G1 and G2 have the same set of characteristics. As to the coefﬁcients jAG1 G2 ;G j2 , then they may be formally interpreted as relative ‘‘weights’’, by which the states jG1 S and jG2 S contribute to the total state jGS. One can see that (1) has a typical structure of the entangled state. For the decomposition of the dual space basis we have, accordingly X /n1 G1 j/n2 G2 jAG1 G2 ;G : ð2Þ /nGj ¼ G1 G2
Assuming that the functions, entering in the decompositions (1) and (2), are orthonormal, one obtains X AG1 G2 ;G AG1 G2 ;G0 ¼ dGG0 : ð3Þ G1 G2
In cases when Eq. (1) is invertible, i.e., when the product of functions jn1 GSjn2 GS can be expressed via jnGS and, analogously, for (2), the genealogical coefﬁcients obey the relation X AG1 G2 ;G AG10 G20 ;G ¼ dG10 G1 dG2 G20 :
In the case, when basis states have irreducible properties, the genealogical coefﬁcients have a very simple geometrical or algebraic interpretation, reduced, for example, to the Clebsch– Gordon coefﬁcients. In typical applications, the genealogical decompositions are used in order to complete matrix representations of operators, depending on the variables of one subsystem. For instance, let us assume that an operator depends on the variables of the second subsystem and label it as O^ 2 O^ 2 ðn1 þ 1; n1 þ 2; . . . ; nÞ:
Then, the matrix elements of Eq. (5), calculated on the state vectors (1), are equal to X IG2 G20 QG2 G20 ðGG0 Þ; ð6Þ /nGjO^ 2 jnG0 S ¼ G2 G20
where the following notations are introduced: IG2 G20 ¼ /n2 G2 jO^ 2 jn2 G20 S
and QG2 G20 ðGG0 Þ ¼
X AG1 G2 ;G AG1 G20 ;G0 :
It is easy to track down an analogy between the expression (6) and the matrix representation of operators in the density matrix formalism used in quantum mechanics. The matrix (8), entering in (6) and deﬁned via the bilinear combination of the genealogical coefﬁcients, can be called the reduced density matrix. This matrix depends on two types of indices—the characteristics G2 and G20 of the second subsystem and the characteristics G and G0 of the total system. Because of this, one cannot regard the matrix (8) as a physical density matrix of the second subsystem, understood in a proper sense of quantum mechanics. It appears in the course of calculation and serves only as a purely mathematical tool, encompassing the irreducible properties of the employed basis states. In the case of stable composite system, the only measurable quantum numbers are G’s. The inner structure of quantum states (1), as one can see from Eq. (6), remains ‘‘invisible’’. For this reason, one can regard the decomposition (1) as a merely technical tool for the construction of basis states of a composite system, so avoiding complications, associated with their interpretation. A similar approach can be applied when considering the case of an unstable composite system. Let us assume that a quantum system A is unstable and decays onto two subsystems 1 and 2: A!1 þ 2. For the sake of simplicity, let us assume that the initial state of the decaying system is its basis state jGS. In the most simple case, the ﬁnal quantum state jGSð1þ2Þ of the system can be expressed via the bilinear combination of basis states of two free subsystems as X jGSð1þ2Þ ¼ B jG1 SjG2 SAG1 G2 ;G ; ð9Þ G1 G2
where B is the decay amplitude of the state jGS, determined by the Hamiltonian of the decay process. The quantity jBj2 is proportional to the rate, by which A, being in the state jGS, decays onto subsystems 1 and 2, or the rate, by which these pairs are generated. Analogously, as in Eq. (1), jAG1 G2 ;G j2 determines a relative ‘‘weight’’, by which a particular pair of states jG1 S and G2 S of the free subsystems contribute to the formation of the total state jGS. Contrary to the stable quantum system case, when the measurable quantum numbers are G’s, here the measurable quantum numbers are G1 and G2 of the free quantum subsystems. Because of this, the interpretation of the state (9) requires some caution. In particular, since the quantum state (9) has a dynamical origin, a set of observables changes. Instead of ordinary Born probabilities and expectation values of a single quantum system observables, here one deals with differential cross sections of interaction, correlations between measured systems’ observables, etc. For example, the simplest observable—the probability rate w, by which detectors count the decay products, i.e., the matrix element of the unit operator E^ (‘‘intensity’’) ^ GSð1þ2Þ ¼ jBj2 ; w¼ð1þ2Þ /GjEj P where E^ ¼ E^ 1 E^ 2 and E^ i ¼ Gi jGi S/Gi j. By normalizing (9) to unit, we shall omit B from the following considerations. Then, a probability to ﬁnd a pair in the state jG1 SjG2 S is equal to pðGÞG1 G2 ¼ jAG1 G2 ;G j2 : In the case of unequivocal correspondence (correlation) between G1 , G2 and G, i.e., when G2 ¼ f ðG; G1 Þ, Eq. (9) reduces to X jGSð1þ2Þ ¼ B jG1 Sjf ðG; G1 ÞSAG1 f ðG;G1 Þ;G : ð10Þ G1
ARTICLE IN PRESS J. Ruzˇa / Physica E 42 (2010) 327–329
The structure of this state suggests, that it is formed by the pairs jG1 S and jf ðG; G1 ÞS of mutually correlated states. As to the probabilities, then, due to this unequivocal correlation, it follows that pðGÞG1 ¼ pðGÞf ðG;G1 Þ ¼ pðGÞG1 f ðG;G1 Þ ¼ jAG1 f ðG;G1 Þ;G j2 : From these equalities, one might conclude that, as soon as the subsystem 1 is found in the state jG1 S, the subsystem 2 will be with certainty found in the state jf ðG; G1 ÞS. However, this statement is not quite correct, because the quantum theory does not predict it. The point is that the determination of even the single probability pðGÞG1 or pðGÞf ðG;G1 Þ presupposes the joint measurement of both subsystems. Before the measurement of the pair is done, one cannot assert that the decay has occurred and the measured subsystems exist at all. Therefore, it would be incorrect to assume, as it has been done in Ref. , that a decay process has occurred in some moment and, before the measurement, we have two noninteracting subsystems 1 and 2, being in the common entangled state. This pictoresque interpretation (10) of the entangled state of two spatially separated quantum systems, independently of how far apart they might be, is inconsistent. The same regards the idea that the measurement of the ﬁrst subsystem determines the result of the measurement of the second one, which would lead to an impression of some ‘‘spooky action’’ at a distance or a manifestation of the non-local nature of the quantum world. This is a redundancy, which follows from the incorrect interpretation of entangled states.
3. Examples There are a lot of physical processes, which generate entangled states. One of the sources, producing two photon entangled states are cascade transitions J ¼ 0-J ¼ 1-J ¼ 0 of calcium . That is, an unstable quantum system, consisting of the calcium atom in the initial state jji mi S ¼ j00S, emits successively two photons, producing ﬁnally a system, consisting on the atom in the state jjf mf S ¼ j00SðaÞ and two photons. Let us denote the state vector of the photon by jLMlS, where L and M are the total momentum and its projection along the chosen axis, l—its ‘‘orbital’’ momentum, which determines photon’s parity ð1Þðlþ1Þ. The state vector j00Sðf Þ of the ﬁnal system, consisting of the atom and two photons, one can obtain by applying twice the formula (10), which gives j00Sðf Þ ¼ B20
X 110 CM C 1 0 1 j1M1 0Sj1M2 0Sj00SðaÞ ; 2 m 0 M1 0 m Mm
i1 j2 j where Cm is the Clebsch–Gordon coefﬁcient of the rotation 1 m1 m group and B0 the negative parity photon’s emission amplitude. This state vector contains all information about angular and polarization correlations of the photons. In particular, one can show that two photons propagate in opposite directions. As to the j00Sðf Þ polarization part, then it is equal to
1 j00SðfpÞ ¼ pﬃﬃﬃðj þ Sð1Þ j þ Sð2Þ þ j Sð1Þ j Sð2Þ Þ; 2
where j þ S and j S are the linear polarization states, labeling vertical and horizontal polarizations with respect to the given orientation. One can see that this state is formed by the pairs of mutually correlated polarization states. If both polarizers have common orientation a, i.e., when the photon detectors 1 2 are characterized by projection operators P^ P^ ¼ jaS/aj jaS/aj, the photon pairs always would be found with equal prob1 abilities Pða; aÞ -- ¼ 2 in the totally corre++ ¼ Pða; aÞ lated states. In fact, these are the only quantities, which one has to determine experimentally in order to make sure that the cascade transitions generate photons pairs in the totally correlated polarization states. If, for some reasons, the photons are measured by the detectors, characterized by projection 1 2 operators, say, P^ P^ ¼ jaS/aj jbS/bj, then one obtains the set 2 of probabilities Pða; bÞ -- ¼ cos ða; bÞ=2 ++ ¼ Pða; bÞ 2 and Pða; bÞþ ¼ Pða; bÞþ ¼ sin ða; bÞ=2. From the physical point of view, the set fPða; bÞij g is equivalent to the set fPða; aÞij g, since they are the results of the measurements of the same quantum process. Atoms and nucleus can be a source of more complicated two photon entangled states, which are produced by the cascade transitions of the general type J1 -J2 -J3 . These entangled states can be formed from various electric EðkÞ and magnetic MðkÞ photons with multipolarity k. In general, the N photon entangled states are produced in the J1 -J2 - -JNþ1 cascade transitions. A rather instructive case for the understanding of the essence of entangled state is the case of positronium decay . To be
deﬁnitive, let us consider the decay of parapositronium (1 S ). The main process, which determines the life-time of the parapositronium is the two photon annihilation 1 S -2g. However, from the theoretical point of view, it can decay to the even number of photons, i.e., 1 S -2g; 4g; . . . . One can see, that, while the measurements of all possible photons of a particular decay mode are not performed, one does not know which quantum state is subjected to the measurement.
4. Conclusion One cannot regard the entangled states as some states of fundamental importance, revealing a new feature of the quantum world. These states appear ordinarily when one constructs basis states of many-body quantum systems from the basis states of their constituent subsystems. The speciﬁc character of the entanglement or the non-separability of these states is due to the irreducible properties of the states, from which they are built. References  A.I. Ahiezer, V.B. Berestetskii, Quantum Electrodynamics, Nauka, Moscow, 1981.  A. Aspect, Physical Letters 54A (1975) 117.  J.S. Bell, Physics 1 (1964) 195.  E. Einstein, B. Podolsky, N. Rosen, Physical Review 47 (1935) 777.  S. Goudsmit, R.F. Bacher, Physical Review 46 (1934) 948.  G. Racah, Physical Review 63 (1943) 367. ¨  E. Schrodinger, Naturwissenschaften 23 (1935) 807.