Two-Photon Entanglement and Quantum Reality

Yanhua Shih    Department of Physics, University of Maryland at Baltimore County, Baltimore, Maryland

I Introduction

One of the most surprising consequences of quantum mechanics has been the entanglement of two or more distant particles. The two-particle entangled state was mathematically formulated by Schrodinger (1935). Consider a pure state for a system composed of two spatially separated subsystems,

^=Ψ〉〈Ψ,Ψ〉=∑a,bcaba〉b〉

where {|a} and {|b} are two sets of orthogonal vectors for subsystems 1 and 2, respectively, and ^ is the density matrix. If c(a, b) does not factor into a product of the form f(a) × g(b), then it follows that the state does not factor into a product state for subsystems 1 and 2:

^≠ρ^1⊗ρ^2

The state was defined by Schrödinger as the entangled state.

The first classic example of a two-particle entangled state was suggested by Einstein, Podolsky, and Rosen (1935) in their famous gedankenexperiment:

〉=∑a,bδa+b−c0a〉b〉

where a and b are the momentum or the position of particles 1 and 2, respectively, and c0 is a constant. A surprising feature of the EPR state is the following: the value of an observable (momentum or position) for neither single subsystem is determinate. However, if one of the subsystems is measured to be at a certain value for that observable, the other one is 100% determined. This point can be easily seen from the delta function in Eq. (2).

A simple yet fundamental question naturally followed, as EPR asked 60 years ago: “Does a single particle have definite momentum in the state of Eq. (2) in the course of its travel, regardless of whether we measure it or not?” Quantum mechanics answers “No!” The memorable quote from Wheeler (1983) “No elementary quantum phenomenon is a phenomenon until it is a recorded phenomenon” summarizes what Copenhagen has been trying to tell us. By 1927, most physicists accepted the Copenhagen interpretation as the standard view of quantum formalism. Einstein, however, refused to compromise. As Pais (1982) recalled vividly: around 1950 during a walk, Einstein suddenly stopped and “asked me if I really believed that the moon exists only if I look at it."

Einstein, Podolsky, and Rosen published their famous paper in 1935: “Can quantum-mechanical description of physical reality be considered complete?” In this paper EPR suggested the classic EPR state, Eq. (2), for a gedankenexperiment, and then give their criteria:

Locality

There is no action-at-a-distance;

Reality

"If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this quantity.” According to EPR, because we can predict with certainty the outcome result of measuring the momentum of particle 1 by measuring the momentum of particle 2, and the measurement of particle 2 cannot cause any disturbance to particle 1, if the measurements are space-like separated events, the momentum of particle 1 must be an element of physical reality. A similar argument shows that the position of particle 1 must be physical reality too. However, this is not allowed by quantum mechanics. Now consider the following.

Completeness

"Every element of the physical reality must have a counterpart in the complete theory.” This leads to the question: “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?"

The state of the signal-idler photon pair of the spontaneous parametric down conversion (SPDC) is a typical entangled EPR state. SPDC is a nonlinear optical process from which a pair of signal-idler photon is generated when a pump laser beam is incident onto an optical nonlinear crystal. Quantum mechanically, the state can be calculated by first-order perturbation theory, see for example, Appendix A, note 1.

=∑s,iΔΩs+Ωi−ΩpΔks+ki−kpas†Ωksai†Ωki0

where ωj, kj (j = s, i, p) are the frequency and wavevectors of the signal (s), idler (i), and pump (p) respectively; ωp and kp can be considered as constants; usually a single mode laser is used for pump; and s† and i† are creation operators for signal and idler photons, respectively. Equation (3) tells us that there are two eigen-modes excited together. The signal or idler photon could be in any modes of its superposition (uncertain); however, if one is known to be in a certain mode the other one is determined with certainty.

(1) Do we have such a state? (2) How special is it physically? In this chapter, we review a “ghost image” experiment (Pittman et al., 1995) and a “ghost interference” experiment (Strekalov et al., 1995) in Section II to answer these questions and to show the striking EPR phenomenon.

Another example of an entangled two-particle system suggested by Bohm (1951) is a singlet state of two spin 1/2 particles:

=12+1−2−−1+2

where the | +  and | −  represent states of spin up and down, respectively, along an arbitrary direction ^. Again for this state, the spin for neither particle is determined; however, if one particle is measured to be spin up along a certain direction, the other one must be spin down along that direction.

Does a single particle in the Bohm state have a definite spin in the course of its travel, regardless of whether we measure it or not? No! The spin for neither particle is defined in Eq. (4). It does not make sense to EPR in the first place: According to EPR, because we can predict with certainty the outcome result of measuring any components of the spin of particle 1 by measuring some component of the spin of particle 2, and the measurement of particle 2 cannot cause any disturbance to particle 1, if the measurements are space-like separated events, the chosen spin component of particle 1 must be an element of physical reality. Following this argument, all of the components of spin of particle 1 must be physical realities associated with it. However, as is well known, this is not allowed by quantum mechanics.

Is it possible to have a “better” theory, which provides correct predictions like quantum mechanics and at the same time respects its description of physical reality by EPR as “complete"? It was Bohm who first attempted a version of a so-called “hidden variable theory,” which seemed to satisfy these requirements (Bohm, 1952a, b, 1957). The hidden variable theory was successfully applied to many different quantum phenomena until 1964, when Bell proofed a theorem to show that an inequality that is violated by certain quantum mechanical statistical predictions can be used to distinguish local hidden variable theory from quantum mechanics (Bell, 1964 and 1987). Since then, the testing of Bell’s inequalities has become a key instrument for the study of...