“When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction, the two representatives (or ψ-functions) have become entangled.” —Erwin Schrödinger (1935) 52 Los Alamos Science Number 27 2002 Quantum State Entanglement Creation, characterization, and application Daniel F. V. James and Paul G. Kwiat ntanglement, a strong and tious technological goal of practical two photons is denoted |HH〉,where inherently nonclassical quantum computation. the first letter refers to Alice’s pho- Ecorrelation between two or In this article, we will describe ton and the second to Bob’s. more distinct physical systems, was what entanglement is, how we have Alice and Bob want to measure described by Erwin Schrödinger, created entangled quantum states of the polarization state of their a pioneer of quantum theory, as photon pairs, how entanglement can respective photons. To do so, each “the characteristic trait of quantum be measured, and some of its appli- uses a rotatable, linear polarizer, a mechanics.” For many years, entan- cations to quantum technologies. device that has an intrinsic trans- gled states were relegated to being mission axis for photons. For a the subject of philosophical argu- given angle φ between the photon’s ments or were used only in experi- Classical Correlation and polarization vector and the polariz- ments aimed at investigating the Quantum State Entanglement er’s transmission axis, the photon fundamental foundations of physics. will be transmitted with a probabili- In the past decade, however, entan- To describe the concept of ty equal to cos2φ. (See the box gled states have become a central quantum entanglement, we are first “Photons, Polarizers, and resource in the emerging field of going to describe what it is not! Projection” on page 76.) Formally, quantum information science, which Let us imagine the simple experi- the polarizer acts like a quantum- can be roughly defined as the appli- ment illustrated in Figure 1. In that mechanical projection operator Pφ cation of quantum physics phenom- experiment, a source S1 continually selecting out the component of the ena to the storage, communication, emits pairs of photons in two direc- photon wave function that lines up and processing of information. tions. As seen in the figure, one with the transmission axis. We say The direct application of entan- photon goes toward an observer that the polarizer “collapses” the gled states to quantum-based tech- named Alice, while the other goes photon wave function to a definite nologies, such as quantum state tele- toward Bob. state of polarization. If, for exam- portation or quantum cryptography, First, imagine that the photons ple, the polarizer is set to an angle θ is being investigated at Los Alamos emitted by S1 are always polarized with respect to the horizontal, then National Laboratory, as well as other in the horizontal direction. a horizontally polarized photon is institutions in the United States and Mathematically, we say that each either projected into the state |θ〉 abroad. These new technologies photon is in the pure state denoted with probability cos2θ or absorbed offer exciting prospects for commer- by the ket |H〉, that is, the “represen- with probability 1 – cos2θ = sin2θ. cial applications and may have tative” of the state Schrödinger The bizarre aspect of quantum important national-security implica- referred to in the quotation on the mechanics is that the projection tions. Furthermore, entanglement is opposite page. Because the photons process is probabilistic.The fate of a sine qua non for the more ambi- are paired, the combined state of the any given photon is completely This article is dedicated to the memory of Professor Leonard Mandel, one of the pioneers of experimental quantum optics, whose profound scientific insights and gentlemanly bearing will be sorely missed. Number 27 2002 Los Alamos Science 53 Quantum State Entanglement Photon source Polarized photons Bob Alice Rotatable polarizer Detector Figure 1. A Simple Two-Photon Correlation Experiment In this experiment, a source emits pairs of photons: One photon is going to Alice and the other to Bob. Each photon passes through a linear polarizer on its way to its respective detector. Both Alice and Bob’s polarizers are rotatable and can be aligned to any angle with respect to the horizontal, but Bob’s is always kept parallel to Alice’s. For a given polarizer setting, Alice and Bob record those instances when they have the same results, that is, when both detect photons or when they don’t. The figure shows the source emitting two horizontal photons in the state |Ψ〉 = |HH〉. The experiment can be performed with other sources to examine differences between other two-photon states. (Picture of Bob is courtesy of Hope Enterprises, Inc.) unknown. Furthermore, any informa- same result, that is, they can examine photon sent to Bob has a 50 percent tion about the photon’s previous the photon–photon correlations. chance of passing through his. polarization state is lost. Suppose Alice has her polarizer The probability is therefore 25 percent Getting back to the experiment, oriented to transmit horizontally that both Alice and Bob detect a we assume that Alice and Bob’s polar- polarized photons. In that case, each photon, 25 percent that neither detects izers are always aligned in the same photon coming to her from S1 will be a photon, and thus 50 percent that they way: When Alice sets her polarizer to transmitted, and her detector will obtain the same result. a certain angle, she communicates her “click,” indicating a photon has The correlation function G is choice to Bob, who uses the same arrived. Subsequent communication shown in Figure 2(a′). It is equal setting. Behind each polarizer is a with Bob would reveal that he also to the product of the independent detector. In our experiment, Alice and detected each photon, so at this probabilities for detecting a photon 2θ) × 2θ Bob rotate their polarizers to a certain polarizer setting, there is a perfect [(cos A (cos )B], plus the prod- angle with respect to the horizontal correlation between Alice’s detection uct of the probabilities for not detecting 2θ) × 2θ and record whether they detect a of a photon and Bob’s. Similarly, one [(sin A (sin )B], where sub- photon. Then, they repeat the proce- by rotating the polarizer to the vertical scripts A and B are for Alice and Bob, dure for different polarizer settings. position, the two would again discover respectively. Thus, Alice and Bob If Alice looks only at her own data a perfect correlation, namely, neither deduce that the two photons are (or Bob looks only at his), she can party would detect his or her photons. independent of each other and the determine the polarization state of the The correlation changes when Alice wave function is in fact separable: photons emitted by the source—see and Bob have their polarizers oriented, |HH〉 = |H〉|H〉. In other words, the Figure 2(a). But Alice and Bob can say, at +45° to the horizontal. In that correlation is entirely consistent with also make a photon-per-photon case, the photon sent to Alice has a classical probability theory. The pho- comparison of their data and deter- 50 percent chance of passing through tons are classically correlated. mine the probability that they have the her polarizer, and independently, the Now, consider performing the 54 Los Alamos Science Number 27 2002 Quantum State Entanglement Alice’s Polarization Measurements Correlation between Alice and Bob’s (a) S1 emits photons in the pure state |HH〉. Alice measures a cos2θ Measurements function for her polarization data (a) (a′) and deduces that photons coming 1.0 1.0 to her are horizontally polarized. (A different linear polarization 0.8 0.8 would shift the curve to the left or A 2 G p = cos θ ′ + right.) (a ) We define the correla- 0.6 0.6 tion function G as the probability θ that both Alice and Bob detect a 0.4 0.4 4 4 Correlation, θθ photon, plus the probability that G = cos + sin neither detects a photon. For this 0.2 0.2 source, G is completely consis- detection probability Alice’s 0 tent with classical probability 45 90 135 1800 45 90 135 180 theory for independent events; Polarizer setting θθ (°) Polarizer setting (°) that is, the correlation function is the product of the detection 2θ. Probability that Alice (or Bob) detects a photon: p+ = cos probability of each photon in 2θ. Probability that Alice (or Bob) does not detect a photon: p– = sin the pair. A × B A × B 4θ 4θ. For independent photons: G = GHH = p+ p+ + p– p– = cos + sin (b) The source S2 emits photons (b) (b′) in the partially mixed state 1.0 1/2(|HH〉〈HH| + |VV〉〈VV|). Photons 1.0 from this source do not have a 0.8 0.8 net polarization. Alice receives at G p A = .5 random either an |H〉 or a |V〉 pho- + 0.6 0.6 ton, so the sum of her measure- ments averages to a 50 percent 0.4 0.4 4 4 detection probability independent Correlation, G = cos θθ + sin of angle. (b′) The correlation func- 0.2 0.2 tion G,however, is the same as detection probability Alice’s in (a), revealing that the photons 0 45 90 135 180 0 45 90 135 180 in each pair are independent of θθ Polarizer setting (°) each other and have polarization Polarizer setting (°) H or V.Therefore, the two photons For this mixed state, exhibit the same classical corre- G = 1/2(G + G ) = G .