Quantum Entanglement and Its Applications Eastern Oregon University
Nam Nguyen March 2013
Nam Nguyen, Quantum Entanglement and Its Applications
Introduction Quantum theory is a new born field of science. Quantum theory separates itself from classical mechanic at the realm of quantum realm of atomic and subatomic length scales. Quantum theory defines the science that we have today. It has become a very acceptable topic by most scientists even though many of its concepts cannot be prove by classical lab experiments. Since it has only been around for the past 100 plus years, there are still a lot to be study about and one of them is entanglement. Quantum entanglement is a phenomenon that occurs between photons, electrons; molecules interact with each other and then become separated but still have connection between them. In another word, quantum entanglement is basically a form of correlation between two or more parties exceeding any classical correlation in magnitude. Entanglement was first discovered in 1935 by an Austrian physicist Erwin Schrodinger but it only began to be taken a serious study by the scientific community at the end of the twentieth century. Today, the applications of quantum theory are on the verge of making its way into everyday life. Concept like quantum cryptography, quantum computer, and teleportation are already nearing practical use. These concepts are all base on entanglement. Therefore, entanglement and its properties will be intensely discussed in this paper.
History Many consider that the German physicist Max Karl Ernst Ludwig Planck (Max Planck) is the father of quantum theory. He is the first one to discover that each energy element (E) is proportional to its frequency. So the foundation of quantum theory was being set up around the mid-twenty century with contributions from many great scientists (Niels Bohr, Werner Heisenberg, Albert Einstein, Arthur Compton, John von Neumann…etc.). Another German physicist Albert Einstein also motivated the science community to start to study this modern science field by published his three famous papers (Photoelectric effect, Brownian motion, Special relativity). There were many people in the science community at the time reject the idea of quantum mechanics and that include one of its founder Albert Einstein. At this time, classical mechanic is a very acceptance field since it can predict everything so accurately and it also can be proven by laboratory experiment. Contrast to classical mechanic, quantum theory does not assign definite values. Instead, it makes a prediction using a probability distribution. As we all know, Albert Einstein believes that God does not play dice with us. He does not believe in statistical-mechanic which using the approach of probability and statistic. So Albert Einstein concludes to himself that quantum mechanics is an incomplete field and there’s a hidden variable somewhere that we haven’t yet found. He stand firm to this believe until his death in 1955. Ironically, quantum entanglement was first study intensify by Albert Einstein himself and his other two colleagues Boris Podolsky, and Nathan Rosen. The three of them published a paper in 1935 known as the EPR paradox which shocked the science community. The EPR paradox is the first published paper that mention about quantum entanglement. Several months after the publication of the EPR-paradox Erwin Schrodinger assigned this physical phenomenon the name “entanglement”.
Nam Nguyen, Quantum Entanglement and Its Applications
Entanglement 1. Non-locality We would often like to think that separated objects are independent of one another. Nonlocality also known as action at a distance is a property of interaction between two objects or more that are separated in space without the intermediate mechanism. Each particle is described by its own position and momentum. Either position or momentum can’t be measure exactly because of the Heisenberg’s uncertainty principle but because they have interacted with each other’s, in the perspective of quantum theory, they have effectively merged to become one interconnected system. Thus, measure the momentum of one of the particle we can immediately deduce the momentum of the other particle without having to do any measurement on it at all. Alternatively, if we choose to measure the position of one of the particle then the position of the other particle will be known immediately. This is true whether the particles are next to each other or in an infinite distance away from one another. Hence, this show that the information transfer must be much faster than the speed of light and this is the core argument in the EPR-paradox. The fact is that entanglement does not have an actual speed since it not traveling through any sort of medium like light does. This phenomenon quantum connection is not meditated by any fields of force. It does not weaken as the particles move apart, because it does not have to travel through a stretch space. So since it’s operating outside of space then it also must operate outside of time, thus if something happen to one then it will effects the other one instantaneously. 2. EPR-Paradox and Bell’s Theorem EPR-paradox also known as the Einstein, Podolosky, Rosen paradox argue that quantum theory is an incomplete theory using entanglement. This is the first time entanglement was being used in the science world. It considered two entangle particles, commonly referred to as Alice and Bob. They pointed out that measuring the quantity of particle Alice will cause the conjugated quantity Bob to become undetermined, even if there was no contact. According to the EPR-paradox there were two explanations. It’s either there were interactions between the two particles even though they are separated or the information about the outcome of all possible measurements were already present in both particles. Einstein, Podolsky and Rosen pointed out that the first argument is not possible since it’s contradicting the theory of relativity because for information to travel faster than the speed of light is absurd. The second argument is encoded with some hidden parameters but since quantum theory does not contain any space for such hidden parameters in its formalism. From there the group concludes that quantum theory is an incomplete theory. In 1960, an Irish physicist known as John Stewart Bell came up with a famous theorem called the “Bell’s Theorem” which shows that EPR-paradox do not hold. His approach was he created a mathematical formulation of EPR specifying locality and hidden variable conditions. He then argues that if EPR is correct that quantum theory is incomplete, then the probability marker of
Nam Nguyen, Quantum Entanglement and Its Applications EPR should equal to the probability marker of quantum theory without hidden variables. Thus if EPR was correct then ( )
( )
Bell shows that they do not equal thus that imply EPR failed.
3. Von Neumann entropy Von Neumann entropy is the entropy of a quantum state discovered by a Hungarian mathematician considered to be the best mathematician to live in his time named John von Neumann. It uses the fact that quantum theory is a probabilistic so the quantum state can be representing as a density matrix. He came up with the formula (
) ( )
( ∑
|
)
∑
|(
Thus this gives
∑
) ∑
( )
Note that the last form of the Neumann’s entropy can be related to the Shannon entropy which is the entropy measure of the uncertainty in random variables. Neumann’s entropy will be one of the main factors to distinguish if the pure state is entangled or not. This concept will be present in a more detail fashion later on in this paper.
4. Tensor Products Tensor product will be extensively use in entanglement; therefore I will give a brief introduction on it. There are many different types of tensor products and for the purpose of this paper we will only worry about the tensor product of vector space. Tensor product commonly denoted as If we have two vectors spaces V and W in a field then the tensor product will be denote as ⨂ Recall a field is a Ring with an extra property of multiplicative inverse for any nonzero element. For instance, ⁄ hence, the tensor product of a Hilbert spaces is a way to extent the construction of the tensor product of two Hilbert Space into another Hilbert Space. Ex)
Nam Nguyen, Quantum Entanglement and Its Applications ⨂ ⨂ Note that this is similar to inner-product. The reason is because Hilbert Space have innerproduct.
5. Quantum States Quantum state is also known as the state of a quantum system. It’s given as the state vector in a vector space. The wave-function of the Schrodinger’s equation is a state vector. |
I.
(|
|
)
Pure States
Any given quantum system is identified in a Hilbert space with finite or infinite dimension. Obviously we can define a pure state as not a mixed state. In another word, in Dirac notation we can represent a pure state of electrons, neutron or photons as| | . In the Stern-Gerlach experiment if you could separate the atoms into only the "spin up" part then that would be in a pure state, not a mixture of spin up / spin down. Pure quantum states are described by the finite dimensional complex vectors | The pure state corresponds to vectors of norm 1. (Notice: this is not the same as the magnitude in a Euclidian space since we are working in Hilbert space). The unit sphere in the Hilbert space is the set of all pure states. Thus the pure state cannot be represented as a mixture of other states. |
II.
√
(|
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Mixed states
A mixed state can be think of as it’s not a truly pure state. For instance, for an electron or photon system it can possess both spin up and spin down. Therefore a mixed state can be represent as (| | ) √ A mixed state is a statistical ensemble of pure states. Therefore a mixed state cannot be described as a ket-vector. It must described by its associated density matrix or density operator. The von Neumann’s entropy for a mixed state will always be positive whereas for the pure state it’s always equal to zero. In a mixed state ( ) |
6. Defining Entanglement in Mathematical Terms In a pure states, it consider as separable if and only if it can be written as a direct product,
Nam Nguyen, Quantum Entanglement and Its Applications
|
|
|
(Recall separable states are states without quantum entanglement.) Else it will be consider as entangled and has the form with at least two non-vanishing complex coefficient . |
∑
|
|
Recall that a pure state of the composite system is any vector of norm 1. Then if quantum mechanical space with basis state {| } {| } | } ⨂ {| Thus, |
∑
|
|
∑
be
|
However, a mixed state of a composite system is commonly represented by the density matrix. So let Then is separable if { } { } are mixed states of the respective and only if there exist subsystem. Thus ∑ Note that
hence ∑
If the above condition is not met, then we said that the mixed state of the composite system represented by density matrix is in entangled state. Now there can be more than two subsystems but because of simplification, I only deal with two subsystems above. Of course we can have and have state space but the idea will remain the same for both pure states and mixed states. We should also note that separable states can also be written as a superposition of several product states, whereas entangled states cannot be written as direct products.
Detecting Entanglement
Nam Nguyen, Quantum Entanglement and Its Applications As mentioned in the previous section that a quantum state is entangle if it cannot be decompose in any of the convex sums of product states. Recall that pure states are different that mixed states; well generally they are simpler. Thus, detecting entanglement in pure states is actually easier than in the mixed states. In the following subsections I will show how we can detect entanglement in both pure states and mixes states.
1) Detection in the pure states It’s pretty simple to detect if a pure state is separable or entangled. It turned out that any pure state | is entangled if and only if ( ) ( ( ) ) | | This can be understood by the understanding that the entropy of a pure state always vanished and entanglement can be considered as information about a composite state that doesn’t apply any of the singlet parties but only to the composite system.
2) Detection in the Mixed states In the mixed states the same idea is apply but the mathematic formalism will be different. The tasks are much more complicated than the pure states. One of the criteria to determine separability is the entanglement witness theorem. Since the set of all separable state S is a convex one from the definition of separable density matrices. Hence, it can be enclosed within hyper-planes separating it from the outside. Thus, for every entangled state there is at least one such hyper-plane separating it from all separable states. This is can visualize from the sketch above, for S is the separable states and P is the entangled state.
P
S Therefore if a quantum system has a state space a trace-class positive operator on the state which has trace 1 is the mixed state we can consider the affine subspace as a function( ), this function ( ) is what called the entangled witness. The function then can be identifying with a Hermitian operator W. Thus, for every entangled state : (
)
(
)
Nam Nguyen, Quantum Entanglement and Its Applications (
|
)
Although this method is capable of detecting all entangled states but it is very challenging to find the appropriate witness operator for the given state. Another criterion to detect entanglement in the mixed state is the PPT (Positive Partial Transpose) criterion also known as the Peres-Horodecki criterion. The PPT criterion is the necessary condition for the joint density matrix ( ) of the two quantum systems to be separable. Hence if we have a state act on |
∑
(
Now if
)
∑
|
|
|
(|
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|)
|
∑
|
|
|
|
is separable then it can be written as: ∑
Thus this immediately follows that if
is separable then: ∑
This implies that is separable then when it has negative eigenvalues.
has non-negative eigenvalues. Thus,
is entangled
Applications of Entanglement Quantum information technology is on its way to make impact on everyday life. There are many uses and applications can be made from entanglement phenomena. Some of the many applications of entanglement will be discuss in brief detail in the following subsections.
1. Quantum Cryptography Cryptography has been around for over thousands of years. It was first used by the Spartans to keep their secret plans from the enemies. Since the beginning of the 20th century, particularly during World War 2; Cryptography took an extensive use of mathematics, information theory, abstract algebra…etc. Since computational power has grown so much and quantum computer is
Nam Nguyen, Quantum Entanglement and Its Applications on the verge of becoming real in the today society, we must use a better way to keep our information secures. As we all know that most banking systems uses RSA algorithm to secure our information but this is becoming more and more breakable with the new improvement has made in mathematics. Once quantum computer become commercial available, people can run the Shor’s algorithm and decode any messages that uses RSA algorithm. Thus, the need for a secure communication is much needed and this is how quantum cryptography started. Quantum cryptography depends on the structures of quantum theory and its phenomena’s. The most developed quantum cryptography method is known as QKD (quantum key distribution). Quantum key distribution is a technique that allows for secure distribution of keys to be used for encrypting and decrypting messages. The original QKD did not used entangled particles but in 1991, Artur Ekert proposed the idea of quantum key distribution using entanglement. Since then when ones mention about QKD, it usually refer to QKD with entanglement. In this method we can construct the S value used in the CHSH (Clauser-Horne-Shimony-Holt) inequality. By finding the value of S, we can determine if there were any eavesdroppers. Thus, the application of quantum entanglement to quantum key distribution is a topic of great relevance in modern quantum cryptography research.
2. Quantum computer Even though it is not commercial available but the idea of quantum computer is very real. Several quantum computer has been made and being tested. Quantum computer is a computer that makes uses of quantum phenomena such as superposition and entanglement. Classical computer recorded data by binary digits (bits), whereas quantum computer uses qubits. A single bit can represent as 0 or 1. Whereas a single qubit can be represent as 0, 1 or any of the superposition of these two states. A quantum computer of n-qubits can be superposition up to states. Quantum computers will be able to solve problems much faster than classical computers since it can process many things at the same time. A classical computer with enough resources can also simulate any quantum algorithm (Shor’s algorithm…etc.) but the problem is it will not be able to store that many values. A classical computer would require to stored complex values to be equivalent to a 500 qubits quantum computer.
3. Teleportation Teleportation involves in dematerializing an object at one point, and sending the details of that object’s to another location within atomic configuration precision where it will be reconstructed. This also means that space and time will be eliminating from travel. Thus this implies that we could be transported to any location instantly, without crossing a physical distance. This may sound fictional but it has become a reality with quantum entanglement. In 1998, physicist at Caltech long with some other scientist has successfully teleporting a photon. They were able to reconstruct the photon at another location and as predicted the original photon no longer existed. It’s the Uncertainty Principle is the barrier for teleportation because if you don’t know the
Nam Nguyen, Quantum Entanglement and Its Applications position of the particle then you can’t teleport it. In order to teleport the photon the group at Caltech used the idea of entanglement to get around the Uncertainty Principle. There were 3 photons needed to do this experiment. Photon A will be the teleported photon, photon B is the transporting photon and photon C is the photon that is entangled with photon B. Without entanglement, if we tried to observe the photon A closely we would bump it, and therefore change it. Since photon B and C are entangled, we can extract some information on photon A. The remaining information would be passed on to B and then on to C by entanglement. Thus, when apply information on photon A to photon C, we can create a replica of photon A but the original will be destroyed. Hence, in the process of teleporting the original system must be destroyed. In 2002, a group of researchers at the Australian National University successfully teleported a laser beam. In 2006, Dr. Eugene Polzik and his research groups were able to teleported information stored in a laser beam into a cloud of atoms at the Niel Bohr Institute. Even though teleportation has become reality but the idea of teleporting a human is still far from being done. A human body is made up more than atoms. Hence to teleport a human body, one must built a machine that can pinpoint and analyze all these atoms. Then this machine would then have to send this information to another location, where the body would be reconstructed with exact precision. A millimeter of miscalculation can lead to a severe neurological or physiological defect.
Nam Nguyen, Quantum Entanglement and Its Applications
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