# Quantum entanglement

Of entanglement is called in quantum physics when a composite physical System z. B. a system with several particles, considered as a whole, assumes a well-defined state without being able to assign each of the sub-systems its own well-defined state.

This phenomenon cannot exist in the field of classical physics. There, composite systems are always separable , i.e. H. Each subsystem has a certain state at all times, which determines its respective behavior, whereby the totality of the states of the individual subsystems and their interaction fully explain the behavior of the entire system. In a quantum-physically entangled state of the system, on the other hand, the subsystems occupy several of their possible states next to one another, with each of these states of a subsystem being assigned a different state of the other subsystems. In order to be able to correctly explain the behavior of the overall system, one must consider all these possibilities that exist side by side together. Nevertheless, when a measurement is carried out on it, each subsystem always shows only one of these possibilities, the probability that this result will occur is determined by a probability distribution . Measurement results on several interlaced subsystems are correlated with one another , i. H. Depending on the measurement result from one subsystem, there is a changed probability distribution for the possible measurement results on the other subsystems. These correlations generated by quantum entanglement are also referred to as quantum correlations .

## overview

Entangled states are common. An entangled state arises every time two subsystems interact with each other (e.g. collide with each other), and there are then different, but coordinated possibilities of how they continue to behave (e.g. in which direction they continue to fly after the collision ). According to quantum mechanics, all these possibilities have a certain probability with which they must be represented in the state of the overall system in a correspondingly coordinated manner up to the moment of the quantum mechanical measurement .

The entanglement is ended as soon as one of the subsystems is set to one of its states. Then another sub-system, which was linked to the first sub-system through the entanglement, immediately changes to the state that was assigned to the state of the first sub-system determined by the observation. The state of the overall system then no longer shows any entanglement, because both sub-systems considered individually are now in their own specific state.

Another example besides the state after a collision process is the ground state of the hydrogen atom , in which the spins of electron and proton add to the atomic spin zero. The participating states of the two particles are those in which they have aligned their spin parallel or anti-parallel to the z – direction. In the ground state of the atom one finds both states for the electron as for the proton with equal probability. If one sets the spin of the electron to one of these possibilities by measuring in the magnetic field, e.g. B. in the (+ z) direction, then the spin of the proton definitely receives a well-determined state - namely the one in the (-z) direction, which can be confirmed by a subsequent measurement on the proton. The state of the atom is then a different, non-entangled state, which in turn can be represented as a superposition of the two entangled states with atomic spin zero and one, each with the same amplitude.

This means that if you have an entangled system in a given state and determine their state through simultaneous measurements on several subsystems, then the measurement results for each individual subsystem are not fixed, but are correlated. The indeterminacy of the states of the entangled subsystems before the observation, together with these correlations between the related observation results, represents one of the greatest problems for understanding quantum physics. Albert Einstein , who in 1935 was the first to make this clear theoretically in a thought experiment (see EPR- Paradoxon ), concluded from this that quantum mechanics could not yet give an accurate picture of physical reality, because of a - so literally - "spooky action at a distance" with which the measurement on one subsystem could influence the result of the measurement on the other around the correlations he did not want to believe.

The correlations caused by entanglement have now been proven by many real experiments. They are independent of how far apart the locations at which the measurements are made on the subsystems are and the time interval between the measurements. This also applies if the measurements are so far apart and are carried out so quickly one after the other (or even at the same time) that the measurement result on one particle cannot have influenced the state of the other in any physical way. In certain experiments, the correlations are so strong that they cannot in principle be explained by any theory which, like classical physics, is based on the physical principle of "local realism"; This means that every subsystem always has a well-defined state that another spatially distant subsystem can only act on at the speed of light. According to Bell's inequality, this also excludes the possibility that such a locally realistic theory could describe the phenomenon of quantum correlation with hypothetical additional hidden variables .

The fact that entanglement (in contrast to classical physics) does not allow a locally realistic interpretation means that either the locality has to be given up (e.g. if the non-local wave function itself is granted a real character - this happens especially in collapse theories in which Many Worlds Interpretation or the De Broglie Bohm Theory ) or the concept of a microscopic reality - or both. This departure from classical realism is represented most radically in the Copenhagen interpretation ; According to this interpretation, which has been the standard for physicists for decades, quantum mechanics is neither real (since a measurement does not determine a state as it existed before the measurement, but rather prepares the state that exists after the measurement) nor local (because the state determines the probability amplitudes for all locations in space at the same time, for example using the wave function ). ${\ displaystyle | \ psi \ rangle}$ ${\ displaystyle \ psi (x, y, z)}$ ## history

Entanglement and its consequences are among those consequences of quantum mechanics that are particularly clearly at odds with classical (everyday) understanding, and have thus provoked most of the resistance to this theory as a whole. Albert Einstein , Boris Podolsky and Nathan Rosen formulated the EPR effect in 1935 , according to which the quantum entanglement would lead to a violation of the classic principle of local realism, which Einstein called "spooky action at a distance" in a famous quote ) was designated. However, the predictions of quantum mechanics could be proven very successfully through experiments.

Many scientists mistakenly attributed this to still unknown, deterministic "hidden variables" which are both subject to local realism and which could explain all quantum phenomena. But in 1964 John Stewart Bell showed theoretically that this question can be decided experimentally. According to Bell's inequality , the correlations through quantum entanglement can be stronger than can be explained with any local-realistic theory with hidden variables. This was confirmed by experiments, so that quantum entanglement is now recognized as a physical phenomenon (apart from a few deviants). Bell also created the illustration of entanglement and the EPR effect based on a comparison with “ Bertlmann's socks ”.

In 2008, Nicolas Gisin's group set a lower limit for the speed of an assumed "spooky action at a distance" in an experiment: According to this, two photons that were entangled in terms of polarization would have to communicate at least 10,000 times the speed of light, if they were that Would send the measurement result of the polarization on one photon to the other. Such communication would blatantly contradict the theory of relativity and mean, among other things, that time loops are possible.

## No information transmission faster than light

The entanglement correlations do not violate the theory of relativity . It is true that the interpretation always suggests that the correlations could only come about through an interaction of the entangled subsystems that is faster than light. However, it is not about an interaction, because no information can be transmitted here. The causality is therefore not violated. There are the following reasons for this:

• Quantum mechanical measurements are probabilistic , that is, not strictly causal.
• The no-cloning theorem prohibits the statistical checking of entangled quantum states without changing them.
• The no-communication theorem states that measurements on a quantum mechanical subsystem cannot be used to transfer information to another subsystem.

Information transfer through entanglement alone is not possible, but with several entangled systems in connection with a classic information channel, see quantum teleportation . Despite this name, no information can be transmitted faster than light because of the classic information channel required.

## Biologically entangled systems

Graham Fleming, Mohan Sarovar and others (Berkeley) believed that they used femtosecond spectroscopy to demonstrate that in the photosystem - the light-collecting complex of the plants, a stable entanglement of photons takes place over the entire complex, which makes the efficient use of light energy without heat loss possible . Among other things, the temperature stability of the phenomenon is remarkable. Sandu Popescu , Hans J. Briegel and Markus Tiersch expressed criticism .

Stuart Hameroff and Roger Penrose propose to explain the astonishing performance of the brain that it is based, among other things, on correlations and entanglements between electronic states of the microtubules that are common in neurons . This was contradicted with physical justification.

## Generation of entangled photons

In the case of photons, the entanglement mostly relates to the polarization . If the polarization of one photon is measured, the polarization of the other photon is fixed (e.g. rotated by 90 ° in the case of linear polarization). However, they can also be entangled with regard to the direction of flight.

The two gamma quanta of the annihilation radiation form an entangled pair of photons. The entanglement concerns both the flight directions, which can be individually arbitrary, but together (in the center of gravity system) are exactly opposite to each other, as well as the circular polarization - equally often for each of the photons right and left, but with both photons always both right or both left. Directional entanglement is the basis of the widespread medical application in positron emission tomography (PET).

Entangled low- energy photons can be generated in non-linear optical crystals by parametric fluorescence (parametric down-conversion) . An entangled pair of photons with half the energy is generated from a photon of higher energy in the crystal. The directions in which these two photons are radiated are strongly correlated with each other and with the direction of the incident photon, so that the entangled photons generated in this way can be used for experiments (and other applications) (see e.g. quantum erasers ).

Certain types of atoms can be excited with the help of a laser in such a way that when they return to their ground state they also emit a pair of polarization-entangled photons. However, these are emitted almost uncorrelated in any spatial direction, so that they cannot be used very efficiently.

## Applications

• With every quantum mechanical measurement , the measurement object is entangled with the measuring apparatus in order to be able to read the state of the measurement object from its "pointer position".
• In the quantum eraser and delayed choice experiment , one of the subsystems is put into its own respective state by one of two complementary observations, which means that the other subsystem is also put into the respective associated state, so that it appears to be without direct influence depending on Choice shows complementary properties.
• Quantum key exchange: Secure exchange of keys between two communication partners for the encrypted transmission of information. The exchange is secure because it is not possible to eavesdrop on it without noticeable interference. The exchanging partners can therefore notice any "listening in" during the key exchange. While the usual quantum key exchange is also possible without entanglement (e.g. with the BB84 protocol ), the use of entangled states allows a secure quantum key exchange even if one does not trust the devices used (one speaks of device-independent or device-independent security).
• Quantum computer : The entanglement of the qubits plays a central role in calculations using qubits on a quantum computer. On the one hand, the main advantage of quantum computers (that some problems can be solved by quantum algorithms with far fewer calculation steps than on conventional computers ) is based on the entanglement of many qubits in the course of the calculation. On the other hand, the methods for quantum error correction , which are necessary to protect the quantum calculations from decoherence , also use entangled states.
• In quantum metrology , entangled states of many particles are used in order to increase the measurement accuracy possible with limited resources (number of particles used).

## Mathematical consideration

The following discussion assumes knowledge of Bra-Ket notation and the general mathematical formulation of quantum mechanics.

Let there be two systems and with the Hilbert spaces and . The Hilbert space of the composite system is the tensor product space . Let the system be in the pure state and the system in the pure state . Then the state of the composite system is also pure and given by: ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle {\ mathcal {H}} _ {\ rm {A}}}$ ${\ displaystyle {\ mathcal {H}} _ {\ rm {B}}}$ ${\ displaystyle {\ mathcal {H}} _ {\ rm {A}} \ otimes {\ mathcal {H}} _ {\ rm {B}}}$ ${\ displaystyle A}$ ${\ displaystyle | \ psi \ rangle _ {\ rm {A}}}$ ${\ displaystyle B}$ ${\ displaystyle | \ phi \ rangle _ {\ rm {B}}}$ ${\ displaystyle | \ psi \ rangle _ {\ rm {A}} \; | \ phi \ rangle _ {\ rm {B}}}$ Pure states that can be written in this form are called separable or product states .

If one chooses orthonormal bases and the Hilbert spaces and , then one can expand the states according to these bases and obtain with complex coefficients and : ${\ displaystyle \ {| i \ rangle _ {\ rm {A}} \}}$ ${\ displaystyle \ {| j \ rangle _ {\ rm {B}} \}}$ ${\ displaystyle {\ mathcal {H}} _ {\ rm {A}}}$ ${\ displaystyle {\ mathcal {H}} _ {\ rm {B}}}$ ${\ displaystyle a_ {i}}$ ${\ displaystyle b_ {j}}$ ${\ displaystyle | \ psi \ rangle _ {\ rm {A}} \; | \ phi \ rangle _ {\ rm {B}} = \ left (\ sum _ {i} a_ {i} | i \ rangle _ {\ rm {A}} \ right) \ left (\ sum _ {j} b_ {j} | j \ rangle _ {\ rm {B}} \ right)}$ A general state on has the form: ${\ displaystyle {\ mathcal {H}} _ {\ rm {A}} \ otimes {\ mathcal {H}} _ {\ rm {B}}}$ ${\ displaystyle \ sum _ {i, j} c_ {ij} | i \ rangle _ {\ rm {A}} \; | j \ rangle _ {\ rm {B}}}$ The separable states of are those whose coefficients allow the representation , i.e. which can be factored as above. If a state is not separable, it is called entangled.${\ displaystyle {\ mathcal {H}} _ {\ rm {A}} \ otimes {\ mathcal {H}} _ {\ rm {B}}}$ ${\ displaystyle c_ {i, j} = a_ {i} b_ {j}}$ For example, let two basis vectors of and two basis vectors of be given. Then the following state, the so-called "singlet state", is entangled:${\ displaystyle \ {| 0 \ rangle _ {\ rm {A}}, | 1 \ rangle _ {\ rm {A}} \}}$ ${\ displaystyle {\ mathcal {H}} _ {\ rm {A}}}$ ${\ displaystyle \ {| 0 \ rangle _ {\ rm {B}}, | 1 \ rangle _ {\ rm {B}} \}}$ ${\ displaystyle {\ mathcal {H}} _ {\ rm {B}}}$ ${\ displaystyle {1 \ over {\ sqrt {2}}} {\ Big (} | 0 \ rangle _ {\ rm {A}} | 1 \ rangle _ {\ rm {B}} - | 1 \ rangle _ {\ rm {A}} | 0 \ rangle _ {\ rm {B}} {\ Big)}}$ If the composite system is in this state, neither nor have a certain state, but their states are superimposed and the systems are entangled in this sense .${\ displaystyle A}$ ${\ displaystyle B}$ Only eigenvalues ​​of Hermitian operators can appear as quantum mechanical measured values . So now let "measurement operators" be given in each of the two subsystems and satisfy the following two eigenvalue equations: ${\ displaystyle \ Omega ^ {(i)}}$ ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle \ Omega ^ {(i)} | 0 \ rangle _ {(i)} = \ lambda _ {0} | 0 \ rangle _ {(i)} {\ text {and}} \ Omega ^ {( i)} | 1 \ rangle _ {(i)} = \ lambda _ {1} | 1 \ rangle _ {(i)}}$ .

Using the tensor product with the unity operator , one can generate an operator on the tensor product space with the above measurement operators of the subsystems, whereby the system on which the measurement is made is then noted in the subscript: ${\ displaystyle I}$ ${\ displaystyle \ Omega _ {A} = \ Omega ^ {(A)} \ otimes I ^ {(B)} {\ text {or}} \ Omega _ {B} = I ^ {(A)} \ otimes \ Omega ^ {(B)}}$ Suppose Alice is observing the system , Bob is observing the system . If Alice takes the measurement , two results can appear with equal probability: ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle \ Omega _ {\ rm {A}}}$ 1. Alice measures , and the state of the system collapses too${\ displaystyle \ lambda _ {0}}$ ${\ displaystyle | 0 \ rangle _ {\ rm {A}} | 1 \ rangle _ {\ rm {B}}.}$ 2. Alice measures , and the condition collapses too${\ displaystyle \ lambda _ {1}}$ ${\ displaystyle | 1 \ rangle _ {\ rm {A}} | 0 \ rangle _ {\ rm {B}}.}$ In the first case, each measurement by Bob can (or could) only ever result, in the second case only ever . So the state of the system was changed by the measurement carried out by Alice, even if and are spatially separated. This is where the EPR paradox lies , and so is the so-called quantum teleportation . ${\ displaystyle \ Omega _ {\ rm {B}}}$ ${\ displaystyle \ lambda _ {1}}$ ${\ displaystyle \ lambda _ {0}}$ ${\ displaystyle A}$ ${\ displaystyle B}$ The result of Alice's measurement is random, she cannot determine the state into which the system is collapsing and therefore cannot transmit any information to Bob through actions on her system. A backdoor that seems possible at first: If Bob can make several exact duplicates of the states he is receiving, he could collect information statistically - but the no-cloning theorem proves the impossibility of cloning states. Therefore - as mentioned above - the causality is not violated.

The degree of entanglement of a state is measured by the Von Neumann entropy of the reduced density operator of the state. The Von Neumann entropy of the reduced density operator of an unentangled state is zero. In contrast, the Von Neumann entropy of a reduced density operator of a maximally entangled state (such as a Bell state ) is maximal. ${\ displaystyle S = - {\ text {tr}} (\ rho _ {\ text {red}} \ ln (\ rho _ {\ text {red}}))}$ It should be pointed out here that in addition to the entangled pure states discussed above (which are opposed to the pure product states - without entanglement), there are also entangled mixed states (which are opposed to the mixed product states - without entanglement).

## Test for entanglement

Whether a given state is entangled can be determined mathematically from its density matrix. There are various methods for this, for example the Peres-Horodecki criterion or the test of whether the Schmidt decomposition of the state has more than one term.

For a purely interlocked state of a system that is composed of a subsystem 1 and a subsystem 2, the following applies . If the partial trace is formed over one of the two systems (e.g. system 1), the reduced density operator is obtained . If you now consider the square of the reduced density operator and if this is not equal , the reduced density operator describes a mixture and thus describes an entangled state. Because with an entangled state, the measurement on one system creates a classic mixture of states in the other system from the point of view of all observers who do not know the measurement result in the first system. If there was a non-entangled state, the measurement on one system would not change the state in the other system. ${\ displaystyle | B \ rangle}$ ${\ displaystyle \ rho = | B \ rangle \ langle B |}$ ${\ displaystyle \ rho _ {2}: = {\ text {track}} _ {1} {\ rho}}$ ${\ displaystyle \ rho _ {2} ^ {2}}$ ${\ displaystyle \ rho _ {2}}$ ${\ displaystyle \ rho}$ For mixed states, the test for entanglement is generally very difficult ( NP difficult ). Partial answers are provided by so-called separability criteria , the violation of which is a sufficient condition for entanglement.

Entanglement can also be quantified, that is, there are more and less strongly entangled states. The degree of entanglement is controlled by a Verschränkungsmaß expressed.

## Others

In 2013, Juan Maldacena and Leonard Susskind hypothesized the equivalence of quantum entangled particle pairs (EPR) and special wormholes in quantum gravity as a way of solving the information paradox of black holes and its intensification in the firewall paradox.

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14. Note: If the eigenvalue was measured, the system is in the state . There is also a postulate for the probabilities of measuring an eigenvalue of an operator, see density operator # properties .${\ displaystyle a}$ ${\ displaystyle {\ frac {{\ hat {\ mathbb {P}}} _ {a} {\ hat {\ rho}} {\ hat {\ mathbb {P}}} _ {a}} {\ operatorname { Tr} ({\ hat {\ mathbb {P}}} _ {a} {\ hat {\ rho}} {\ hat {\ mathbb {P}}} _ {a})}}}$ 15. Naresh Chandra, Rama Ghosh: Quantum Entanglement in Electron Optics: Generation, Characterization, and Applications. Springer, 2013, ISBN 3-642-24070-4 , p. 43. Google Books.
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17. For pure states the density operator only consists of a projector and is therefore idempotent .
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