LOG#215. Entanglement is the key?

Posted on by August 7, 2020

Hi everyone!

Is entanglement the key? A tribute to Ant-Man and Hawking today as the preamble, Quantum Chess playing for heroes like you:

Entanglement is the subject we have today. Entanglement is that spooky weird feature of the quantum realm that stunned Einstein and realist scientists believing that reality is a preexistent “thing/stuff/entity”. Going more precise, entangled states are related to quantum states that are the product of complex systems made of parts or quantum states being composed from states of single systems. Let me introduce a little bit terminology:

  • Pure states.
  • Mixed states.
  • Separable states.
  • Entangled (non-separable!) states.

Entanglement is related to the above 4 types os states. Quantum Mechanics as we know it today is based on some basic axioms:

  1. Superposition (linearity) of quantum states.
  2. Heisenberg uncertainty principle (HUP).
  3. Unitarity.
  4. Projection postulate.
  5. Quantum composite systems or states can be made up from tensor products of single systems. What a tensor product is? It is a way to create some matrix states from two or more ingle matrices. It is not the only way, but it is the one that works.

Take for instante N=2 (two party, two subsystems creating a big one system). The Hilbert space of the composite two-party quantum system is made from the tensor product of H=H_A\otimes H_B, i.e., the Hilbert space is the tensor product of the two sybsystem Hilbert spaces. Then, the quantum states of the composite system are given by:

    \[\vert\Psi>_{AB}=\vert AB>=\sum c_{ij}\vert i>_A\vert j>_B\]

Then, it is that a state is separable IFF you can find out vector \vert c_i>_A and \vert c_j>_B such as c_{ij}=c_i(A)c_j(B) in the previous expansion. That is, if you can factorize the state as a single product of the two single systems quantum states the state is separable, otherwise the state is ENTANGLED. You can generalize the above definition to any number of systems (parties!). The general n-party quantum state is defined as certain tensor product of the subsystem quantum states as follows:

    \[H=\bigotimes_{i=1}^n H_i\]

(1)   \begin{equation*} \vert A_1\;A_2\;\cdots A_n>=\sum c_{i_1i_2\cdots i_n}\vert i_1>_{A_1}\vert i_2>_{A_2}\cdots \vert i_n>_{A_n}\end{equation*}

That’s entanglement!!!! You would say, then, why is it “hard”? Well, there are several reasons why entanglement is hard and why entanglement does matter  A LOT in QM affairs. Let me start for the first item. Why entanglement is hard? A list:

  • Entanglement is a subtle non-separability meaning certain non-locality compatible with special relativity. Yes! It is true. Entangled states have certain abilities that allow you to do magic at very large distances but causality and finite propagation of signals are not violated.
  • Bell’s theorem (more on this later). Bell found out that the existence of entangled state in QM allows you to test the existence of hidden variable theories. It yields that QM holds superb. Unchallenged. Bell experiment kills any hope for local realist theories. You need a very special type of theories if you can mimic QM results of Bell-type experiments. They need to be contextual. Reality is not independent from the way we measure it, and indeed, there are systems with act as if they were not independent from their parts even when separated to km of distance. Chinese people have indeed build up a satellite using entanglement to secure communication.
  • Currently, the EPR (Einstein-Podolski-Rosen) experiment, the type of experiment Bell inded realized has been focused by quantum gravity theorists due to the black hole information problem and the nature of gravity. Van Ramsdook proposes that gravity “is” entanglement, and Susskind and collaborators are developing an idea summarized in the formal equation ER=EPR. ER is Einstein Rosen bridge in General Relativity. ER=EPR states that quantum entanglement is caused by two (or more) quantum particles being connected by (micro)wormholes (Einstein-Rosen bridges!). That quantum entanglement could be caused by non-simple connected quantum microwormholes is just quite an statement. Hard to experimentally test. Van Ramsdook indeed suggests the gravity itself is caused by entanglement.

The relationship between gravity (“classicality”) and entanglement is an old friend. In fact, there is another point where this idea arises, but I am not sure my readers will know it. Some time ago, Rigolin’s proved that a high number of entangled particles can beat the Heisenberg Uncertainty Principle bound. Even more, he conjectured that in the limit of an infinite number of entangled particles, you get “classical” zero dispersion. That is, with an infinite number of entangled particles, you could in principle ban the uncertainty relationship. From this viewpoint, (the amount of) entanglement REDUCES uncertainty. Reciprocally, separability enlarges uncertainty. You can read the Rigolin original work here http://cds.cern.ch/record/499980/files/0105057.pdf. Wait, what if you modify the HUP by some generalized form of it like EHUP, GUP or EGUP? Logical thoughts impose here: EHUP and EGUP or GUP make the system more quantum and less classical, enhancing the bounds reacting against the reduction of uncertainty of very large number of entanglement particles. GUP, EHUP and EGUP have the opposite effect to entanglement and make more uncertain the entangled states. See about this here https://arxiv.org/pdf/1706.10013.pdf

You can also read that noncommutativeness (as a bonus) makes entanglement and nonclassicality more evident in the paper: https://arxiv.org/pdf/1506.08901.pdf 

And now? We return to some vocabulary! N-level pure states are defined formally as quantum states

    \[\vert \Psi>=\sum_{i=0}^{N-1}c_i\vert i>\]

Thus, pure states are simple linelar superpositions of quantum states! You can bet qubits with N=2, qutrits with N=3, and qu\inftyits with N=\infty (quantum fields!). Even more, you could add a continuous term as well and spoil the finite term sum. Of course, entanglement of infinite dimensional systems is not usual in standard discussions of quantum computing, but it can be added without generality loss. What about mixed states? Well, we need a new gadget to explain mixed states. This new device is the density matrix. For pure states, the density matrix is a set with copies of the N-level system. For pure states the density matrix reads

    \[\rho=\sum_i w_i\vert i><i\vert\]

where \sum w_i=1 by probability conservation. Now, take a N=2 party system. If separable, then you can write by definition the density matrix as the following tensor product:

    \[\rho = w_i\left[\overline{c}_{ij}c_{ij}\vert ij>(A) (A)<ij\vert \vert ij>(B) (B)<ij\vert\right]=\sum_i w_i\rho_i (A)\otimes\rho_i (B)\]

and where \sum_j\vert c_{ij}\vert^2=2 and we can generalize this to N-party systems as

(2)   \begin{equation*}\rho=\sum_i\omega_i\rho_{i_1}^{A_1}\cdots\rho_{i_n}^{A_n}\end{equation*}

for separable states with

    \[\sum_j\vert c_{ij}\vert^2=\sum_i\omega_i=1\]

by probability conservation once again.

Next step is to define the so-called reduced density matrix. It is a density matrix created from the big one tracing over a simple or more subsystems. For a single reduction:

    \[\rho_T=\vert\Psi><\Psi\vert\]

and for the reduced density matrix tracing over A (N=2 party case) you get

    \[\rho_A=\sum_j <j\vert_B\left(\vert\Psi><\Psi\vert\right)\vert j>_B=\mbox{Tr}_B\rho_T\]

and similarly you could get the reduced density matrix tracing by A states.

Entanglement example 1. Bell states.

Take N=2, two level system. H_A=\left[\vert 0>_A,\vert 1>_A\right] is the A basis and H_B=\left[\vert 0>_B,\vert 1>_B\right] the basis for quantum states of the B system. For the composite system, tensor product, you can find out 4 interesting Bell states that are entangled and can not be decomposed into single products of basis states. They are:

(3)   \begin{equation*}\vert BELL>_1=\dfrac{1}{\sqrt{2}}\left[\vert 0>_A\vert 0>_B+\vert 1>_A\vert 1>_B\right]\end{equation*}

(4)   \begin{equation*}\vert BELL>_2=\dfrac{1}{\sqrt{2}}\left[\vert 0>_A\vert 0>_B-\vert 1>_A\vert 1>_B\right]\end{equation*}

(5)   \begin{equation*}\vert BELL>_3=\dfrac{1}{\sqrt{2}}\left[\vert 0>_A\vert 1>_B+\vert 1>_A\vert 0>_B\right]\end{equation*}

(6)   \begin{equation*}\vert BELL>_4=\dfrac{1}{\sqrt{2}}\left[\vert 0>_A\vert 1>_B-\vert 1>_A\vert 0>_B\right]\end{equation*}

They are indeed special in a sense. They are maximally entangled states, i.e., they are the states with the greatest degree of entanglement possible within the composite system.

Entanglement example 2. Bell 4 reduced density matrix.

Take the 4th Bell state:

(7)   \begin{equation*}\vert BELL>_4=\dfrac{1}{\sqrt{2}}\left[\vert 0>_A\vert 1>_B-\vert 1>_A\vert 0>_B\right]\end{equation*}

Trace over the B subsystem:

    \[\rho_A=\mbox{Tr}_B\rho_T(\Psi)=\dfrac{1}{2}\left(\vert 0>_A<0\vert_A+\vert 1>_A<1\vert_A\right)\]

Then, you see that the reduced density matrix for entangled pure ensemble IS a mixed ensemble or state. This result is general, in bipartite systems, \rho is entangled iff the reduced states are mixed rather than pure!

Entanglement example 3. Other entangled states.

For M>2 parties, with two levels, there is a very interesting generalization of Bell states. It is called the GHZ state:

(8)   \begin{equation*}\vert GHZ>=\dfrac{1}{\sqrt{2}}\left(\vert 0>^{\otimes M}+\vert 1>^{\otimes M}\right)\end{equation*}

There are also the so called spin squeezed states, a special set or type of squeezed coherent states. They are important in optics. For 2 bosonic modes, there is the NOON state:

(9)   \begin{equation*}\vert NOON>=\dfrac{\vert N>_A\vert 0>_B+\vert 0>_A\vert N>_B}{\sqrt{2}}\end{equation*}

This is similar to Bell states excepting that the instead the 0,1 kets you have N,0 kets. That is, you have N-excited or N-photons in one mode and 0 photons in the other mode. Well, it shows that Bell states, GHZ states and NOON states are maximally entangled. However, there are other non maximally entangled states. For instance, the previously mentioned spin squeezed states or the twin Fock states. NOON states can also be “phased”, such as you build up a modulated NOON as

    \[\vert NOON>=\dfrac{\vert N>_A\vert 0>_B+e^{iN\theta}\vert 0>_A\vert N>_B}{\sqrt{2}}\]

This state represents the superposition of N-particles in a mode A, with 0-particles in the mode B, and viceversa, shifted by a phase factor. NOON states are useful objects in quantum metrology since they are capable to make precision phase measurements in optical interferometers. Build up the observable A NOON as follows:

    \[A=\vert N,O><O,N\vert+\vert O,N><N,O\vert\]

Then, you can easily prove that the expectation value of A in a NOON state switches between +1 and -1 if phase changes from 0 to \pi/N. Moreover, the error in the phase measurement IS inded

    \[\delta \theta=\dfrac{\delta A}{\vert \dfrac{d<A>}{d\theta}\vert}=\dfrac{1}{N}\]

This is the so-called Heisenberg limit, in fact an improvement over the standard quantum limit (SQL) given by

    \[\delta_{SQL}=\sqrt{\dfrac{\hbar \theta}{M}}\]

The simplest non-Bell GHZ state is made with M=3 parties. GHZ states are used in very important applications:

  • Quantum communication protocols.
  • Quantum cryptography protocols.
  • Secret key sharing.

There is no standard measurement, in a standard way, of multipartite entanglement because, as we saw, there are different types of multipartite entanglement. Indeed, entanglement is not generally mutually convertible. The GHZ state is maximally entangled. For M=3, take

    \[\vert GHZ^3>=\dfrac{\vert 000>+\vert 111>}{\sqrt{2}}\]

    \[\rho_3=\mbox{Tr}_3\left(\dfrac{\vert 000>+\vert 111>}{\sqrt{2}}\right)\left(\dfrac{< 000\vert+< 111\vert}{\sqrt{2}}\right)\]

so you get an unentangled mixed state:

(10)   \begin{equation*}\rho_3=\left(\dfrac{\vert 00><00\vert+\vert 11><11\vert}{2}\right)\end{equation*}

 

Thus, this GHZ state has certain 2-particle quantum correlations but there are of “a classical nature” somehow. GHZ leads to striking non-classical correlations too. They allow you to test the internal inconsitencies of the EPR elements of reality. The generalized GHZ state for d-levels is given by the state

(11)   \begin{equation*}\boxed{\vert GHZ^d>=\dfrac{1}{\sqrt{d}}\sum_{j=0}^{d-1}\vert j>^{\otimes q}}\end{equation*}

Maybe you want to experiment with quantum states and quantum entanglement. There is a MATLAB toolbox for exploring quantum entanglement theory. It is called QETLAB. I do not checked, and I am sure there are other similar toys and apps out there. Let me know any way!

Entanglement example 4. W-states.

There is a 3 qubit interesting entangled quantum state called W-state. It is interesting for storage of quantum memories. It reads:

    \[\vert W>=\dfrac{1}{\sqrt{3}}\left(\vert 001>+\vert 010>+\vert 100>\right)\]

For N-qubits the W-state is

    \[\vert W>_N=\dfrac{1}{\sqrt{N}}\left(\vert 0\cdots 1>+\cdots+\vert 1\cdots 0>\right)\]

The W-state is just a linear quantum superposition of all possible pure states with exactly one excited state and the others being in the ground state, weighted with the same probability.

Multipartite entanglement is much more complicated. M>2 entanglement is richer in possibilities than M=2 entanglement. With M=2 there are fully entangled (maximally entangled) and fully separable states. However, things go wild in M>2 parties. You can also have partially separable or partially entangled states. The full M-partite separability

    \[\rho_{A_1\cdots A_M}=\sum_i P_i\rho^i_{A_1}\otimes \cdots \otimes \rho^i_{A_M}\]

is fully entangled when written in this way. But there are also pure states

    \[\vert A_1\cdots A_M>=\vert A_1>\otimes\cdots\vert A_M>\]

and, partially entangled states beyond the fully (maximally) entangled states.

There are other cool measurements of entanglement related to states. They have weird names like tangles or hyperdeterminants! However, in the end, all are expressed in term of pure or mixed states, with certain amount of partial or maximal (or null) entanglement!

What else? Beyond Bell, GHZ, W, NOON and similar states, there are many interesting topics related to entanglement these days. These subjects include:

  • Going from multipartite to bipartite entanglement.
  • Entropy bounds related to entanglement and entanglement entropy.
  • Quantum channels and quantum channel capacities.
  • LOCC=Local Operations and Classical Communication observables.
  • Entanglement distillation (yet, you can distillate entanglement, we have seen an example before!).
  • Quantum teleportation (it is not just like beam me up, Scotty, but it rocks).
  • Quantum cryptography and quantum communication, quantum key sharing.
  • Quantum game theory.
  • Black hole information paradoxes.
  • The EPR=ER and Gravity=Entanglement ideas.
  • Hyperentanglement, i.e., the simultaneous entanglement between multiple degrees of freedom of 2 or more entangled systems.

In summary, quantum entanglement is a fascinating topic. I am sure many of you knew many of these things before. Perhaps, Rigolin’s works and the research involving how to spoil or enhance uncertainty from entanglement and or noncommutativity (GUP,EHUP,EGUP) is the strangest topic I discussed here today. Did you enjoy it? I hope so!

Challenge question: could entanglement affect to gravity and then to time/space measurements? How could you know if your time or space is entangled to mine or to the local time/space measurements in other parts of the observable Universe?

See you in another blog post!!!!!

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