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By fundamental definition of a entangled system we can say that if we know the quantum state of one subsystem then we can describe the state of another subsystem. A particle possess wave-particle duality. If one experiment verify the wave nature of particle then we can not see its particle behaviour in same exp and vice-versa. Can we say that wave and particle behaviour are in some sort of entanglement state.

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No they are different things.

Entanglement:

Imagine that you are in lab, you have two electron spin measuring devices and you have a particle which has spin 0. after few seconds it decays into two electrons. After you will measure first electron's spin and for example device says that it has spin $\uparrow$ then according to spin conservation: sum of spins should equal $0$ and because of that second electron will have spin $\downarrow$ ($S=\uparrow+\downarrow=0$), no matter how many times you will do experiment you always will get same result: if one electron's spin is down second's will be up and vice versa (unless you are measuring spin of electrons in different axes in other words EPR paradox). Here's Sample image of that experiment about entanglement: enter image description here

Wave-particle duality:

Wave-particle duality in simple words means that any matter can behave like a wave and a particle. Wavelength of that wave can be calculated using De-Broglie wavelength: $$ \lambda=\frac{h}{mv} $$ Where $\lambda$ is wavelength, $h$ is Plank's Constant, $m$ is mass and $v$ is velocity. So for example imagine youngs double slit experiment, when you aren't observing light it behaves like a wave and you will get interference pattern but when you will observe it (in other words you will know that in which slit each photon passed through) interference pattern will dissapear. Here's sample image of young's double slit experiment.

Young's Double slit experiment

Conclusion:

So in short, Quantum Entanglement means that when you have two entangled particles, and you will measure one particle's spin you will "collapse" wavefunction and second particle will definitely have opposite spin or in other words in you know state of one of two entangled particles you will definitely know state of the other entangled particle (so it means that state of one of two(or $n$) entangled particles depends on the state of the other and vice versa), and Wave-Particle duality means that matter can sometimes behave like a wave and sometimes behave like a particle, in aforementioned example about wave-particle duality you can see that when you don't observe particle (in example case you don't know in which slit photon passed though) light will behave like a wave but when you will start observing it (Using polarizers (It's called Quantum Eraser or Delayed choice Quantum Eraser)) it will behave like a particle. So as you can see entanglement and wave-particle duality are different things. So we can't say that wave and particle behaviour are in some sort of entanglement state.

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  • $\begingroup$ We can have entanglement between different degree of freedom of single particle, don't we? If we know that electron is behaving like a wave in double slit exp then we can conclude that it will not show any particle property in same setup. $\endgroup$ – Rahul kumar walia Apr 25 '14 at 5:06
  • $\begingroup$ Rahulkumarwalia electron behaves like a wave when we aren't observing it (in case of double slit experiment we don't know in which slit electron passed through). But when we will observe it, it will behave like a particle. So in double slit experiment it won't show particle properties while we don't know in which slit it passed though. To see quantum states see @user45232's answer. $\endgroup$ – Gigi Butbaia Apr 25 '14 at 11:47
  • $\begingroup$ ,I know that in double slit exp wave nature will be disappear as soon as we place a detector to find electron position. But point that i raised is that both nature are complementary to each other(means we can not get interference while placing a detector) so can we have a correlation between this duality and entanglement. yes i know that entanglement make correlation between quantum state while duality is in between behaviour(nature) but complementary. $\endgroup$ – Rahul kumar walia May 15 '14 at 17:52
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To see the quantum states of Gigi10012's answer.
1. The state of the electrons is $|\psi >=\frac{|+-\rangle+|-+\rangle}{\sqrt{2}}$, when you measure one electron$|\psi\rangle$ will collapse into $|+-\rangle$ or $|-+\rangle$ then if you know the spin of the first electron you can know the second one. Before measurement you cannot describe one of the electrons individually. This is entanglement.
2. In the Double Split Experiment, let $|\phi\rangle$ describe the photon. $$|\phi\rangle=\int |x\rangle \langle x|\phi\rangle=\int \phi(x)|x\rangle$$
So when you measure the photon, $|\phi\rangle$ collapses into one of the $|x\rangle$, which looks like a particle in a particular position. The Wave-Particle duality is just something used to vividly describe some phenomena.

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  • $\begingroup$ In first part about electrons, shouldn't $|\psi\rangle$ be $\frac{|+-\rangle + |-+\rangle}{\sqrt{2}}$ or $\alpha|+-\rangle + \beta|-+\rangle$ and not just $|+-\rangle + |-+\rangle$. $\endgroup$ – Gigi Butbaia Apr 25 '14 at 13:06
  • $\begingroup$ @Gigi10012 Yes, you are right. I ignored the normalization where I wrote it. I have corrected it. Thank you for pointing out. $\endgroup$ – Nahc Apr 25 '14 at 15:15

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