# What is Pure state: Definition and 25 Discussions

In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior. A mixture of quantum states is again a quantum state. Quantum states that cannot be written as a mixture of other states are called pure quantum states, while all other states are called mixed quantum states. A pure quantum state can be represented by a ray in a Hilbert space over the complex numbers, while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces.Pure states are also known as state vectors or wave functions, the latter term applying particularly when they are represented as functions of position or momentum. For example, when dealing with the energy spectrum of the electron in a hydrogen atom, the relevant state vectors are identified by the principal quantum number n, the angular momentum quantum number l, the magnetic quantum number m, and the spin z-component sz. For another example, if the spin of an electron is measured in any direction, e.g. with a Stern–Gerlach experiment, there are two possible results: up or down. The Hilbert space for the electron's spin is therefore two-dimensional, constituting a qubit. A pure state here is represented by a two-dimensional complex vector

(
α
,
β
)

{\displaystyle (\alpha ,\beta )}
, with a length of one; that is, with

|

α

|

2

+

|

β

|

2

=
1
,

{\displaystyle |\alpha |^{2}+|\beta |^{2}=1,}
where

|

α

|

{\displaystyle |\alpha |}
and

|

β

|

{\displaystyle |\beta |}
are the absolute values of

α

{\displaystyle \alpha }
and

β

{\displaystyle \beta }
. A mixed state, in this case, has the structure of a

2
×
2

{\displaystyle 2\times 2}
matrix that is Hermitian and positive semi-definite, and has trace 1. A more complicated case is given (in bra–ket notation) by the singlet state, which exemplifies quantum entanglement:

|
ψ

=

1

2

(

|

↑↓

|

↓↑

)

,

{\displaystyle \left|\psi \right\rangle ={\frac {1}{\sqrt {2}}}{\big (}\left|\uparrow \downarrow \right\rangle -\left|\downarrow \uparrow \right\rangle {\big )},}
which involves superposition of joint spin states for two particles with spin 1⁄2. The singlet state satisfies the property that if the particles' spins are measured along the same direction then either the spin of the first particle is observed up and the spin of the second particle is observed down, or the first one is observed down and the second one is observed up, both possibilities occurring with equal probability.
A mixed quantum state corresponds to a probabilistic mixture of pure states; however, different distributions of pure states can generate equivalent (i.e., physically indistinguishable) mixed states. The Schrödinger–HJW theorem classifies the multitude of ways to write a given mixed state as a convex combination of pure states. Before a particular measurement is performed on a quantum system, the theory gives only a probability distribution for the outcome, and the form that this distribution takes is completely determined by the quantum state and the linear operators describing the measurement. Probability distributions for different measurements exhibit tradeoffs exemplified by the uncertainty principle: a state that implies a narrow spread of possible outcomes for one experiment necessarily implies a wide spread of possible outcomes for another.

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1. ### I Qbit pure vs mixed state space

According to this Wikipedia entry a quantum pure qbit state is a ray in the Hilbert space ##\mathbb H_2## of dimension 2. In other words a qbit pure quantum state is a point in the Hilbert projective line. Now my question: is an arbitrary vector in ##\mathbb H_2## actually a "mixed" state for...
2. ### I Entanglement, Mixed or Pure State?

I have a source that says when two particles are entangled, we must describe them using the density operator because it is a mixed state. But I have another source that says that the singlet state of two spins is an entangled state, but that has a wavefunction. So could someone explain what I am...
3. ### I Mixed states from entangled state

I'm an undergrad in physics, and have been asking myself the following question recently. Suppose you have a pure quantum state p (von neumann entropy=0), made of 2 sub-states p1 and p2 that are entangled. Because they are entangled, p \neq p1 x p2. Hence the entanglement entropy of p (=0) is...
4. ### B Is a pure state a kind of mixed states?

Hi. 1. Does a pure state belong to mixed states \hat{\rho}=\sum_k p_k|\psi_k><\psi_k| where ##p_k=1## for k=i and otherwise 0 ? 2. Does quantum jump by observation work for both mixed and pure states ? Your teachings will be appreciated.
5. ### I Mixed states vs pure states - physical POV

Hey guys, I am having issues with understanding the physical nature of pure and mixed states. Maybe you can help me out? 1) A pure state - superposition is a state that consists of different states at the same time. It's like having several waves, each one belonging to an Eigenstate of the...
6. ### I Entangled system pure state or not?

According to atty.. entangled system are pure (can be in superposition) while according to Bill.. entangled system are not pure (not in superposition) Here's the prove of what they stated: atty said...
7. ### B Understanding Pure States in Quantum Mechanics: From Atoms to Electrons

I learned about pure state, mixed state, reduce density matrix, etc. from this now famous paper http://philsci-archive.pitt.edu/5439/1/Decoherence_Essay_arXiv_version.pdf (thanks to Bill) I'd like to know something. Atty said somewhere: "A pure state means that we have prepared many copies of...
8. ### A Is the Quantum State in a Dressed Atom System Pure?

Consider a simple two level atom system,|1>,|2> (not degenerate) interacted with monochromatic laser, of which the frequency is exactly resonant with the |1> to |2> transition. The evolution start from {1>. As far as I understand, the dressed degenerate state |1>|n> and |2>|n-1> will get...
9. ### B Does weak measurement cause mixed state?

Does a 'weak measurement' on the spin of an electron in a pure state put the electron in a mixed state of the previous state and the state of the measurement axis of the measurement?
10. ### I Difference between superposition and mixed state

I get that a if we have complete information of the state of the system (i.e. all the possible knowledge we could have about it: the values its observables can take and their corresponding probabilities), then it is a pure state and can be represented by a vector (ket), ##\lvert\psi\rangle## in...
11. ### I Density matrices, pure states and mixed states

I got (very) confused about the concept of states, pure states and mixed states. Is it correct that a linear combination of pure states is another pure state? Can pure (and mixed) states only be expressed in density matrices? Is a pure state expressed in a single density matrix, whereas mixed...
12. ### Fidelity between a pure state and an arbitrary state

\Homework Statement Show that the Fidelity between one pure state |\Psi\rangle and one arbitrary state \rho is given by F(|\Psi\rangle , \rho)=\sqrt{\langle\Psi|\rho|\Psi\rangle} . Homework Equations The quantum mechanical fidelity is defined as \begin{equation*}...
13. ### What is the concept of a pure state in the electromagnetic field?

In Ballentine's Quantum Mechanics book, as part of a discussion of pure states vs non-pure (mixed) states, he says Polarized monochromatic light produced by a laser can approximate a pure state of the electromagnetic field. Unpolarized monochromatic radiation and black body radiation are...
14. ### Is it possible to distinguish between pure and mixed states?

Is it possible, in principle, for an experiment to distinguish between an ensemble of pure states and an ensemble of mixed states? If so, how? In particular, I am thinking of an ensemble of particles whose spin has been measured, one at a time, on the "Vertical" axis. The ensemble consists of...
15. ### Pure state, mixed state and measurement

Hello guys, Homework Statement the problem goes as follows: "Which measurement should you do on a statistical ensemble of qubits in order to distinguish between the pure state |Ψ>= cos(θ)|0> + sin(θ)|1> and the mixed state ρ=cos^2(θ)|0><0| + sin^2(θ)|1><1| " Homework Equations I am not...
16. ### Pure to mixed and back to pure state

Let's say you prepare a pure state in a single electron at a time double slit experiment meaning the electron interferes with itself, then you hit the electron with a one photon at the sides (serving as environment) causing decoherence, so the double slit setup decohere and becomes mixed state...
17. ### [Quantum Mechanics] Quantum Fisher Information for a Pure State

Hi everyone. Homework Statement We are given N spins 1/2. A rotation is defined as \rho_\theta=e^{-i\theta J_n}\rho_\theta e^{i\theta J_n} on an Hilbert Space H, with J_n=n_xJ_x+n_yJ_y+n_zJ_z\:,\quad n_x^2+n_y^2+n_z^2=1, and \theta isn't related to any observable. Given a quantum state...
18. ### Mixed state vs. pure state

In quantum mechanics, regarding light (photon), how to tell that a wavefunction is in a pure state or mixed state? I am learning these stuffs for my first time. I have attempted to answer that question but I am not sure: a wavefunction can be wrtitten as a linear combination of linear...
19. ### Decoherence, Coherence, Pure State, Mixed State

Hi, I'm confused by subtle differences between the concept. Let's take the example of a Schrodinger Cat. Supposed you could make a box that can isolate anything inside from say gravity, microwave radiation, is in 0 kelvin, etc. or let's just accept (for sake of discussion) that a box can totally...
20. ### Pure State without Superposition possible?

Please give an example of pure state without superposition present, is this possible at all?
21. ### Pure state versus superposition

I am a little bit versed in the formalism of Hilbert space, the state vector and the matricies and their eigenvector basis of observables in QM. I understand that a pure state in one observable basis may be (is) a superposition in another basis (spin xUP) is superposition of spinyUP, spin yDOWN...
22. ### Density Matrix (pure state) Property

Hello. I need some help to prove the first property of the density matrix for a pure state. According to this property, the density matrix is definite positive (or semi-definite positive). I've been trying to prove it mathematically, but I can't. I need to prove that |a|^2 x |c|^2 +...
23. ### Producing Pure States in Practical Systems: Challenges and Compromises

After reading some introudction of quantum states (pure and mixed), I know that states produced statistically are called mixed state. My question is: since there always exists stochastic process, so in practical system, how can we produce "pure" state?
24. ### About mixed state and pure state

I don't have any problems dealing with mechanical calculations(I think), but yet I have some conceptual problems with mixed state(= statistical mixture?) and pure state in QM. - pure state : |Φ> = 1/sqrt2 ( |↑> + |↓> ) - mixed state : ρ = 1/2 ( |↑><↑| + |↑><↑| ) What is the difference...
25. ### What is a Pure State and Mixed State?

What is a pure state and a mixed state?