# What is Quantum state: Definition and 96 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 Quantum state of a macroscopic object?

(Mods, I posted a lot on the MWI yesterday, but this seemed different enough to warrant its own thread. If you disagree, I apologize.) The Stanford Encyclopedia of Philosophy says the following in its article on the Many Worlds Interpretation: Is this backed by science? It reminds me of...
2. ### I Bra-Ket Notation Manipulations: Quantum State Expansion

I suspect it will help if you know about my background: I did some linear algebra in university but never used it and am now in my mid 60s. I am interested in understanding the mathematics of quantum physics. I have read a number of layman's texts on quantum mechanics, but they all gloss over...
3. ### I The collapse of a quantum state as a joint probability construction

The titular paper can be found here, https://doi.org/10.1088/1751-8121/ac6f2f, and on arXiv as https://arxiv.org/abs/2101.10931 (which is paginated differently, but the text and equation and section numbers are the same). Please see the abstract, but in part this 24 page paper argues that we...
4. ### I The quantum state of the Universe

The dimension of the space of quantum states of multiple particles grows exponentially as the number of particles increases. I would have expected to more likely find the quantum state of many particles in a strange state (such as an entangled one) but it is not so, why? Why isn't the universe...
5. ### A Homodyne detection quantum state tomography

In balanced homodyne detection, it is claimed that one can do state tomography. I understand most of the derivation except one part. Here is a figure describing homodyne detection. the operator that is being measured is $$R=N_{1}-N_{2}=a^{\dagger} b+b^{\dagger} a$$. taking the mode b to be...
6. ### I Quantum phase estimation - Question regarding rewriting the state

In Nielsen and Chuang p.223 we have the following situation: $$\frac{1}{2^t} \sum\limits_{k,l=0}^{2^t-1} e^{\frac{-2\pi i k l}{2^t}} e^{2 \pi i \varphi k} |l\rangle$$ Which results from applying the inverse quantum Fourier transform to state ##\frac{1}{2^{t/2}} \sum\limits_{k=0}^{2^t-1}...
7. ### I Measurement of a qubit in the computational basis - Phase estimation

Hello, I have a question about the measurement of a qubit in the computational basis. I would like to first state what I know so far and then ask my actual question at the end.What I know: Let's say we have a qubit in the general state of ##|\psi\rangle = \alpha|0\rangle + \beta|1\rangle##. Now...
8. ### A What Does Constant Fidelity in Quantum States Imply?

I have a fidelity between the initial state and final state is 1 at t=0,1,2...etc. What does it mean physically? Does it mean that the quantum state is not evolved here. But In quantum dynamics every physical system evolves with time.
9. ### A Fidelity for quantum state at t=0

fidelity for pure state with respect to t=0 is 1. My teacher told me this. But I am not getting this. This is my detailed question the initial state(t=0)##|\psi\rangle=|\alpha\rangle|0\rangle## the final state (t) ##|\chi\rangle= |i\alpha\sin(t)\rangle|cos(t)\alpha\rangle## Fidelity between the...
10. ### A Restricted Boltzmann machine for Quantum state tomography

I'm struggling with my Final Degree Project. I would like to perform a quantum simulation and perform quantum tomography for a single-qubit using a resrticted Boltzmann machine. In order to do so I'm trying to follow the recipe in the paper "Neural Network quantum state tomography, Giacomo...
11. ### I Extraction of a particular quantum state from a quantum circuit

Hello everyone! I'm trying to implement a quantum circuit that yields a superposition state $$\frac{1}{\sqrt{2}} (|01 \rangle + |10 \rangle)$$ I'm using parameterized gates to achieve this. I have been able to create the state $$\frac{1}{\sqrt{2}}(|01\rangle + e^{i\phi} |10 \rangle)$$ Is there...
12. ### B Time-reversal of an unknown quantum state

A very interesting paper was recently released that's a follow up to the paper that talked about Time Reversal to a known state. If you remember a lot of papers talked about how they reversed time. Here's more from the new article. Basically, Schrodinger's equation is reversable and there's no...
13. ### Electron excited from the ground state to a quantum state

Hi there, popping by here to check my answer because another online platform has already answered it but my answer appears to be wrong. I can't seem to understand why though :/ Since I can find the energy at a state to be ##E_{n}=\dfrac {-13.6z^{2}}{n^{2}}eV## At ground state where n=1...
14. E

### B The use of the dx in the quantum state vector integral

As a simple example, the probability of measuring the position between x and x + dx is |\psi(x)|^{2} dx since |\psi(x)|^{2} is the probability density. So summing |\psi(x)|^{2} dx between any two points within the boundaries yields the required probability. The integral I'm confused about is...
15. ### I Quantum state variable for entangled system

(This question is on the elementary side...) In the Schrödinger picture, the state is dependent on time. If you have a state composed of several particles, generally you can break them up, with each one depending on local time. But in an entangled system, say of two particles, you can no longer...
16. ### I Quantum states: only vectors?

Elementary question: Is there ever a case where the solutions for a wave equation turn out not to be a vector (in Hilbert space of infinite complex-valued dimensions, or a restriction to a subspace thereof) , but something else -- say, (higher-order) tensors or bivectors, or some such? My...
17. ### A Applying General Lorentz Boost to Multipartite Quantum State

I would like to apply a General Lorentz Boost to some Multi-partite Quantum State. I have read several papers (like this) on the theory of boosting quantum states, but I have a hard time applying this theory to concrete examples. Let us take a ##|\Phi^+\rangle## Bell State as an example, and...
18. ### A Can Work Be Defined for a Quantum System with a Time-Dependent Hamiltonian?

I have a Hamiltonian ##H_{\lambda(t)}##, where ##\lambda (t)## characterizes a time dependent path in parameter space. The parameter is changed in finite time from ##\lambda (t_i)## to ##\lambda(t_f)## . At ##t = t_i## the system is in the intial state ##|\Psi>##. What is the work done on the...
19. ### I Is the Quantum State Holographic?

I'm reading Tim Maudlin latest book "Philosophy of Physics: Quantum Theory". In the following descriptions, it is not akin to holographic? "In the context of the quantum recipe, the mathematics of the wavefunction suggests that the quantum state (whatever it is) is a fundamentally global sort...
20. ### A Definition of bi-local measurement by Masanes et al.

Dear experts, I'm currently working my way through the paper Masanes, Galley, Müller, https://arxiv.org/abs/1811.11060. On page 3, they define what they call a bi-local measurement: If we have two systems a and b and we define an outcome probability function for some measurement f on system a...
21. ### I Phase shift after double-slit for entangled photons

Hi all, I'm trying to understand how to describe the quantum state of entangled photons, including their phase, if one of them encounters a double-slit. Here's a simple example: Suppose you have two polarization-entangled photons A and B in the following Bell state: ...
22. ### Quantum state of entangled photons

Homework Statement Suppose two polarization-entangled photons A and B in the following Bell state: $$\Phi=\frac{1}{\sqrt{2}}\bigl(\left|H_{A},H_{B}\right\rangle + \left| V_{A},V_{B}\right\rangle\bigr)$$ 1. What is the state if the photon A passes through a...
23. ### I Quantum State: Electron, Proton, and Neutron Explained

Can anyone tell me, What quantum state really is? Is it applicable for all sub atomic particles? Then, Can anyone explain how two electron are never in same quantum state. And Does proton or neutron follows the same law as electron for obtaining unique quantum state.
24. ### Calculating the quantum state of an electron

Homework Statement An electron in a hydrogen atom falling from an excited state to the ground state has the same wavelength than an electron moving at a speed of 7821 ms^-1. From which excited state did the electron fall from? Homework Equations I used the kinetic energy equation: K = (m...
25. ### A generic quantum state and the charge of an electron

I'm watching a lecture and the professor is talking about generic quantum states as |\psi> He's making the point that this state is very generic. It can represent anything. He references some examples like the polarization of a photon and the path of a photon and the spin of an electron...
26. ### Quantum state of system after measurement

> Operator $$\hat{A}$$ has two normalized eigenstates $$\psi_1,\psi_2$$ with > eigenvalues $$\alpha_1,\alpha_2$$. Operator $$\hat{B}$$ has also two > normalized eigenstates $$\phi_1,\phi_2$$ with eigenvalues > $$\beta_1,\beta_2$$. Eigenstates satisfy: > $$\psi_1=(\phi_1+2\phi_2)/\sqrt{5}$$ >...
27. ### I Definition and Rules of Quantum State Observation

I was wondering how the rules work for observation in a quantum system. Particularly, about what happens if two separate entities try measuring at the same time. And also, what kinds of interactions are happening all the time that are considered measurements, for example in quantum...
28. ### B Library of Babel quantum state

Hello! If the Library of Babel has 10^(2,000,000) books, does anyone think that it is possible to create a quantum state (with a quantum computer) that represents this Library? I think that in a classical way it is impossible, but in a quantum way? I find it quite interesting! What about you? :)
29. ### Electron in a Quantum State: Finding Eigenvectors and Eigenvalues

Homework Statement If possible could someone have a look at my working for this problem, I am not sure if I have carried out part b) correctly. I have done all three problem and carried through my solution to b) just to see if it did simplify out, which it didn’t which make me think I may have...
30. ### Validity of Two-Fermion System Wavefunction with Quantum Numbers a and b

Homework Statement Is the statement ”Given a two-fermion system, and two orbitals φ labeled by quantum numbers a, b, the two-body wavefunction (1,2 represent the particle variables) $$\psi(1,2) = \phi_a(1) \phi_a(2) - \phi_b(1) \phi_b(2) + \phi_a(1) \phi_b(2) - \phi_b(1) \phi_a(2)$$...
31. ### B Quantum state of the Universe?

I am trying to read through this paper discussing what quantum fluctuations mean in their various contexts, particularly in de Sitter space. I have come across this passage and am curious to what it actually means? https://arxiv.org/pdf/1405.0298.pdf pg. 10, second paragraph: "If a quantum...
32. ### Time evolution of quantum state with time ind Hamiltonian

Homework Statement Part e) Homework Equations I know that the time evolution of a system is governed by a complex exponential of the hamiltonian: |psi(t)> = Exp(-iHt) |psi(0)> I know that |psi(0)> = (0, -2/Δ) The Attempt at a Solution I'm stuck on part e. I was told by my professor...
33. ### Calculating Average Energy of a quantum state

Homework Statement Given a wave function that is the super position of the two lowest energies of a particle in an infinite square well ##\Psi = \frac{\sqrt{2}}{\sqrt{3}}\psi _1 + \frac{1}{\sqrt{3}}\psi _2##, find ##\langle E \rangle##. Homework EquationsThe Attempt at a Solution I'm not sure...
34. ### A Experimental tests on the reality of the quantum state

Does this mean that we will one day have an answer to the old age question on the ontology of the wave function: Towards optimal experimental tests on the reality of the quantum state http://iopscience.iop.org/article/10.1088/1367-2630/aa54ab
35. ### B Need Listing of Photon and Particle Quantum States

Hi, I am learning quantum entanglement. I am interested to create an up to date list of all known : - Photon Quantum States - Particle Quantum States - Classically entagled photon states I guess that there is an organization out there that already have this info. If someone can point me into...
36. ### How can I make singular matrix become nonsingular matrix?

<< Mentor Note -- thread moved from Homework Help forums to General Math >> Good day, I run coding in Mathematica. But, I get singular matrix A at certain loop. In theory, how can I make matrix A become orthogonal A=\begin{pmatrix} 0& 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0& 0 & 0 & 0 & 0 & 0 & 0 &...
37. ### How to calculate negativity measurement of quantum state?

Good day, From my reading according to negativity for tripartite state, it is given as below; $$N_{ABC}(\rho)=(N_{A-BC}N_{B-AC}N_{C-AB})^{1/3}$$ with $$N_{I-JK}=-2\Sigma_i\sigma_i(\rho^{TI})$$ where $$\sigma_i(\rho^{TI})$$ being the negative eigenvalues of $$\rho^{TI}$$, the partial...
38. ### I What is the relation between Temperature and Quantum State?

Hi folks, Let's pick a simple example, the H atom. We can calculate all spherical armonics, all quantum numbers so we are able to know which are all the possible states of the electron. We know all the values this observables can take. But the question is, let's say we have a handbook of...
39. ### Calculating the momentum operator in a quantum state

Homework Statement A gaussian wave packet is given by the formula: Ψ(x)=(1/(π1/4d1/2))eikx-(x2/2d2) Calculate the expectation value in this quantum state of the momentum squared. Homework Equations <p2>=-ħ∫Ψ*(X) (d2Ψ(x)/dx2) dx ∫e(-x2/d2) dx= d√π ∫xe(-x2/d2) dx =0 ∫x2e(-x2/d2) dx = (d3√π)/2...
40. ### B Multi or single quantum state?

Is Bose condensation state a multiparticle state or many single identical states? Can we apply Pauli principle to two electrons that one is on the Earth and the other is on the Moon?
41. ### I What is meant by the terminology "single particle state"?

I'm currently reading through a set of notes on statistical mechanics, and when it comes to deriving the Fermi-Dirac and Bose-Einstein distributions it uses the terminology single-particle state. By this, is it meant that if the particles can be assumed independent, then each particle can be...
42. ### B When are particles in a quantum state?

These are just a few quick and simple questions. When are particles in a quantum state? Are they always in a quantum state in-between interactions, or once interacted with are they never in a quantum state again? At the instant of the big bang was everything in a quantum state? If so, what...
43. ### Quantum state vector and physical state

Interesting new work on the link between quantum state vectors are physical states: Can different quantum state vectors correspond to the same physical state? An experimental test Daniel Nigg et al 2016 New J. Phys. 18 013007 Abstract A century after the development of quantum theory, the...
44. ### Why do the off-diagonal terms not matter in density matrices?

can all quantum state be entangled without any exception even if their phases don't coincide? is the term to call this mixed state entanglement accurate? does it have to do with Fourier addition? this is related to environmental entanglement... when you are shaking hands with another person...
45. ### Understanding Pure and Mixed Quantum States in Entanglement

Quick Question. I've recently been told that a pure quantum state is the only state that can be entangled in quantum entanglement and display the results that is predicted with QM. The mixed states would lose their superposition immediately after entanglement. Is this true?
46. ### Operator works on a quantum state yields another state?

The well-known eigen value expression A(a)=a(a) assuming the operator which represents a physical phenomena acts on a quantum state which is represented by an eigen vector, (a) corresponds to an observed value a. But I am wondering if the same operator A can act on (a) and produce another eigen...
47. ### What exactly is a quantum state?

What is a quantum state of a system? I keep hearing it, but I'm not able to fully understand what it means, especially in relation to Bose-Einstein Condensate, and the Pauli Exclusion Principle.
48. ### Quantum State of Cards: Is the Top Card Set Before Viewing?

I can understand that the state of a particle is not known or even determined until it is viewed. What about a deck of cards that have been thoroughly shuffled and not looked at. Can the top card take on the value of any card before they are looked at or are they set at the time they are...
49. ### Quantum state of system before measurement

Hello! If we consider a single-particle system, I understand that the measurement of an observable on this system will collapse the wave function of the system onto an eigenstate of the (observable) operator. Therefore, we know the state of the system immediately after the measurement. But as...
50. ### Vector representation of a Quantum State

In Griffith books of introduction to QM, it say that Quantum State, mathematically represented as a vector. My problem is with understanding what are the components of such a vector. Do I understand it correctly, that, say, in case of a particle in a box, Quantum State, as a vector is a...