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Density matrix

From Wikipedia, the free encyclopedia
Mathematical tool in quantum physics
Part of a series of articles about
Quantum mechanics
iℏddt|Ψ⟩=H^|Ψ⟩{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }

Inquantum mechanics, adensity matrix (ordensity operator) is amatrix used in calculating theprobabilities of the outcomes ofmeasurements performed onphysical systems.[1] It is a generalization of the state vectors orwavefunctions: while those can only representpure states, density matrices can also represent mixed ensembles of states.[2]: 73 [3]: 100  These arise in quantum mechanics in two different situations:

  1. when the preparation of a system can randomly produce different pure states, and thus one must deal with the statistics of the ensemble of possible preparations; and
  2. when one wants to describe a physical system that isentangled with another, without describing their combined state. This case is typical for a system interacting with some environment (e.g.decoherence). In this case, the density matrix of an entangled system differs from that of an ensemble of pure states that, combined, would give the same statistical results upon measurement.

Density matrices are thus crucial tools in areas of quantum mechanics that deal with mixed states (not to be confused withsuperposed states), such asquantum statistical mechanics,open quantum systems andquantum information.

Definition and motivation

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The density matrix is a representation of alinear operator called thedensity operator. The density matrix is obtained from the density operator by a choice of anorthonormalbasis in the underlying space.[4] In practice, the termsdensity matrix anddensity operator are often used interchangeably.

Pick a basis with states|0⟩{\displaystyle |0\rangle },|1⟩{\displaystyle |1\rangle } in a two-dimensionalHilbert space, then the density operator is represented by the matrix(ρij)=(ρ00ρ01ρ10ρ11)=(p0ρ01ρ01∗p1){\displaystyle (\rho _{ij})=\left({\begin{matrix}\rho _{00}&\rho _{01}\\\rho _{10}&\rho _{11}\end{matrix}}\right)=\left({\begin{matrix}p_{0}&\rho _{01}\\\rho _{01}^{*}&p_{1}\end{matrix}}\right)} where the diagonal elements arereal numbers that sum to one (also called populations of the two states|0⟩{\displaystyle |0\rangle },|1⟩{\displaystyle |1\rangle }). The off-diagonal elements arecomplex conjugates of each other (also called coherences); they are restricted in magnitude by the requirement that(ρij){\displaystyle (\rho _{ij})} be apositive semi-definite operator, see below.

A density operator is apositive semi-definite,self-adjoint operator oftrace one acting on theHilbert space of the system.[5][6][7] This definition can be motivated by considering a situation where some pure states|ψj⟩{\displaystyle |\psi _{j}\rangle } (which are not necessarily orthogonal) are prepared with probabilitypj{\displaystyle p_{j}} each.[8] This is known as anensemble of pure states. The probability of obtainingprojective measurement resultm{\displaystyle m} when usingprojectorsΠm{\displaystyle \Pi _{m}} is given by[3]: 99 p(m)=∑jpj⟨ψj|Πm|ψj⟩=tr⁡[Πm(∑jpj|ψj⟩⟨ψj|)],{\displaystyle p(m)=\sum _{j}p_{j}\left\langle \psi _{j}\right|\Pi _{m}\left|\psi _{j}\right\rangle =\operatorname {tr} \left[\Pi _{m}\left(\sum _{j}p_{j}\left|\psi _{j}\right\rangle \left\langle \psi _{j}\right|\right)\right],}which makes thedensity operator, defined asρ=∑jpj|ψj⟩⟨ψj|,{\displaystyle \rho =\sum _{j}p_{j}\left|\psi _{j}\right\rangle \left\langle \psi _{j}\right|,}a convenient representation for the state of this ensemble. It is easy to check that this operator is positive semi-definite, self-adjoint, and has trace one. Conversely, it follows from thespectral theorem that every operator with these properties can be written as∑jpj|ψj⟩⟨ψj|{\textstyle \sum _{j}p_{j}\left|\psi _{j}\right\rangle \left\langle \psi _{j}\right|} for some states|ψj⟩{\displaystyle \left|\psi _{j}\right\rangle } and coefficientspj{\displaystyle p_{j}} that are non-negative and add up to one.[9][3]: 102  However, this representation will not be unique, as shown by theSchrödinger–HJW theorem.

Another motivation for the definition of density operators comes from considering local measurements on entangled states. Let|Ψ⟩{\displaystyle |\Psi \rangle } be a pure entangled state in the composite Hilbert spaceH1⊗H2{\displaystyle {\mathcal {H}}_{1}\otimes {\mathcal {H}}_{2}}. The probability of obtaining measurement resultm{\displaystyle m} when measuring projectorsΠm{\displaystyle \Pi _{m}} on the Hilbert spaceH1{\displaystyle {\mathcal {H}}_{1}} alone is given by[3]: 107 p(m)=⟨Ψ|(Πm⊗I)|Ψ⟩=tr⁡[Πm(tr2⁡|Ψ⟩⟨Ψ|)],{\displaystyle p(m)=\left\langle \Psi \right|\left(\Pi _{m}\otimes I\right)\left|\Psi \right\rangle =\operatorname {tr} \left[\Pi _{m}\left(\operatorname {tr} _{2}\left|\Psi \right\rangle \left\langle \Psi \right|\right)\right],}wheretr2{\displaystyle \operatorname {tr} _{2}} denotes thepartial trace over the Hilbert spaceH2{\displaystyle {\mathcal {H}}_{2}}. This makes the operatorρ=tr2⁡|Ψ⟩⟨Ψ|{\displaystyle \rho =\operatorname {tr} _{2}\left|\Psi \right\rangle \left\langle \Psi \right|}a convenient tool to calculate the probabilities of these local measurements. It is known as thereduced density matrix of|Ψ⟩{\displaystyle |\Psi \rangle } on subsystem 1. It is easy to check that this operator has all the properties of a density operator. Conversely, theSchrödinger–HJW theorem implies that all density operators can be written astr2⁡|Ψ⟩⟨Ψ|{\displaystyle \operatorname {tr} _{2}\left|\Psi \right\rangle \left\langle \Psi \right|} for some state|Ψ⟩{\displaystyle \left|\Psi \right\rangle }.

Pure and mixed states

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A pure quantum state is a state that can not be written as a probabilistic mixture, orconvex combination, of other quantum states.[7] There are several equivalent characterizations of pure states in the language of density operators.[2]: 73  A density operator represents a pure state if and only if:

It is important to emphasize the difference between a probabilistic mixture (i.e. an ensemble) of quantum states and thesuperposition of two states. If an ensemble is prepared to have half of its systems in state|ψ1⟩{\displaystyle |\psi _{1}\rangle } and the other half in|ψ2⟩{\displaystyle |\psi _{2}\rangle }, it can be described by the density matrix:

ρ=12(1001),{\displaystyle \rho ={\frac {1}{2}}{\begin{pmatrix}1&0\\0&1\end{pmatrix}},}

where|ψ1⟩{\displaystyle |\psi _{1}\rangle } and|ψ2⟩{\displaystyle |\psi _{2}\rangle } are assumed orthogonal and of dimension 2, for simplicity. On the other hand, aquantum superposition of these two states with equalprobability amplitudes results in the pure state|ψ⟩=(|ψ1⟩+|ψ2⟩)/2,{\displaystyle |\psi \rangle =(|\psi _{1}\rangle +|\psi _{2}\rangle )/{\sqrt {2}},} with density matrix

|ψ⟩⟨ψ|=12(1111).{\displaystyle |\psi \rangle \langle \psi |={\frac {1}{2}}{\begin{pmatrix}1&1\\1&1\end{pmatrix}}.}

Unlike the probabilistic mixture, this superposition can displayquantum interference.[3]: 81 

In theBloch sphere representation of aqubit, each point on the unit sphere stands for a pure state. All other density matrices correspond to points in the interior.

Geometrically, the set of density operators is aconvex set, and the pure states are theextremal points of that set. The simplest case is that of a two-dimensional Hilbert space, known as aqubit. An arbitrary mixed state for a qubit can be written as alinear combination of thePauli matrices, which together with the identity matrix provide a basis for2×2{\displaystyle 2\times 2}self-adjoint matrices:[10]: 126 

ρ=12(I+rxσx+ryσy+rzσz),{\displaystyle \rho ={\frac {1}{2}}\left(I+r_{x}\sigma _{x}+r_{y}\sigma _{y}+r_{z}\sigma _{z}\right),}

where the real numbers(rx,ry,rz){\displaystyle (r_{x},r_{y},r_{z})} are the coordinates of a point within theunit ball and

σx=(0110),σy=(0−ii0),σz=(100−1).{\displaystyle \sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\quad \sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.}

Points withrx2+ry2+rz2=1{\displaystyle r_{x}^{2}+r_{y}^{2}+r_{z}^{2}=1} represent pure states, while mixed states are represented by points in the interior. This is known as theBloch sphere picture of qubit state space.

Example: light polarization

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The incandescent light bulb (1) emits completely random polarized photons (2) with mixed state density matrix:
After passing through vertical plane polarizer (3), the remaining photons are all vertically polarized (4) and have pure state density matrix:

An example of pure and mixed states islight polarization. An individualphotoncan be described as having right or leftcircular polarization, described by the orthogonal quantum states|R⟩{\displaystyle |\mathrm {R} \rangle } and|L⟩{\displaystyle |\mathrm {L} \rangle } or asuperposition of the two: it can be in any stateα|R⟩+β|L⟩{\displaystyle \alpha |\mathrm {R} \rangle +\beta |\mathrm {L} \rangle } (with|α|2+|β|2=1{\displaystyle |\alpha |^{2}+|\beta |^{2}=1}), corresponding tolinear,circular, orelliptical polarization. Consider now a vertically polarized photon, described by the state|V⟩=(|R⟩+|L⟩)/2{\displaystyle |\mathrm {V} \rangle =(|\mathrm {R} \rangle +|\mathrm {L} \rangle )/{\sqrt {2}}}. If we pass it through acircular polarizer that allows either only|R⟩{\displaystyle |\mathrm {R} \rangle } polarized light, or only|L⟩{\displaystyle |\mathrm {L} \rangle } polarized light, half of the photons are absorbed in both cases. This may make itseem like half of the photons are in state|R⟩{\displaystyle |\mathrm {R} \rangle } and the other half in state|L⟩{\displaystyle |\mathrm {L} \rangle }, but this is not correct: if we pass(|R⟩+|L⟩)/2{\displaystyle (|\mathrm {R} \rangle +|\mathrm {L} \rangle )/{\sqrt {2}}} through alinear polarizer there is no absorption whatsoever, but if we pass either state|R⟩{\displaystyle |\mathrm {R} \rangle } or|L⟩{\displaystyle |\mathrm {L} \rangle } half of the photons are absorbed.

Unpolarized light (such as the light from anincandescent light bulb) cannot be described asany state of the formα|R⟩+β|L⟩{\displaystyle \alpha |\mathrm {R} \rangle +\beta |\mathrm {L} \rangle } (linear, circular, or elliptical polarization). Unlike polarized light, it passes through a polarizer with 50% intensity loss whatever the orientation of the polarizer; and it cannot be made polarized by passing it through anywave plate. However, unpolarized lightcan be described as a statistical ensemble, e. g. as each photon having either|R⟩{\displaystyle |\mathrm {R} \rangle } polarization or|L⟩{\displaystyle |\mathrm {L} \rangle } polarization with probability 1/2. The same behavior would occur if each photon had either vertical polarization|V⟩{\displaystyle |\mathrm {V} \rangle } or horizontal polarization|H⟩{\displaystyle |\mathrm {H} \rangle } with probability 1/2. These two ensembles are completely indistinguishable experimentally, and therefore they are considered the same mixed state. For this example of unpolarized light, the density operator equals[2]: 75 

ρ=12|R⟩⟨R|+12|L⟩⟨L|=12|H⟩⟨H|+12|V⟩⟨V|=12(1001).{\displaystyle \rho ={\frac {1}{2}}|\mathrm {R} \rangle \langle \mathrm {R} |+{\frac {1}{2}}|\mathrm {L} \rangle \langle \mathrm {L} |={\frac {1}{2}}|\mathrm {H} \rangle \langle \mathrm {H} |+{\frac {1}{2}}|\mathrm {V} \rangle \langle \mathrm {V} |={\frac {1}{2}}{\begin{pmatrix}1&0\\0&1\end{pmatrix}}.}

There are also other ways to generate unpolarized light: one possibility is to introduce uncertainty in the preparation of the photon, for example, passing it through abirefringent crystal with a rough surface, so that slightly different parts of the light beam acquire different polarizations. Another possibility is using entangled states: a radioactive decay can emit two photons traveling in opposite directions, in the quantum state(|R,L⟩+|L,R⟩)/2{\displaystyle (|\mathrm {R} ,\mathrm {L} \rangle +|\mathrm {L} ,\mathrm {R} \rangle )/{\sqrt {2}}}. The joint state of the two photonstogether is pure, but the density matrix for each photon individually, found by taking the partial trace of the joint density matrix, is completely mixed.[3]: 106 

Equivalent ensembles and purifications

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Main article:Schrödinger–HJW theorem

A given density operator does not uniquely determine which ensemble of pure states gives rise to it; in general there are infinitely many different ensembles generating the same density matrix.[11] Those cannot be distinguished by any measurement.[12] The equivalent ensembles can be completely characterized: let{pj,|ψj⟩}{\displaystyle \{p_{j},|\psi _{j}\rangle \}} be an ensemble. Then for any complex matrixU{\displaystyle U} such thatU†U=I{\displaystyle U^{\dagger }U=I} (apartial isometry), the ensemble{qi,|φi⟩}{\displaystyle \{q_{i},|\varphi _{i}\rangle \}} defined by

qi|φi⟩=∑jUijpj|ψj⟩{\displaystyle {\sqrt {q_{i}}}\left|\varphi _{i}\right\rangle =\sum _{j}U_{ij}{\sqrt {p_{j}}}\left|\psi _{j}\right\rangle }

will give rise to the same density operator, and all equivalent ensembles are of this form.

A closely related fact is that a given density operator has infinitely many differentpurifications, which are pure states that generate the density operator when a partial trace is taken. Let

ρ=∑jpj|ψj⟩⟨ψj|{\displaystyle \rho =\sum _{j}p_{j}|\psi _{j}\rangle \langle \psi _{j}|}

be the density operator generated by the ensemble{pj,|ψj⟩}{\displaystyle \{p_{j},|\psi _{j}\rangle \}}, with states|ψj⟩{\displaystyle |\psi _{j}\rangle } not necessarily orthogonal. Then for all partial isometriesU{\displaystyle U} we have that

|Ψ⟩=∑jpj|ψj⟩U|aj⟩{\displaystyle |\Psi \rangle =\sum _{j}{\sqrt {p_{j}}}|\psi _{j}\rangle U|a_{j}\rangle }

is a purification ofρ{\displaystyle \rho }, where|aj⟩{\displaystyle |a_{j}\rangle } is an orthogonal basis, and furthermore all purifications ofρ{\displaystyle \rho } are of this form.

Measurement

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LetA{\displaystyle A} be anobservable of the system, and suppose the ensemble is in a mixed state such that each of the pure states|ψj⟩{\displaystyle \textstyle |\psi _{j}\rangle } occurs with probabilitypj{\displaystyle p_{j}}. Then the corresponding density operator equals

ρ=∑jpj|ψj⟩⟨ψj|.{\displaystyle \rho =\sum _{j}p_{j}|\psi _{j}\rangle \langle \psi _{j}|.}

Theexpectation value of themeasurement can be calculated by extending from the case of pure states:

⟨A⟩=∑jpj⟨ψj|A|ψj⟩=∑jpjtr⁡(|ψj⟩⟨ψj|A)=tr⁡(∑jpj|ψj⟩⟨ψj|A)=tr⁡(ρA),{\displaystyle \langle A\rangle =\sum _{j}p_{j}\langle \psi _{j}|A|\psi _{j}\rangle =\sum _{j}p_{j}\operatorname {tr} \left(|\psi _{j}\rangle \langle \psi _{j}|A\right)=\operatorname {tr} \left(\sum _{j}p_{j}|\psi _{j}\rangle \langle \psi _{j}|A\right)=\operatorname {tr} (\rho A),}

wheretr{\displaystyle \operatorname {tr} } denotestrace. Thus, the familiar expression⟨A⟩=⟨ψ|A|ψ⟩{\displaystyle \langle A\rangle =\langle \psi |A|\psi \rangle } for pure states is replaced by

⟨A⟩=tr⁡(ρA){\displaystyle \langle A\rangle =\operatorname {tr} (\rho A)}

for mixed states.[2]: 73 

Moreover, ifA{\displaystyle A} has spectral resolution

A=∑iaiPi,{\displaystyle A=\sum _{i}a_{i}P_{i},}

wherePi{\displaystyle P_{i}} is theprojection operator into theeigenspace corresponding to eigenvalueai{\displaystyle a_{i}}, the post-measurement density operator is given by[13][14]

ρi′=PiρPitr⁡[ρPi]{\displaystyle \rho _{i}'={\frac {P_{i}\rho P_{i}}{\operatorname {tr} \left[\rho P_{i}\right]}}}

when outcomei is obtained. In the case where the measurement result is not known the ensemble is instead described by

ρ′=∑iPiρPi.{\displaystyle \;\rho '=\sum _{i}P_{i}\rho P_{i}.}

If one assumes that the probabilities of measurement outcomes are linear functions of the projectorsPi{\displaystyle P_{i}}, then they must be given by the trace of the projector with a density operator.Gleason's theorem shows that in Hilbert spaces of dimension 3 or larger the assumption of linearity can be replaced with an assumption ofnon-contextuality.[15] This restriction on the dimension can be removed by assuming non-contextuality forPOVMs as well,[16][17] but this has been criticized as physically unmotivated.[18]

Entropy

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Thevon Neumann entropyS{\displaystyle S} of a mixture can be expressed in terms of the eigenvalues ofρ{\displaystyle \rho } or in terms of thetrace andlogarithm of the density operatorρ{\displaystyle \rho }. Sinceρ{\displaystyle \rho } is a positive semi-definite operator, it has aspectral decomposition such thatρ=∑iλi|φi⟩⟨φi|{\displaystyle \rho =\textstyle \sum _{i}\lambda _{i}|\varphi _{i}\rangle \langle \varphi _{i}|}, where|φi⟩{\displaystyle |\varphi _{i}\rangle } are orthonormal vectors,λi≥0{\displaystyle \lambda _{i}\geq 0}, and∑λi=1{\displaystyle \textstyle \sum \lambda _{i}=1}. Then the entropy of a quantum system with density matrixρ{\displaystyle \rho } is

S=−∑iλiln⁡λi=−tr⁡(ρln⁡ρ).{\displaystyle S=-\sum _{i}\lambda _{i}\ln \lambda _{i}=-\operatorname {tr} (\rho \ln \rho ).}

This definition implies that the von Neumann entropy of any pure state is zero.[19]: 217  Ifρi{\displaystyle \rho _{i}} are states that have support on orthogonal subspaces, then the von Neumann entropy of a convex combination of these states,

ρ=∑ipiρi,{\displaystyle \rho =\sum _{i}p_{i}\rho _{i},}

is given by the von Neumann entropies of the statesρi{\displaystyle \rho _{i}} and theShannon entropy of the probability distributionpi{\displaystyle p_{i}}:

S(ρ)=H(pi)+∑ipiS(ρi).{\displaystyle S(\rho )=H(p_{i})+\sum _{i}p_{i}S(\rho _{i}).}

When the statesρi{\displaystyle \rho _{i}} do not have orthogonal supports, the sum on the right-hand side is strictly greater than the von Neumann entropy of the convex combinationρ{\displaystyle \rho }.[3]: 518 

Given a density operatorρ{\displaystyle \rho } and a projective measurement as in the previous section, the stateρ′{\displaystyle \rho '} defined by the convex combination

ρ′=∑iPiρPi,{\displaystyle \rho '=\sum _{i}P_{i}\rho P_{i},}

which can be interpreted as the state produced by performing the measurement but not recording which outcome occurred,[10]: 159  has a von Neumann entropy larger than that ofρ{\displaystyle \rho }, except ifρ=ρ′{\displaystyle \rho =\rho '}. It is however possible for theρ′{\displaystyle \rho '} produced by ageneralized measurement, orPOVM, to have a lower von Neumann entropy thanρ{\displaystyle \rho }.[3]: 514 

Von Neumann equation for time evolution

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See also:Liouville's theorem (Hamiltonian) § Quantum Liouville equation

Just as theSchrödinger equation describes how pure states evolve in time, thevon Neumann equation (also known as theLiouville–von Neumann equation) describes how a density operator evolves in time. The von Neumann equation dictates that[20][21][22]

iℏddtρ=[H,ρ] ,{\displaystyle i\hbar {\frac {d}{dt}}\rho =[H,\rho ]~,}

where the brackets denote acommutator.

This equation only holds when the density operator is taken to be in theSchrödinger picture, even though this equation seems at first look to emulate the Heisenberg equation of motion in theHeisenberg picture, with a crucial sign difference:

iℏddtAH=−[HH,AH] ,{\displaystyle i\hbar {\frac {d}{dt}}A_{\text{H}}=-[H_{\text{H}},A_{\text{H}}]~,}

whereAH(t){\displaystyle A_{\text{H}}(t)} is someHeisenberg picture operator; but in this picture the density matrix isnot time-dependent, and the relative sign ensures that the time derivative of the expected value⟨A⟩{\displaystyle \langle A\rangle } comes outthe same as in the Schrödinger picture.[7]

If the Hamiltonian is time-independent, the von Neumann equation can be easily solved to yield

ρ(t)=e−iHt/ℏρ(0)eiHt/ℏ.{\displaystyle \rho (t)=e^{-iHt/\hbar }\rho (0)e^{iHt/\hbar }.}

For a more general Hamiltonian, ifG(t){\displaystyle G(t)} is the wavefunction propagator over some interval, then the time evolution of the density matrix over that same interval is given by

ρ(t)=G(t)ρ(0)G(t)†.{\displaystyle \rho (t)=G(t)\rho (0)G(t)^{\dagger }.}

If one enters theinteraction picture, choosing to focus on some componentH1{\displaystyle H_{1}} of the HamiltonianH=H0+H1{\displaystyle H=H_{0}+H_{1}}, the equation for the evolution of the interaction-picture density operatorρI(t){\displaystyle \rho _{\,\mathrm {I} }(t)} possesses identical structure to the von Neumann equation, except the Hamiltonian must also be transformed into the new picture:

iℏddtρI(t)=[H1,I(t),ρI(t)],{\displaystyle {\displaystyle i\hbar {\frac {d}{dt}}\rho _{\text{I}}(t)=[H_{1,{\text{I}}}(t),\rho _{\text{I}}(t)],}}

whereH1,I(t)=eiH0t/ℏH1e−iH0t/ℏ{\displaystyle {\displaystyle H_{1,{\text{I}}}(t)=e^{iH_{0}t/\hbar }H_{1}e^{-iH_{0}t/\hbar }}}.

Wigner functions and classical analogies

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Main article:Phase-space formulation

The density matrix operator may also be realized inphase space. Under theWigner map, the density matrix transforms into the equivalentWigner function,

W(x,p) =def 1πℏ∫−∞∞ψ∗(x+y)ψ(x−y)e2ipy/ℏdy.{\displaystyle W(x,p)\,\ {\stackrel {\mathrm {def} }{=}}\ \,{\frac {1}{\pi \hbar }}\int _{-\infty }^{\infty }\psi ^{*}(x+y)\psi (x-y)e^{2ipy/\hbar }\,dy.}

The equation for the time evolution of the Wigner function, known asMoyal equation, is then the Wigner-transform of the above von Neumann equation,

∂W(x,p,t)∂t=−{{W(x,p,t),H(x,p)}},{\displaystyle {\frac {\partial W(x,p,t)}{\partial t}}=-\{\{W(x,p,t),H(x,p)\}\},}

whereH(x,p){\displaystyle H(x,p)} is the Hamiltonian, and{{⋅,⋅}}{\displaystyle \{\{\cdot ,\cdot \}\}} is theMoyal bracket, the transform of the quantumcommutator.

The evolution equation for the Wigner function is then analogous to that of its classical limit, theLiouville equation ofclassical physics. In the limit of a vanishing Planck constantℏ{\displaystyle \hbar },W(x,p,t){\displaystyle W(x,p,t)} reduces to the classical Liouville probability density function inphase space.

Example applications

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Density matrices are a basic tool of quantum mechanics, and appear at least occasionally in almost any type of quantum-mechanical calculation. Some specific examples where density matrices are especially helpful and common are as follows:

C*-algebraic formulation of states

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It is now generally accepted that the description of quantum mechanics in which all self-adjoint operators represent observables is untenable.[28][29] For this reason, observables are identified with elements of an abstractC*-algebraA (that is one without a distinguished representation as an algebra of operators) andstates are positivelinear functionals onA. However, by using theGNS construction, we can recover Hilbert spaces that realizeA as a subalgebra of operators.

Geometrically, a pure state on a C*-algebraA is a state that is an extreme point of the set of all states onA. By properties of the GNS construction these states correspond toirreducible representations ofA.

The states of the C*-algebra ofcompact operatorsK(H) correspond exactly to the density operators, and therefore the pure states ofK(H) are exactly the pure states in the sense of quantum mechanics.

The C*-algebraic formulation can be seen to include both classical and quantum systems. When the system is classical, the algebra of observables become an abelian C*-algebra. In that case the states become probability measures.

History

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This formalism of the operators and matrices was introduced in 1927 byJohn von Neumann[30] and independently, but less systematically, byLev Landau[31] and later in 1946 byFelix Bloch.[32] Von Neumann introduced a matrix in order to develop both quantum statistical mechanics and a theory of quantum measurements. The termdensity was introduced by Dirac in 1931 when he used von Neumann's operator to calculate electron density clouds.[33][34]

Nowadays the term "density matrix" obtained a significance of its own, and corresponds to a classicalphase-spaceprobability measure (probability distribution of position and momentum) in classicalstatistical mechanics, which was introduced byEugene Wigner in 1932.[5]

In contrast, the motivation that inspired Landau was the impossibility of describing a subsystem of a composite quantum system by a state vector.[31]

See also

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Notes and references

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  1. ^Shankar, Ramamurti (2014).Principles of quantum mechanics (2. ed., [19. corrected printing] ed.). New York, NY: Springer.ISBN 978-0-306-44790-7.
  2. ^abcdPeres, Asher (1995).Quantum Theory: Concepts and Methods. Kluwer.ISBN 978-0-7923-3632-7.OCLC 901395752.
  3. ^abcdefghNielsen, Michael; Chuang, Isaac (2000),Quantum Computation and Quantum Information,Cambridge University Press,ISBN 978-0-521-63503-5.
  4. ^Ballentine, Leslie (2009). "Density Matrix".Compendium of Quantum Physics. Berlin, Heidelberg: Springer Berlin Heidelberg. p. 166.doi:10.1007/978-3-540-70626-7_51.ISBN 978-3-540-70622-9.
  5. ^abFano, U. (1957). "Description of States in Quantum Mechanics by Density Matrix and Operator Techniques".Reviews of Modern Physics.29 (1):74–93.Bibcode:1957RvMP...29...74F.doi:10.1103/RevModPhys.29.74.
  6. ^Holevo, Alexander S. (2001).Statistical Structure of Quantum Theory. Lecture Notes in Physics. Springer.ISBN 3-540-42082-7.OCLC 318268606.
  7. ^abcHall, Brian C. (2013). "Systems and Subsystems, Multiple Particles".Quantum Theory for Mathematicians. Graduate Texts in Mathematics. Vol. 267. pp. 419–440.doi:10.1007/978-1-4614-7116-5_19.ISBN 978-1-4614-7115-8.
  8. ^Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2019).Quantum Mechanics, Volume 1. Weinheim, Germany: John Wiley & Sons. pp. 301–303.ISBN 978-3-527-34553-3..
  9. ^Davidson, Ernest Roy (1976).Reduced Density Matrices in Quantum Chemistry.Academic Press, London.
  10. ^abWilde, Mark M. (2017).Quantum Information Theory (2nd ed.). Cambridge University Press.arXiv:1106.1445.doi:10.1017/9781316809976.001.ISBN 978-1-107-17616-4.OCLC 973404322.S2CID 2515538.
  11. ^Kirkpatrick, K. A. (February 2006). "The Schrödinger-HJW Theorem".Foundations of Physics Letters.19 (1):95–102.arXiv:quant-ph/0305068.Bibcode:2006FoPhL..19...95K.doi:10.1007/s10702-006-1852-1.ISSN 0894-9875.S2CID 15995449.
  12. ^Ochs, Wilhelm (1981-11-01). "Some comments on the concept of state in quantum mechanics".Erkenntnis.16 (3):339–356.doi:10.1007/BF00211375.ISSN 1572-8420.S2CID 119980948.
  13. ^Lüders, Gerhart (1950). "Über die Zustandsänderung durch den Messprozeß".Annalen der Physik.443 (5–8): 322.Bibcode:1950AnP...443..322L.doi:10.1002/andp.19504430510. Translated by K. A. Kirkpatrick asLüders, Gerhart (2006-04-03). "Concerning the state-change due to the measurement process".Annalen der Physik.15 (9):663–670.arXiv:quant-ph/0403007.Bibcode:2006AnP...518..663L.doi:10.1002/andp.200610207.S2CID 119103479.
  14. ^Busch, Paul; Lahti, Pekka (2009), Greenberger, Daniel; Hentschel, Klaus; Weinert, Friedel (eds.), "Lüders Rule",Compendium of Quantum Physics, Springer Berlin Heidelberg, pp. 356–358,doi:10.1007/978-3-540-70626-7_110,ISBN 978-3-540-70622-9
  15. ^Gleason, Andrew M. (1957)."Measures on the closed subspaces of a Hilbert space".Indiana University Mathematics Journal.6 (4):885–893.doi:10.1512/iumj.1957.6.56050.MR 0096113.
  16. ^Busch, Paul (2003). "Quantum States and Generalized Observables: A Simple Proof of Gleason's Theorem".Physical Review Letters.91 (12) 120403.arXiv:quant-ph/9909073.Bibcode:2003PhRvL..91l0403B.doi:10.1103/PhysRevLett.91.120403.PMID 14525351.S2CID 2168715.
  17. ^Caves, Carlton M.; Fuchs, Christopher A.; Manne, Kiran K.; Renes, Joseph M. (2004). "Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements".Foundations of Physics.34 (2):193–209.arXiv:quant-ph/0306179.Bibcode:2004FoPh...34..193C.doi:10.1023/B:FOOP.0000019581.00318.a5.S2CID 18132256.
  18. ^Andrzej Grudka; Paweł Kurzyński (2008). "Is There Contextuality for a Single Qubit?".Physical Review Letters.100 (16) 160401.arXiv:0705.0181.Bibcode:2008PhRvL.100p0401G.doi:10.1103/PhysRevLett.100.160401.PMID 18518167.S2CID 13251108.
  19. ^Rieffel, Eleanor G.; Polak, Wolfgang H. (2011-03-04).Quantum Computing: A Gentle Introduction. MIT Press.ISBN 978-0-262-01506-6.
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  21. ^Schwabl, Franz (2002),Statistical mechanics, Springer, p. 16,ISBN 978-3-540-43163-3
  22. ^Müller-Kirsten, Harald J.W. (2008),Classical Mechanics and Relativity, World Scientific, pp. 175–179,ISBN 978-981-283-251-1
  23. ^Kardar, Mehran (2007).Statistical Physics of Particles.Cambridge University Press.ISBN 978-0-521-87342-0.OCLC 860391091.
  24. ^Schlosshauer, M. (2019). "Quantum Decoherence".Physics Reports.831:1–57.arXiv:1911.06282.Bibcode:2019PhR...831....1S.doi:10.1016/j.physrep.2019.10.001.S2CID 208006050.
  25. ^Granade, Christopher; Combes, Joshua; Cory, D. G. (2016-01-01). "Practical Bayesian tomography".New Journal of Physics.18 (3) 033024.arXiv:1509.03770.Bibcode:2016NJPh...18c3024G.doi:10.1088/1367-2630/18/3/033024.ISSN 1367-2630.S2CID 88521187.
  26. ^Ardila, Luis; Heyl, Markus; Eckardt, André (28 December 2018). "Measuring the Single-Particle Density Matrix for Fermions and Hard-Core Bosons in an Optical Lattice".Physical Review Letters.121 (260401): 6.arXiv:1806.08171.Bibcode:2018PhRvL.121z0401P.doi:10.1103/PhysRevLett.121.260401.PMID 30636128.S2CID 51684413.
  27. ^Kittel, Charles (1963).Quantum theory of solids. New York: Wiley. p. 101.
  28. ^See appendix,Mackey, George Whitelaw (1963),Mathematical Foundations of Quantum Mechanics, Dover Books on Mathematics, New York:Dover Publications,ISBN 978-0-486-43517-6{{citation}}:ISBN / Date incompatibility (help)
  29. ^Emch, Gerard G. (1972),Algebraic methods in statistical mechanics and quantum field theory,Wiley-Interscience,ISBN 978-0-471-23900-0
  30. ^von Neumann, John (1927),"Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik",Göttinger Nachrichten,1:245–272
  31. ^ab"The Damping Problem in Wave Mechanics (1927)".Collected Papers of L.D. Landau. 1965. pp. 8–18.doi:10.1016/B978-0-08-010586-4.50007-9.ISBN 978-0-08-010586-4.
  32. ^Fano, Ugo (1995). "Density matrices as polarization vectors".Rendiconti Lincei.6 (2):123–130.doi:10.1007/BF03001661.S2CID 128081459.
  33. ^Dirac, P. A. M. (July 1930)."Note on Exchange Phenomena in the Thomas Atom".Mathematical Proceedings of the Cambridge Philosophical Society.26 (3):376–385.doi:10.1017/S0305004100016108.ISSN 0305-0041.
  34. ^Dirac, P. A. M. (April 1931)."Note on the Interpretation of the Density Matrix in the Many-Electron Problem".Mathematical Proceedings of the Cambridge Philosophical Society.27 (2):240–243.doi:10.1017/S0305004100010343.ISSN 0305-0041.
Matrix classes
Explicitly constrained entries
Constant
Conditions oneigenvalues or eigenvectors
Satisfying conditions onproducts orinverses
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