von Neumann entropy

{{main|Density matrix}}

In quantum mechanics, probabilities for the outcomes of experiments made upon a system are calculated from the quantum state describing that system. Each physical system is associated with a vector space, or more specifically a Hilbert space. The dimension of the Hilbert space may be infinite, as it is for the space of square-integrable functions on a line, which is used to define the quantum physics of a continuous degree of freedom. Alternatively, the Hilbert space may be finite-dimensional, as occurs for spin degrees of freedom. A density operator, the mathematical representation of a quantum state, is a positive semi-definite, self-adjoint operator of trace one acting on the Hilbert space of the system.{{cite journal |doi=10.1103/RevModPhys.29.74 |title=Description of States in Quantum Mechanics by Density Matrix and Operator Techniques |journal=Reviews of Modern Physics |volume=29 |issue=1 |pages=74–93 |year=1957 |last1=Fano |first1=U. |author-link=Ugo Fano |bibcode=1957RvMP...29...74F }}{{sfn|Holevo|2001|pages=1,15}}{{cite book |doi=10.1007/978-1-4614-7116-5_19 |chapter=Systems and Subsystems, Multiple Particles |title=Quantum Theory for Mathematicians |volume=267 |pages=419–440 |series=Graduate Texts in Mathematics |year=2013 |last1=Hall |first1=Brian C. |isbn=978-1-4614-7115-8 |publisher=Springer}} A density operator that is a rank-1 projection is known as a pure quantum state, and all quantum states that are not pure are designated mixed. Pure states are also known as wavefunctions. Assigning a pure state to a quantum system implies certainty about the outcome of some measurement on that system (i.e., $P(x)\; =\; 1$ for some outcome $x$). The state space of a quantum system is the set of all states, pure and mixed, that can be assigned to it. For any system, the state space is a convex set: Any mixed state can be written as a convex combination of pure states, though not in a unique way.{{Cite journal|last=Kirkpatrick |first=K. A. |date=February 2006 |title=The Schrödinger-HJW Theorem |journal=Foundations of Physics Letters |volume=19 |issue=1 |pages=95–102 |doi=10.1007/s10702-006-1852-1 |issn=0894-9875 |arxiv=quant-ph/0305068|bibcode=2006FoPhL..19...95K |s2cid=15995449 }} The von Neumann entropy quantifies the extent to which a state is mixed.{{sfn|Holevo|2001|page=15}}

The prototypical example of a finite-dimensional Hilbert space is a qubit, a quantum system whose Hilbert space is 2-dimensional. An arbitrary state for a qubit can be written as a linear combination of the Pauli matrices, which provide a basis for $2\; \backslash times\; 2$ self-adjoint matrices:{{sfnm|1a1=Wilde|1y=2017|1p=126 |2a1=Zwiebach|2y=2022|2at=§22.2}}

$$\backslash rho\; =\; \backslash tfrac\{1\}\{2\}\backslash left(I\; +\; r\_x\; \backslash sigma\_x\; +\; r\_y\; \backslash sigma\_y\; +\; r\_z\; \backslash sigma\_z\backslash right),$$

where the real numbers $(r\_x,\; r\_y,\; r\_z)$ are the coordinates of a point within the unit ball and

$$$$

\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} .

The von Neumann entropy vanishes when $\backslash rho$ is a pure state, i.e., when the point $(r\_x,\; r\_y,\; r\_z)$ lies on the surface of the unit ball, and it attains its maximum value when $\backslash rho$ is the maximally mixed state, which is given by $r\_x\; =\; r\_y\; =\; r\_z\; =\; 0$.{{sfnm|1a1=Rieffel|1a2=Polak|1y=2011|1pp=216–217 |2a1=Zwiebach|2y=2022|2at=§22.2}}

Some properties of the von Neumann entropy:

• {{math|S(ρ)}} is zero if and only if {{math|ρ}} represents a pure state.{{sfnm|1a1=Holevo|1y=2001|1p=15 |2a1=Nielsen|2a2=Chuang|2y=2010|2p=513 |3a1=Rau|3y=2017|3p=32}}
• {{math|S(ρ)}} is maximal and equal to $\backslash ln\; N$ for a maximally mixed state, {{math|N}} being the dimension of the Hilbert space.{{sfnm|1a1=Holevo|1y=2001|1p=15 |2a1=Nielsen|2a2=Chuang|2y=2010|2p=513}}
• {{math|S(ρ)}} is invariant under changes in the basis of {{math|ρ}}, that is, {{math|S(ρ) {{=}} S(UρU^†)}}, with {{math|U}} a unitary transformation.{{sfnm|1a1=Rau|1y=2017|1p=32 |2a1=Wilde|2y=2017|2at=§11.1.1}}
• {{math|S(ρ)}} is concave, that is, given a collection of positive numbers {{math|λ_i}} which sum to unity ($\backslash Sigma\_i\; \backslash lambda\_i\; =\; 1$) and density operators {{math|ρ_i}}, we have{{sfnm|1a1=Holevo|1y=2001|1p=15 |2a1=Nielsen|2a2=Chuang|2y=2010|2p=516 |3a1=Rau|3y=2017|3p=32}}

$$S\backslash bigg(\backslash sum\_\{i=1\}^k\; \backslash lambda\_i\; \backslash rho\_i\; \backslash bigg)\; \backslash geq\; \backslash sum\_\{i=1\}^k\; \backslash lambda\_i\; S(\backslash rho\_i).$$

• {{math|S(ρ)}} is additive for independent systems. Given two density matrices {{math| ρ_A , ρ_B}} describing independent systems A and B, we have{{sfnm|1a1=Nielsen|1a2=Chuang|1y=2010|1p=514 |2a1=Rau|2y=2017|2pp=32–33}}

$$S(\backslash rho\_A\; \backslash otimes\; \backslash rho\_B)=S(\backslash rho\_A)+S(\backslash rho\_B).$$

• {{math|S(ρ)}} is strongly subadditive. That is, for any three systems A, B, and C:{{sfn|Nielsen|Chuang|2010|p=519}}

$$S(\backslash rho\_\{ABC\})\; +\; S(\backslash rho\_\{B\})\; \backslash leq\; S(\backslash rho\_\{AB\})\; +\; S(\backslash rho\_\{BC\}).$$

:This automatically means that {{math|S(ρ)}} is subadditive:

$$S(\backslash rho\_\{AC\})\; \backslash leq\; S(\backslash rho\_\{A\})\; +S(\backslash rho\_\{C\}).$$

Below, the concept of subadditivity is discussed, followed by its generalization to strong subadditivity.

If {{math|ρ_A, ρ_B}} are the reduced density matrices of the general state {{math|ρ_AB}}, then

$$\backslash left|\; S(\backslash rho\_A)\; -\; S(\backslash rho\_B)\; \backslash right|\; \backslash leq\; S(\backslash rho\_\{AB\})\; \backslash leq\; S(\backslash rho\_A)\; +\; S(\backslash rho\_B)\; .$$

The right hand inequality is known as subadditivity, and the left is sometimes known as the triangle inequality.{{sfn|Nielsen|Chuang|2010|pp=515–516}} While in Shannon's theory the entropy of a composite system can never be lower than the entropy of any of its parts, in quantum theory this is not the case; i.e., it is possible that {{math| S(ρ_AB) {{=}} 0}}, while {{math| S(ρ_A) {{=}} S(ρ_B) > 0}}. This is expressed by saying that the Shannon entropy is monotonic but the von Neumann entropy is not.{{sfn|Bengtsson|Życzkowski|2017|p=361}} For example, take the Bell state of two spin-1/2 particles:

$$\backslash left|\; \backslash psi\; \backslash right\backslash rangle\; =\; \backslash left|\; \backslash uparrow\; \backslash downarrow\; \backslash right\backslash rangle\; +\; \backslash left|\; \backslash downarrow\; \backslash uparrow\; \backslash right\backslash rangle\; .$$

This is a pure state with zero entropy, but each spin has maximum entropy when considered individually, because its reduced density matrix is the maximally mixed state. This indicates that it is an entangled state;{{sfnm|1a1=Rieffel|1a2=Polak|1y=2011|1p=220 |2a1=Rau|2y=2021|2p=236}} the use of entropy as an entanglement measure is discussed further below.

{{main|Strong subadditivity of quantum entropy}}

The von Neumann entropy is also strongly subadditive.{{sfn|Bengtsson|Życzkowski|2017|p=364}} Given three Hilbert spaces, A, B, C,

$$S(\backslash rho\_\{ABC\})\; +\; S(\backslash rho\_\{B\})\; \backslash leq\; S(\backslash rho\_\{AB\})\; +\; S(\backslash rho\_\{BC\}).$$

By using the proof technique that establishes the left side of the triangle inequality above, one can show that the strong subadditivity inequality is equivalent to the following inequality:

$$S(\backslash rho\_\{A\})\; +\; S(\backslash rho\_\{C\})\; \backslash leq\; S(\backslash rho\_\{AB\})\; +\; S(\backslash rho\_\{BC\})$$

where {{math| ρ_AB}}, etc. are the reduced density matrices of a density matrix {{math|ρ_ABC}}.{{cite journal|first=Mary Beth |last=Ruskai |author-link=Mary Beth Ruskai |title=Inequalities for Quantum Entropy: A Review with Conditions for Equality |journal=Journal of Mathematical Physics |volume=43 |pages=4358–4375 |year=2002 |arxiv=quant-ph/0205064 |doi=10.1063/1.1497701}} If we apply ordinary subadditivity to the left side of this inequality, we then find

$$S(\backslash rho\_\{AC\})\; \backslash leq\; S(\backslash rho\_\{AB\})\; +\; S(\backslash rho\_\{BC\}).$$

By symmetry, for any tripartite state {{math| ρ_ABC}}, each of the three numbers {{math|S(ρ_AB), S(ρ_BC), S(ρ_AC)}} is less than or equal to the sum of the other two.{{cite journal|first=Elliott H. |last=Lieb |author-link=Elliott H. Lieb |title=Some convexity and subadditivity properties of entropy |journal=Bulletin of the American Mathematical Society |volume=81 |number=1 |date=January 1975 |doi=10.1090/S0002-9904-1975-13621-4 |mr=0356797|doi-access=free }}

Given a quantum state and a specification of a quantum measurement, we can calculate the probabilities for the different possible results of that measurement, and thus we can find the Shannon entropy of that probability distribution. A quantum measurement can be specified mathematically as a positive operator valued measure, or POVM.{{cite journal|last1=Peres |first1=Asher |author-link1=Asher Peres |last2=Terno |first2=Daniel R. |title=Quantum information and relativity theory |journal=Reviews of Modern Physics |volume=76 |number=1 |year=2004 |pages=93–123 |arxiv=quant-ph/0212023 |doi=10.1103/RevModPhys.76.93 |bibcode=2004RvMP...76...93P|s2cid=7481797 }} In the simplest case, a system with a finite-dimensional Hilbert space and measurement with a finite number of outcomes, a POVM is a set of positive semi-definite matrices $\backslash \{F\_i\backslash \}$ on the Hilbert space that sum to the identity matrix,{{sfn|Nielsen|Chuang|2010|p=90}}

$$\backslash sum\_\{i=1\}^n\; F\_i\; =\; \backslash operatorname\{I\}.$$

The POVM element $F\_i$ is associated with the measurement outcome $i$, such that the probability of obtaining it when making a measurement on the quantum state $\backslash rho$ is given by

$$\backslash text\{Prob\}(i)\; =\; \backslash operatorname\{tr\}(\backslash rho\; F\_i).$$

A POVM is rank-1 if all of the elements are proportional to rank-1 projection operators. The von Neumann entropy is the minimum achievable Shannon entropy, where the minimization is taken over all rank-1 POVMs.{{sfn|Wilde|2017|at=§11.1.2}}

{{see also|Holevo's theorem}}

If {{math|ρ_i}} are density operators and {{math|λ_i}} is a collection of positive numbers which sum to unity ($\backslash Sigma\_i\; \backslash lambda\_i\; =\; 1$), then

$$\backslash rho\; =\; \backslash sum\_\{i=1\}^k\; \backslash lambda\_i\; \backslash rho\_i$$

is a valid density operator, and the difference between its von Neumann entropy and the weighted average of the entropies of the {{math|ρ_i}} is bounded by the Shannon entropy of the {{math|λ_i}}:

$$S\backslash bigg(\backslash sum\_\{i=1\}^k\; \backslash lambda\_i\; \backslash rho\_i\; \backslash bigg)\; -\; \backslash sum\_\{i=1\}^k\; \backslash lambda\_i\; S(\backslash rho\_i)\; \backslash leq\; -\backslash sum\_\{i=1\}^k\; \backslash lambda\_i\; \backslash log\; \backslash lambda\_i.$$

Equality is attained when the supports of the {{math|ρ_i}} – the spaces spanned by their eigenvectors corresponding to nonzero eigenvalues – are orthogonal. The difference on the left-hand side of this inequality is known as the Holevo χ quantity and also appears in Holevo's theorem, an important result in quantum information theory.{{sfn|Nielsen|Chuang|2010|pp=531–534}}

The time evolution of an isolated system is described by a unitary operator:

$$\backslash rho\; \backslash to\; U\; \backslash rho\; U^\backslash dagger.$$

Unitary evolution takes pure states into pure states,{{sfnm|1a1=Nielsen|1a2=Chuang|1y=2010|1p=102 |2a1=Zwiebach|2y=2022|2at=§22.3}} and it leaves the von Neumann entropy unchanged. This follows from the fact that the entropy of $\backslash rho$ is a function of the eigenvalues of $\backslash rho$.{{sfnm|1a1=Peres|1y=1993|1p=267 |2a1=Wilde|2y=2017|2at=§11.1.1}}

A measurement upon a quantum system will generally bring about a change of the quantum state of that system. Writing a POVM does not provide the complete information necessary to describe this state-change process.{{sfn|Wilde|2017|at=§4.2.1}} To remedy this, further information is specified by decomposing each POVM element into a product:

$$E\_i\; =\; A^\backslash dagger\_\{i\}\; A\_\{i\}.$$

The Kraus operators $A\_\{i\}$, named for Karl Kraus, provide a specification of the state-change process. They are not necessarily self-adjoint, but the products $A^\backslash dagger\_\{i\}\; A\_\{i\}$ are. If upon performing the measurement the outcome $E\_i$ is obtained, then the initial state $\backslash rho$ is updated to

$$\backslash rho\; \backslash to\; \backslash rho\text{'}\; =\; \backslash frac\{A\_\{i\}\; \backslash rho\; A^\backslash dagger\_\{i\}\}\{\backslash mathrm\{Prob\}(i)\}\; =\; \backslash frac\{A\_\{i\}\; \backslash rho\; A^\backslash dagger\_\{i\}\}\{\backslash operatorname\{tr\}\; (\backslash rho\; E\_i)\}.$$

An important special case is the Lüders rule, named for Gerhart Lüders.{{cite journal|first=Gerhart |last=Lüders |author-link=Gerhart Lüders |title=Über die Zustandsänderung durch den Messprozeß |journal=Annalen der Physik |volume=443 |year=1950 |issue=5–8 |page=322|doi=10.1002/andp.19504430510 |bibcode=1950AnP...443..322L }} Translated by K. A. Kirkpatrick as {{Cite journal|last=Lüders|first=Gerhart|author-link=Gerhart Lüders |date=3 April 2006|title=Concerning the state-change due to the measurement process|journal=Annalen der Physik|volume=15|issue=9|pages=663–670|arxiv=quant-ph/0403007|bibcode=2006AnP...518..663L|doi=10.1002/andp.200610207|s2cid=119103479}}{{Citation|last1=Busch|first1=Paul|author-link=Paul Busch (physicist) |title=Lüders Rule|date=2009|work=Compendium of Quantum Physics|pages=356–358|editor-last=Greenberger|editor-first=Daniel|publisher=Springer Berlin Heidelberg|language=en|doi=10.1007/978-3-540-70626-7_110|isbn=978-3-540-70622-9|last2=Lahti|first2=Pekka|editor2-last=Hentschel|editor2-first=Klaus|editor3-last=Weinert|editor3-first=Friedel}} If the POVM elements are projection operators, then the Kraus operators can be taken to be the projectors themselves:

$$\backslash rho\; \backslash to\; \backslash rho\text{'}\; =\; \backslash frac\{\backslash Pi\_i\; \backslash rho\; \backslash Pi\_i\}\{\backslash operatorname\{tr\}\; (\backslash rho\; \backslash Pi\_i)\}.$$

If the initial state $\backslash rho$ is pure, and the projectors $\backslash Pi\_i$ have rank 1, they can be written as projectors onto the vectors $|\backslash psi\backslash rangle$ and $|i\backslash rangle$, respectively. The formula simplifies thus to

$$\backslash rho\; =\; |\backslash psi\backslash rangle\backslash langle\backslash psi|\; \backslash to\; \backslash rho\text{'}\; =\; \backslash frac$$

i\rangle\langle i | \psi\rangle\langle\psi | i \rangle\langle i

{|\langle i |\psi \rangle|^2} = |i\rangle\langle i|.

We can define a linear, trace-preserving, completely positive map, by summing over all the possible post-measurement states of a POVM without the normalisation:

$$\backslash rho\; \backslash to\; \backslash sum\_i\; A\_i\; \backslash rho\; A^\backslash dagger\_i.$$

It is an example of a quantum channel,{{sfn|Wilde|2017|at=§4.4.1}} and can be interpreted as expressing how a quantum state changes if a measurement is performed but the result of that measurement is lost.{{sfn|Wilde|2017|at=§4.5.1}} Channels defined by projective measurements can never decrease the von Neumann entropy; they leave the entropy unchanged only if they do not change the density matrix.{{sfn|Nielsen|Chuang|2010|p=515}} A quantum channel will increase or leave constant the von Neumann entropy of every input state if and only if the channel is unital, i.e., if it leaves fixed the maximally mixed state. An example of a channel that decreases the von Neumann entropy is the amplitude damping channel for a qubit, which sends all mixed states towards a pure state.{{sfn|Bengtsson|Życzkowski|2017|pp=380–381}}

The quantum version of the canonical distribution, the Gibbs states, are found by maximizing the von Neumann entropy under the constraint that the expected value of the Hamiltonian is fixed. A Gibbs state is a density operator with the same eigenvectors as the Hamiltonian, and its eigenvalues are

$$\backslash lambda\_i\; =\; \backslash frac\{1\}\{Z\}\; \backslash exp\backslash left(-\backslash frac\{E\_i\}\{k\_B\; T\}\backslash right),$$

where T is the temperature, $k\_B$ is the Boltzmann constant, and Z is the partition function.{{sfnm|1a1=Peres|1y=1993|1pp=266–267 |2a1=Rau |2y=2017 |2p=37}}{{cite journal|last1=Yunger Halpern |first1=Nicole |display-authors=etal |title=Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges |url=https://authors.library.caltech.edu/69093/1/ncomms12051.pdf |journal=Nature Communications |date=2016-07-07 |doi=10.1038/ncomms12051 |volume=7 |page=12051}} The von Neumann entropy of a Gibbs state is, up to a factor $k\_B$, the thermodynamic entropy.{{sfnm|1a1=Peres|1y=1993|1p=267 |2a1=Rau|2y=2017|2pp=51}}

Let $\backslash rho\_\{AB\}$ be a joint state for the bipartite quantum system AB. Then the conditional von Neumann entropy $S(A|B)$ is the difference between the entropy of $\backslash rho\_\{AB\}$ and the entropy of the marginal state for subsystem B alone:

$$S(A|B)\; =\; S(\backslash rho\_\{AB\})\; -\; S(\backslash rho\_B).$$

This is bounded above by $S(\backslash rho\_A)$. In other words, conditioning the description of subsystem A upon subsystem B cannot increase the entropy associated with A.{{sfn|Wilde|2017|at=§11.4}}

Quantum mutual information can be defined as the difference between the entropy of the joint state and the total entropy of the marginals:

$$S(A:B)\; =\; S(\backslash rho\_A)\; +\; S(\backslash rho\_B)\; -\; S(\backslash rho\_\{AB\}),$$

which can also be expressed in terms of conditional entropy:{{sfn|Nielsen|Chuang|2010|p=514}}

$$S(A:B)\; =\; S(A)\; -\; S(A|B)\; =\; S(B)\; -\; S(B|A).$$

{{main|Quantum relative entropy}}

Let $\backslash rho$ and $\backslash sigma$ be two density operators in the same state space. The relative entropy is defined to be

$$S(\backslash sigma|\backslash rho)\; =\; \backslash operatorname\{tr\}\; [\backslash rho(\backslash log\; \backslash rho\; -\; \backslash log\backslash sigma)].$$

The relative entropy is always greater than or equal to zero; it equals zero if and only if $\backslash rho\; =\; \backslash sigma$.{{sfnm|1a1=Peres|1y=1993|1p=264 |2a1=Nielsen|2a2=Chuang|2y=2010|2p=511}} Unlike the von Neumann entropy itself, the relative entropy is monotonic, in that it decreases (or remains constant) when part of a system is traced over:{{sfn|Nielsen|Chuang|2010|p=524}}

$$S(\backslash sigma\_\{A\}\; |\; \backslash rho\_\{A\})\; \backslash leq\; S(\backslash sigma\_\{AB\}\; |\; \backslash rho\_\{AB\}).$$

Just as energy is a resource that facilitates mechanical operations, entanglement is a resource that facilitates performing tasks that involve communication and computation.{{sfnm|1a1=Nielsen|1a2=Chuang|1y=2010|1p=106 |2a1=Rieffel|2a2=Polak|2y=2011|2p=218 |3a1=Bengtsson|3a2=Życzkowski|3y=2017|3p=435}} The mathematical definition of entanglement can be paraphrased as saying that maximal knowledge about the whole of a system does not imply maximal knowledge about the individual parts of that system.{{sfn|Rau|2021|p=131}} If the quantum state that describes a pair of particles is entangled, then the results of measurements upon one half of the pair can be strongly correlated with the results of measurements upon the other. However, entanglement is not the same as "correlation" as understood in classical probability theory and in daily life. Instead, entanglement can be thought of as potential correlation that can be used to generate actual correlation in an appropriate experiment.{{cite book |first=Christopher A. |last=Fuchs |title=Coming of Age with Quantum Information |date=6 January 2011 |publisher=Cambridge University Press |isbn=978-0-521-19926-1 |page=130}} The state of a composite system is always expressible as a sum, or superposition, of products of states of local constituents; it is entangled if this sum cannot be written as a single product term.{{sfn|Rieffel|Polak|2011|p=39}} Entropy provides one tool that can be used to quantify entanglement.{{cite journal|last1=Plenio |first1=Martin B. |first2=Shashank |last2=Virmani|title=An introduction to entanglement measures|year=2007|pages=1–51|volume=1|journal=Quantum Information and Computation |arxiv=quant-ph/0504163|bibcode=2005quant.ph..4163P}}{{cite journal | last = Vedral | first = Vlatko |author-link = Vlatko Vedral | doi = 10.1103/RevModPhys.74.197 | arxiv = quant-ph/0102094 | bibcode=2002RvMP...74..197V | volume=74 | issue = 1 | title=The role of relative entropy in quantum information theory | year=2002 | journal=Reviews of Modern Physics | pages=197–234 | s2cid = 6370982 }} If the overall system is described by a pure state, the entropy of one subsystem can be used to measure its degree of entanglement with the other subsystems. For bipartite pure states, the von Neumann entropy of reduced states is the unique measure of entanglement in the sense that it is the only function on the family of states that satisfies certain axioms required of an entanglement measure.{{sfn|Holevo|2001|p=31}}{{cite journal |last1=Hill |first1=S. |last2=Wootters |first2=W. K. |author-link2=William Wootters |title=Entanglement of a Pair of Quantum Bits |journal=Physical Review Letters |arxiv=quant-ph/9703041 |doi =10.1103/PhysRevLett.78.5022 |year=1997 |volume=78 |issue=26 |pages=5022–5025 |bibcode=1997PhRvL..78.5022H }} It is thus known as the entanglement entropy.{{sfn|Bengtsson|Życzkowski|2017|p=447}}

It is a classical result that the Shannon entropy achieves its maximum at, and only at, the uniform probability distribution {{mset|1/n, ..., 1/n}}.{{sfn|Nielsen|Chuang|2010|p=505}} Therefore, a bipartite pure state {{math|ρ ∈ H_A ⊗ H_B}} is said to be a maximally entangled state if the reduced state of each subsystem of {{mvar|ρ}} is the diagonal matrix{{Cite journal |last1=Enríquez |first1=M. |last2=Wintrowicz |first2=I. |last3=Życzkowski |first3=K. |author-link3=Karol Życzkowski |date=March 2016 |title=Maximally Entangled Multipartite States: A Brief Survey |url=https://iopscience.iop.org/article/10.1088/1742-6596/698/1/012003 |journal=Journal of Physics: Conference Series |volume=698 |issue=1 |pages=012003 |doi=10.1088/1742-6596/698/1/012003 |bibcode=2016JPhCS.698a2003E |doi-access=free }}

For mixed states, the reduced von Neumann entropy is not the only reasonable entanglement measure.{{sfnm|1a1=Holevo|1y=2001|1p=31 |2a1=Bengtsson|2a2=Życzkowski|2y=2017|2p=471}} Some of the other measures are also entropic in character. For example, the relative entropy of entanglement is given by minimizing the relative entropy between a given state $\backslash rho$ and the set of nonentangled, or separable, states.{{sfn|Bengtsson|Życzkowski|2017|p=474}} The entanglement of formation is defined by minimizing, over all possible ways of writing of $\backslash rho$ as a convex combination of pure states, the average entanglement entropy of those pure states.{{sfn|Bengtsson|Życzkowski|2017|p=475}} The squashed entanglement is based on the idea of extending a bipartite state $\backslash rho\_\{AB\}$ to a state describing a larger system, $\backslash rho\_\{ABE\}$, such that the partial trace of $\backslash rho\_\{ABE\}$ over E yields $\backslash rho\_\{AB\}$. One then finds the infimum of the quantity

$$\backslash frac\{1\}\{2\}[S(\backslash rho\_\{AE\})\; +\; S(\backslash rho\_\{BE\})\; -\; S(\backslash rho\_E)\; -\; S(\backslash rho\_\{ABE\})],$$

over all possible choices of $\backslash rho\_\{ABE\}$.{{sfn|Bengtsson|Życzkowski|2017|p=477}}

Just as the Shannon entropy function is one member of the broader family of classical Rényi entropies, so too can the von Neumann entropy be generalized to the quantum Rényi entropies:

$$S\_\backslash alpha(\backslash rho)\; =\; \backslash frac\{1\}\{1-\backslash alpha\}\; \backslash ln[\backslash operatorname\{tr\}\; \backslash rho^\backslash alpha]\; =\; \backslash frac\{1\}\{1-\backslash alpha\}\; \backslash ln\; \backslash sum\_\{i=1\}^N\; \backslash lambda\_i^\backslash alpha.$$

In the limit that $\backslash alpha\; \backslash to\; 1$, this recovers the von Neumann entropy. The quantum Rényi entropies are all additive for product states, and for any $\backslash alpha$, the Rényi entropy $S\_\backslash alpha$ vanishes for pure states and is maximized by the maximally mixed state. For any given state $\backslash rho$, $S\_\backslash alpha(\backslash rho)$ is a continuous, nonincreasing function of the parameter $\backslash alpha$. A weak version of subadditivity can be proven:

$$S\_\backslash alpha(\backslash rho\_A)\; -\; S\_0(\backslash rho\_B)\; \backslash leq\; S\_\backslash alpha(\backslash rho\_\{AB\})\; \backslash leq\; S\_\backslash alpha(\backslash rho\_A)\; +\; S\_0(\backslash rho\_B).$$

Here, $S\_0$ is the quantum version of the Hartley entropy, i.e., the logarithm of the rank of the density matrix.{{sfn|Bengtsson|Życzkowski|2017|pp=369–370}}

The density matrix was introduced, with different motivations, by von Neumann and by Lev Landau. The motivation that inspired Landau was the impossibility of describing a subsystem of a composite quantum system by a state vector.{{Cite journal | last1 = Landau | first1 = L. | author-link = Lev Landau | title = Das Daempfungsproblem in der Wellenmechanik | doi = 10.1007/BF01343064 | journal = Zeitschrift für Physik | volume = 45 | issue = 5–6 | pages = 430–464 | year = 1927 |bibcode = 1927ZPhy...45..430L | s2cid = 125732617 }} On the other hand, von Neumann introduced the density matrix in order to develop both quantum statistical mechanics and a theory of quantum measurements.{{Cite book |last=Von Neumann |first=John |author-link=John von Neumann |title=Mathematische Grundlagen der Quantenmechanik |year=1932 |publisher=Springer |location=Berlin |isbn=3-540-59207-5 }}; {{Cite book |last=Von Neumann |first=John |author-link=John von Neumann |title=Mathematical Foundations of Quantum Mechanics | year=1955|publisher=Princeton University Press | isbn= 978-0-691-02893-4 }} He introduced the expression now known as von Neumann entropy by arguing that a probabilistic combination of pure states is analogous to a mixture of ideal gases.{{sfn|Peres|1993|p=271}}{{cite journal|first=Leah |last=Henderson |title=The Von Neumann Entropy: A reply to Shenker |journal=British Journal for the Philosophy of Science |volume=54 |year=2003 |pages=291–296 |doi=10.1093/bjps/54.2.291 |jstor=3541968}} Von Neumann first published on the topic in 1927.{{cite book|first=D. |last=Petz |author-link=Dénes Petz |chapter=Entropy, von Neumann and the von Neumann entropy |title=John von Neumann and the Foundations of Quantum Physics |editor-first1=M. |editor-last1=Rédei |editor-link1=Miklós Rédei |editor-first2=M. |editor-last2=Stöltzner |publisher=Kluwer |year=2001 |arxiv=math-ph/0102013 |isbn=978-0-7923-6812-0 |doi=10.1007/978-94-017-2012-0_7 |pages=83–96}} His argument was built upon earlier work by Albert Einstein and Leo Szilard.{{cite journal|first=Albert |last=Einstein |author-link=Albert Einstein |title=Beiträge zur Quantentheorie |journal=Deutsche Physikalische Gesellschaft. Verhandlungen |volume=16 |year=1914 |page=820}} Translated in {{cite book|title=The Collected Papers of Albert Einstein, Volume 6 |translator-first=Alfred |translator-last=Engel |publisher=Princeton University Press |year=1997 |isbn=0-691-01734-4 |pages=20–26}}{{cite journal|first=Leo |last=Szilard |author-link=Leo Szilard |title=Über die Ausdehnung der phänomenologischen Thermodynamik auf die Schwankungserscheinungen |journal=Zeitschrift für Physik |date=December 1925 |volume=32 |pages=753–788 |doi=10.1007/BF01331713}}{{cite journal|first=Alfred |last=Wehrl |title=General properties of entropy |doi=10.1103/RevModPhys.50.221 |journal=Reviews of Modern Physics |volume=50 |number=2 |pages=221–260 |date=April 1978}}

Max Delbrück and Gert Molière proved the concavity and subadditivity properties of the von Neumann entropy in 1936. Quantum relative entropy was introduced by Hisaharu Umegaki in 1962.{{sfn|Bengtsson|Życzkowski|2017|pp=361,365}}{{cite web|url=https://resolver.caltech.edu/CaltechOH:OH_Delbruck_M |title=Max Delbrück Oral History Interview |first1=Max |last1=Delbrück |author-link1=Max Delbrück |first2=Carolyn |last2=Harding |website=Caltech Archives Oral History Project |access-date=2024-12-30}} The subadditivity and triangle inequalities were proved in 1970 by Huzihiro Araki and Elliott H. Lieb.{{cite journal

| last1=Araki | first1=Huzihiro | authorlink1=Huzihiro Araki

| last2=Lieb | first2=Elliott H. | authorlink2=Elliott H. Lieb

| title=Entropy Inequalities

| journal=Communications in Mathematical Physics

| volume=18

| pages=160–170

| date=1970

| issue=2 | doi=10.1007/BF01646092 | bibcode=1970CMaPh..18..160A | s2cid=189832417 | doi-access=}} Strong subadditivity is a more difficult theorem. It was conjectured by Oscar Lanford and Derek Robinson in 1968.{{cite journal|first1=Oscar E. |last1=Lanford |author-link1=Oscar Lanford |first2=Derek W. |last2=Robinson |author-link2=Derek W. Robinson |title=Mean entropy of states in quantum-statistical mechanics |journal=Journal of Mathematical Physics |date=July 1968 |volume=9 |number=7 |pages=1120–1125 |doi=10.1063/1.1664685 |url=https://cds.cern.ch/record/936663/files/CM-P00057584.pdf}} Lieb and Mary Beth Ruskai proved the theorem in 1973,{{cite journal

| last1=Lieb | first1=Elliott H. | authorlink1=Elliott H. Lieb

| last2=Ruskai | first2=Mary Beth | authorlink2=Mary Beth Ruskai

| title=Proof of the Strong Subadditivity of Quantum-Mechanical Entropy

| journal=Journal of Mathematical Physics

| volume=14

| pages=1938–1941

| date=1973

| issue=12 | doi=10.1063/1.1666274| bibcode=1973JMP....14.1938L | url=http://www.numdam.org/item/RCP25_1973__19__A5_0/ | doi-access=free}}{{cite web |last1=Ruskai |first1=Mary Beth |title=Evolution of a Fundamental Theorem on Quantum Entropy |url=https://www.youtube.com/watch?v=P3-xI1u1Y2s |archive-url=https://ghostarchive.org/varchive/youtube/20211221/P3-xI1u1Y2s |archive-date=2021-12-21 |url-status=live|website=youtube.com |date=10 January 2014 |publisher=World Scientific |access-date=20 August 2020}} Invited talk at the Conference in Honour of the 90th Birthday of Freeman Dyson, Institute of Advanced Studies, Nanyang Technological University, Singapore, 26–29 August 2013.{{cbignore}} using a matrix inequality proved earlier by Lieb.{{cite journal

| last1=Lieb | first1=Elliott H. | authorlink1=Elliott H. Lieb

| title=Convex Trace Functions and the Wigner–Yanase–Dyson Conjecture

| journal=Advances in Mathematics

| volume=11

| issue=3

| pages=267–288

| date=1973

| doi=10.1016/0001-8708(73)90011-X | doi-access=free}}{{sfn|Nielsen|Chuang|2010|p=527}}

{{reflist}}

• {{cite book|first1=Ingemar |last1=Bengtsson |first2=Karol |last2=Życzkowski |author-link2=Karol Życzkowski |title=Geometry of Quantum States: An Introduction to Quantum Entanglement |title-link=Geometry of Quantum States |year=2017 |publisher=Cambridge University Press |edition=2nd |isbn=978-1-107-02625-4}}
• {{cite book|first=Alexander S. |last=Holevo |author-link=Alexander Holevo |title=Statistical Structure of Quantum Theory |publisher=Springer |series=Lecture Notes in Physics. Monographs |year=2001 |isbn=3-540-42082-7}}
• {{cite book |last1=Nielsen |first1=Michael A. |author-link1=Michael Nielsen |title=Quantum Computation and Quantum Information |title-link=Quantum Computation and Quantum Information |last2=Chuang |first2=Isaac L. |author-link2=Isaac Chuang |publisher=Cambridge Univ. Press |year=2010 |isbn=978-0-521-63503-5 |edition=10th anniversary|location=Cambridge}}
• {{cite book|first=Asher |last=Peres |author-link=Asher Peres |title=Quantum Theory: Concepts and Methods |title-link=Quantum Theory: Concepts and Methods |publisher=Kluwer |year=1993 |isbn=0-7923-2549-4 }}
• {{cite book|first=Jochen |last=Rau |title=Statistical Physics and Thermodynamics |year=2017 |publisher=Oxford University Press |isbn=978-0-19-959506-8}}
• {{cite book|first=Jochen |last=Rau |title=Quantum Theory: An Information Processing Approach |publisher=Oxford University Press |year=2021 |isbn=978-0-19-289630-8}}
• {{Cite book |last1=Rieffel |first1=Eleanor |author-link1=Eleanor Rieffel |title=Quantum Computing: A Gentle Introduction |title-link=Quantum Computing: A Gentle Introduction |last2=Polak |first2=Wolfgang |date=2011 |publisher=MIT Press |isbn=978-0-262-01506-6 |series=Scientific and engineering computation |location=Cambridge, Mass}}
• {{cite book|last=Wilde |first=Mark M. |author-link=Mark Wilde |title=Quantum Information Theory |edition=2nd |publisher=Cambridge University Press |year=2017 |doi=10.1017/9781316809976 |isbn=9781316809976 |arxiv=1106.1445}}
• {{cite book|first=Barton |last=Zwiebach |title=Mastering Quantum Mechanics: Essentials, Theory, and Applications |author-link=Barton Zwiebach |publisher=MIT Press |year=2022 |isbn=978-0-262-04613-8}}

{{Statistical mechanics topics}}

Category:Quantum mechanical entropy

Category:John von Neumann