Wehrl entropy
In quantum information theory, the Wehrl entropy,[1] named after Alfred Wehrl, is a classical entropy of a quantum-mechanical density matrix. It is a type of quasi-entropy defined for the Husimi Q representation of the phase-space quasiprobability distribution. See [2] for a comprehensive review of basic properties of classical, quantum and Wehrl entropies, and their implications in statistical mechanics.
Definitions
The Husimi function[3] is a "classical phase-space" function of position x and momentum p, and in one dimension is defined for any quantum-mechanical density matrix ρ by
where φ is a "(Glauber) coherent state", given by
(It can be understood as the Weierstrass transform of the Wigner quasi-probability distribution.)
The Wehrl entropy is then defined as
The definition can be easily generalized to any finite dimension.
Properties
Such a definition of the entropy relies on the fact that the Husimi Q representation remains non-negative definite,[4] unlike other representations of quantum quasiprobability distributions in phase space. The Wehrl entropy has several important properties:
- It is always positive, like the full quantum von Neumann entropy, but unlike the classical differential entropy which can be negative at low temperature. In fact, the minimum value of the Wehrl entropy is 1, i.e. as discussed below in the section "Werhl's conjecture".
- The entropy for the tensor product of two systems is always greater than the entropy of one system. In other words, for a state on a Hilbert space , we have , where . Note that the quantum von Neumann entropy, , does not have this property, as can be clearly seen for a pure maximally entangled state.
- The Wehrl entropy is strictly lower bounded by a von Neumann entropy, . There is no known upper or lower bound (other than zero) for the difference .
- The Wehrl entropy is not invariant under all unitary transformations, unlike the von Neumann entropy. In other words, for a general unitary U. It is, however, invariant under certain unitary transformations.[1]
Wehrl's conjecture
In his original paper [1] Wehrl posted a conjecture that the smallest possible value of Wehrl entropy is 1, and it occurs if and only if the density matrix is a pure state projector onto any coherent state, i.e. for all choices of ,
- .
Soon after the conjecture was posted, E. H. Lieb proved [5] that the minimum of the Wehrl entropy is 1, and it occurs when the state is a projector onto any coherent state.
In 1991 E. Carlen proved [6] the uniqueness of the minimizer, i.e. the minimum of the Wehrl entropy occurs only when the state is a projector onto any coherent state.
The analog of the Wehrl conjecture for systems with a classical phase space isomorphic to the sphere (rather than the plane) is the Lieb conjecture.
Discussion
However, it is not the fully quantum von Neumann entropy in the Husimi representation in phase space, − ∫ Q ★ log★Q dx dp: all the requisite star-products ★ in that entropy have been dropped here. In the Husimi representation, the star products read
and are isomorphic[7] to the Moyal products of the Wigner–Weyl representation.
The Wehrl entropy, then, may be thought of as a type of heuristic semiclassical approximation to the full quantum von Neumann entropy, since it retains some ħ dependence (through Q) but not all of it.
Like all entropies, it reflects some measure of non-localization,[8] as the Gauss transform involved in generating Q and the sacrifice of the star operators have effectively discarded information. In general, as indicated, for the same state, the Wehrl entropy exceeds the von Neumann entropy (which vanishes for pure states).
Wehrl entropy for Bloch coherent states
Wehrl entropy can be defined for other kinds of coherent states. For example, it can be defined for Bloch coherent states, that is, for angular momentum representations of the group for quantum spin systems.
Bloch coherent states
Consider a space with . We consider a single quantum spin of fixed angular momentum J, and shall denote by the usual angular momentum operators that satisfy the following commutation relations: and cyclic permutations.
Define , then and .
The eigenstates of are
For the state satisfies: and .
Denote the unit sphere in three dimensions by
- ,
and by the space of square integrable function on Ξ with the measure
- .
The Bloch coherent state is defined by
- .
Taking into account the above properties of the state , the Bloch coherent state can also be expressed as
where , and
is a normalised eigenstate of satisfying .
The Bloch coherent state is an eigenstate of the rotated angular momentum operator with a maximum eigenvalue. In other words, for a rotation operator
- ,
the Bloch coherent state satisfies
- .
Wehrl entropy for Bloch coherent states
Given a density matrix ρ, define the semi-classical density distribution
- .
The Wehrl entropy of for Bloch coherent states is defined as a classical entropy of the density distribution ,
- ,
where is a classical differential entropy.
Wehrl's conjecture for Bloch coherent states
The analogue of the Wehrl's conjecture for Bloch coherent states was proposed in [5] in 1978. It suggests the minimum value of the Werhl entropy for Bloch coherent states,
- ,
and states that the minimum is reached if and only if the state is a pure Bloch coherent state.
In 2012 E. H. Lieb and J. P. Solovej proved [9] a substantial part of this conjecture, confirming the minimum value of the Wehrl entropy for Bloch coherent states, and the fact that it is reached for any pure Bloch coherent state. The uniqueness of the minimizers was proved in 2022 by R. L. Frank[10] and A. Kulikov, F. Nicola, J. Ortega-Cerda' and P. Tilli.[11]
Generalized Wehrl's conjecture
In [9] E. H. Lieb and J. P. Solovej proved Wehrl's conjecture for Bloch coherent states by generalizing it in the following manner.
Generalized Wehrl's conjecture
For any concave function (e.g. as in the definition of the Wehrl entropy), and any density matrix ρ, we have
- ,
where ρ0 is a pure coherent state defined in the section "Wehrl conjecture".
Generalized Wehrl's conjecture for Bloch coherent states
Generalized Wehrl's conjecture for Glauber coherent states was proved as a consequence of the similar statement for Bloch coherent states. For any concave function , and any density matrix ρ we have
- ,
where is any point on a sphere.
The uniqueness of the minimizers was proved in the aforementioned papers [10] and.[11]
See also
- Coherent state
- Entropy
- Information theory and measure theory
- Lieb conjecture
- Quantum information
- Quantum mechanics
- Spin
- Statistical mechanics
- Von Neumann entropy
References
- ^ a b c Wehrl, A. (1979). "On the relation between classical and quantum-mechanical entropy". Reports on Mathematical Physics. 16 (3): 353–358. Bibcode:1979RpMP...16..353W. doi:10.1016/0034-4877(79)90070-3.
- ^ Wehrl, A. (1978). "General properties of entropy". Reviews of Modern Physics. 50 (2): 221–260. Bibcode:1978RvMP...50..221W. doi:10.1103/RevModPhys.50.221.
- ^ Kôdi Husimi (1940). "Some Formal Properties of the Density Matrix". Proceedings of the Physico-Mathematical Society of Japan. 3. 22 (4): 264–314. doi:10.11429/ppmsj1919.22.4_264.
- ^ Cartwright, N. D. (1975). "A non-negative Wigner-type distribution". Physica A: Statistical Mechanics and Its Applications. 83 (1): 210–818. Bibcode:1976PhyA...83..210C. doi:10.1016/0378-4371(76)90145-X.
- ^ a b Lieb, Elliott H. (1978). "Proof of an entropy conjecture of Wehrl". Communications in Mathematical Physics. 62 (1). Springer Science and Business Media LLC: 35–41. Bibcode:1978CMaPh..62...35L. doi:10.1007/bf01940328. ISSN 0010-3616. S2CID 189836756.
- ^ Carlen, E. (1991). "Some integral identities and inequalities for entire functions and their application to the coherent state transform". Journal of Functional Analysis. 97: 231–249. doi:10.1016/0022-1236(91)90022-W.
- ^ C. Zachos, D. Fairlie, and T. Curtright, “Quantum Mechanics in Phase Space” (World Scientific, Singapore, 2005) ISBN 978-981-238-384-6 .
- ^ Gnutzmann, Sven; Karol Zyczkowski (2001). "Rényi–Wehrl entropies as measures of localization in phase space". J. Phys. A: Math. Gen. 34 (47): 10123. arXiv:quant-ph/0106016. Bibcode:2001JPhA...3410123G. doi:10.1088/0305-4470/34/47/317. S2CID 7327137.
- ^ a b Lieb, E.H.; Solovej, J.P. (2014). "Proof of an entropy conjecture for Bloch coherent spin states and its generalizations". Acta Mathematica. 212 (2): 379–398. arXiv:1208.3632. doi:10.1007/s11511-014-0113-6. S2CID 119166106.
- ^ a b Frank, R.L. (2023). "Sharp inequalities for coherent states and their optimizers". Advanced Nonlinear Studies. 23 (1): Paper No. 20220050, 28. arXiv:2210.14798. doi:10.1515/ans-2022-0050.
- ^ a b Kulikov, A.; Nicola, F.; Ortega-Cerda', J.; Tilli, P. (2022). "A monotonicity theorem for subharmonic functions on manifolds". arXiv:2212.14008 [math.CA].