வெப்ப இயல் ஆற்றலை அப்படியே அந்த நிலைகளில் அளவுபடுத்தப்பட முடியாத தன்மை இயற்பியல் கணிதமொழியில் நான் ஈக்குலிபிரியம் தெர்மொடைனமிக் என்ட்ரோபி எனப்படும்.இதுவே குவாண்டம் இயக்கவியலின் அளவீடுக்ளில் அதன் புள்ளியியல் கணக்கு விவரப்படி அந்த வெப்ப் இயக்கவியலை (தெர்மோடைனாமிக்ஸ்) அளவீடு செய்ய முயலுகையில் குவாண்டம் கார்ரலேஷன் என்ட்ரோபி ஆகி விடும்.இதை தமிழ்ப்படுத்தினால் "அளபடைய நேர்த்தொடர்பியல் வழுவாற்றல்" ஆகும்.ஆற்றல் அளவு பாட்டுக்குள் வர இயலாமல் குறைவு படும் அல்லது வழுவிவிடும் இயல்பே இங்கு என்ட்ரோபி எனப்படுகிறது.
புள்ளிவிவர இயல் அடிப்படையில் உள்ள நேர்த்தொடர்பியல் மரபு நிலை நேர்த்தொடர்பியல் ஆகும்.(க்ளாஸ்ஸிகல் கார்ரலேஷன் ஆகும். இது குவாண்ட இயலில் அணுகப்படும்போது தான் குவாண்டம் கார்ரலேஷன் ஆகிறது.இது இருவகைகளில் ஆராயப்படுகிறது
1.ரிலேடிவ் என்ட்ரோபி ஆஃப் குவாண்டம்னெஸ்
2.ஸீரோ வே குவாண்டம் டெஃபிசிட்.
முதல் வகை "அளபடையத்தன்மையை அவற்றின் மரபு நிலை நேர்த்தொர்டபியல்களின் தூரம் அதாவது வேறுபாடுகளை வைத்து கணிப்பது ஆகும்
.இரண்டாம் வகை அவற்றின் அந்தந்த இடத்து செயல்பாடுகளிலிருந்து பெறப்படும் செயல்திறனைப்பற்றியது ஆகும்.
அளவுபாட்டியல் (மெஷர் தியரி) இந்த குவாண்டம் கார்ரலேஷனில் ஒரு புள்ளிவிவர நோக்கில் தான் செல்லுகிறது.இந்த அளவுகள் சமத்தன்மை வாய்ந்தனவா? அதற்கு நாம் உற்று நோக்கவேண்டியது புள்ளிவிவர இயல் இயக்கவியலை (ஸ்டாடிஸ்டிகல் மெகானிக்ஸ்) பற்றி அறிவதும் அவசியம்.
இதில் "ஸ்டாஸ்டிஸ்டிகல் கோர்ஸ்_க்ரெய்னிங்" என்னும் கணிதப்பாடு மிக நுட்பமானது.
,
,
he uncertainty of local measurements, and directly contributes to non-equilibrium thermodynamic entropy.
The quantum correlation entropy, as well as being a
statistical/thermodynamic entropy, is a measure of total
nonclassical correlations. As discussed further in Sec. VI,
it is equal to two other such measures: the relative entropy of quantumness [23, 42, 47, 56, 57] (a measure of
distance from the set of classically correlated states), and
the zero-way quantum deficit [42, 58, 59] (a measure of
work extractable by certain local operations). The equivalence of these measures, each with quite different meanings, suggests that together they have a general and distinguished role. With this in mind, this paper aims to
provide a self-contained treatment in terms of the statistical mechanics of coarse-graining.
Entanglement entropy, by its usual definition, is defined only for bipartite pure states, where entanglement
and total nonclassical correlation are equivalent. The
generalization S
qc can be nonzero on separable states,
but is zero precisely on classically correlated states. This
clarifies that it measures total nonclassical correlation,
not entanglement. In contrast, no entanglement measure
is known to arise from statistical mechanics.3
Analysis in terms of coarse-graining leads to a distinction between three types of entropy:4
• von Neumann entropy is inherent to the state ρ,
and quantifies fundamental uncertainty in a system
due to being in a mixed state;
• quantum correlation entropy (equivalently, where
it is defined, entanglement entropy) depends on a
partition into subsystems, and quantifies the additional uncertainty in a multipartite system if one
can only make subsystem-local measurements;
• coarse-grained entropy depends on a division of the
state space into macrostates, and quantifies uncertainty associated with describing a system in terms
of these macrostates.
The first two each contribute to the third: the entropy
of any possible coarse-graining is bounded below by von
3 This raises a subtle point. The term “entanglement entropy” suggests that “entanglement” (i.e. non-separability) and “entropy”
are closely related. This seems to be true only in the special case
of bipartite pure states. More generally statistical mechanical entropy appears (based on this work) tied to quantum correlations,
not entanglement, when the two are inequivalent.
4 With this distinction the terms “von Neumann entropy” and
“entanglement entropy” should not be applied interchangeably.
While it is true that von Neumann entropy may arise in a system
(say, a joint system described by ρAB) because of its entanglement with some external system (say, system C), this is a fundamentally different concept than quantum correlation entropy
(i.e. entanglement entropy) within the system (that is, between
A and B).
Neumann entropy, while the entropy of any local coarsegraining is bounded below by the sum of von Neumann
and quantum correlation entropies.5
In this way quantum correlation entropy provides a
key piece to a unified treatment of quantum statistical/thermodynamic entropy, along with a direct link to
important measures in quantum information theory.
II. QUANTUM CORRELATION ENTROPY
FROM COARSE-GRAINED ENTROPY
In the theory of quantum coarse-grained entropy [49–
53], a coarse-graining C = {Pˆ
i} is a collection of Hermitian (Pˆ†
i = Pˆ
i) orthogonal projectors (Pˆ
iPˆ
j = Pˆ
i δij )
forming a partition of unity (P
i Pˆ
i = 1). A coarsegraining is the set of outcomes of some projective measurement. Each subspace generated by Pˆ
i
is called a
“macrostate.”
Given a coarse-graining C the “coarse-grained entropy”
(or “observational entropy”) of a density operator ρ is
SC(ρ) = −
X
i
pi
log
pi
Vi
, (1)
where pi = tr(Pˆ
iρ) is the probability to find ρ in each
macrostate, and Vi = tr(Pˆ
i) is the volume of each
macrostate. The coarse-grained entropy is defined both
in and out of equilibrium, obeys a second law, and (with
a properly chosen coarse-graining) is equal to thermodynamic entropy in appropriate cases [50–55, 60–62].
One way to specify a coarse-graining is via the spectral
decomposition of an observable operator Qˆ =
P
q
q Pˆ
q,
with associated coarse-graining CQˆ = {Pˆ
q}. If Qˆ has
a full spectrum of distinct eigenvalues, then SCQˆ
(ρ) is
merely the Shannon entropy of measuring Qˆ. On the
other hand Qˆ may have larger eigenspaces. If ρ has a definite value q then SCQˆ
(ρ) is the log of the dimension of the
q eigenspace of Qˆ (i.e. the volume of the q macrostate),
a quantum analog of the Boltzmann entropy. Evidently
the coarse-grained entropy provides a quantum generalization of both the Shannon and Boltzmann entropies of
a measurement, and represents the uncertainty an observer making measurements assigns to the system.
Given a density operator ρ, the minimum value
of coarse-grained entropy, minimized over all coarsegrainings C, is
inf
C
SC(ρ)
= SCρ
(ρ) = S
vn(ρ), (2)
the von Neumann entropy [49, 51, 52]. The second
equality states that the von Neumann entropy S
vn(ρ) =
5 Recalling its connection to local energy coarse-graining, this sum
is then a bound on non-equilibrium thermodynamic entropy.
3
−tr(ρ log ρ) is equal to the coarse-grained entropy SCρ
(ρ)
in the coarse-graining Cρ consisting of eigenspaces of ρ.
Thus (2) expresses that no measurement can be more
informative than measuring the density matrix itself.
Now consider an arbitrary multipartite system
AB . . . C, whose Hilbert space is the tensor product
H = HA ⊗ HB ⊗ . . . ⊗ HC .
In this background one can consider a subclass of
coarse-grainings, the “local” coarse-grainings, defined by
CA ⊗ CB ⊗ . . . ⊗ CC = {PˆA
l ⊗ PˆB
m ⊗ . . . ⊗ PˆC
n
}, (3)
where CA = {PˆA
l
} is a coarse-graining of A, and so on
for the other subsystems. These are precisely the coarsegrainings describing local measurements (i.e. consisting
of only local operators). Applying the definition (1) in
such a coarse-graining yields the entropy
SCA⊗...⊗CC (ρ) = −
X
lm...n
plm...n log
plm...n
Vlm...n
, (4)
where plm...n = tr(PˆA
l ⊗ PˆB
m ⊗ . . . ⊗ PˆC
n ρ) are the probabilities to find the system in each macrostate, and
Vlm...n = tr(PˆA
l ⊗ PˆB
m ⊗ . . . ⊗ PˆC
n
) are the volumes of
each macrostate.
One can now ask: what is the minimum entropy of any
set of local measurements? That is, what is the infimum
value
inf
C=CA⊗...⊗CC
SC(ρ)
(5)
of the coarse-grained entropy, if the minimum is taken
only over local coarse-grainings?
There are two possibilities. Either the minimum value
S
vn(ρ) can be saturated by local coarse-grainings, or it
cannot. Which of these is the case depends on the density matrix. If the minimum fails to be saturated, then
there is an entropy gap ∆S above the minimum which is
inherent to any local measurements.
A natural question is then: how is this entropy gap, associated with restricting to local coarse-grainings, related
to entanglement entropy? Two observations provide a
foundation for answering this question. The first, quite
nontrivial, observation is that the entropy gap is equal to
the entanglement entropy for bipartite pure states (see
Property Ia). The second is that the entropy gap is zero
for any product state (see Property II). These facts suggest that entanglement entropy should in general be identified with this entropy gap. The aim of this article is to
make precisely that identification and show that it leads
to an intuitive and useful framework.
The observations above motivate the definition
S
qc
AB...C (ρ) ≡ inf
C=CA⊗...⊗CC
SC(ρ)
குவாண்டம் கார்ரிலெஷன் என்ட்ரோபி
vn(ρ) (6)
of the quantum correlation (quarrelation) entropy
S
qc
AB...C (ρ). The subscript denotes the partition into subsystems, allowing various partitions of the same system.
In other words, quarrelation entropy is the difference
in coarse-grained entropy between the best possible local coarse-graining and the best possible global coarsegraining. This definition can be evaluated exactly for a
variety of states using the properties introduced below
கருத்துகள் இல்லை:
கருத்துரையிடுக