Quantum states can be subjected to classical measurements, whose
incompatibility, or uncertainty, can be quantified by a comparison of certain
entropies. There is a long history of such entropy inequalities between
position and momentum. Recently these inequalities have been generalized to the
tensor product of several Hilbert spaces and we show here how their derivations
can be shortened to a few lines and how they can be generalized. All the
recently derived uncertainty relations utilize the strong subadditivity (SSA)
theorem; our contribution relies on directly utilizing the proof technique of
the original derivation of SSA.Comment: 4 page

C. Davis

D. Deutsch

E.H. Lieb

Elliott H. Lieb

G. Lindblad

H. Maassen

I.I. Hirschman Jr

J.M. Renes

K. Kraus

M. Berta

M. Rumin

M. Tomamichel

M.J.W. Hall

P.J. Coles

R.L. Frank

Rupert L. Frank

S. Wehner

English

arXiv

Quantum states can be subjected to classical measurements, whose incompatibility, or uncertainty, can be quantified by a comparison of certain entropies. There is a long history of such entropy inequalities between position and momentum. Recently these inequalities have been generalized to the tensor product of several Hilbert spaces and we show here how their derivations can be shortened to a few lines and how they can be generalized. Our proofs utilize the technique of the original derivation of strong subadditivity of the von Neumann entropy

Frank, Rupert L.

Lieb, Elliott H.

Caltech Authors - Main

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Extended quantum conditional entropy and quantum uncertainty inequalities∗
Rupert L. Frank1 and Elliott H. Lieb1, 2
1Department of Mathematics, Princeton University, Princeton, NJ 08544, USA
2Department of Physics, Princeton University, P. O. Box 708, Princeton, NJ 08542, USA
Quantum states can be subjected to classical measurements, whose incompatibility, or uncertainty,
can be quantified by a comparison of certain entropies. There is a long history of such entropy
inequalities between position and momentum. Recently these inequalities have been generalized
to the tensor product of several Hilbert spaces and we show here how their derivations can be
shortened to a few lines and how they can be generalized. All the recently derived uncertainty
relations utilize the strong subadditivity (SSA) theorem; our contribution relies on directly utilizing
the proof technique of the original derivation of SSA.
PACS numbers: 03.67.-a, 03.67.Mn, 03.67.Hk, 03.65.Ta
A celebrated inequality of Maassen–Uffink [1], based on
earlier work in [2–4], relates the ‘classical’ entropy of a
quantum density matrix in two different bases and shows
that although one of them could be very small, the sum
of the two is bounded below by a positive constant. The
word ‘classical’ refers to H = −
∑
j pj ln pj , where pj is
the expectation of a density matrix in some orthonormal
basis. The inequality is
H(A) +H(B) ≥ −2 ln sup
j,k
|〈aj , bk〉| ,
where A and B represent two orthonormal bases (aj) and
(bk). This can be generalized to continuous bases, like
position x and momentum p, in which case the inequality
becomes (with p = 2pik)
−
∫
ddx ρ(x,x) ln ρ(x,x) −
∫
ddk ρ̂(k,k) ln ρ̂(k,k) ≥ 0 .
These inequalities have subsequently been improved; see
the review [5] and the recent papers [6, 7, 12]. Only
purely classical analysis needs to be utilized to prove
these inequalities and only one Hilbert space is involved.
Another direction was opened up by the conjectures of
Renes and Boileau [8], in which more than one Hilbert
space appears. The analogous inequalities become more
difficult mathematically because of the well-known en-
tanglement problems in quantum mechanics. Indeed, as
noted in [8], the Lieb–Ruskai strong subadditivity (SSA)
theorem [9], or one of its equivalents, would ultimately
be needed to prove the conjectures. Berta et al. [10] then
proved a special 2–space version of the conjecture, that
can be called the rank-one version, and used [9] again
(this time the concavity of conditional entropy, which is
equivalent to SSA) to deduce a 3–space version of the
uncertainty principle. This quantum version of the un-
certainty principle has attracted a great deal of attention.
Subsequently, Coles et al. [6] and Tomamichel and
Renner [11] were able to eliminate the rank one condition
for the 3–space version and partially removed it for the
2–space version. The significant developments in [6, 10]
came at the cost of rather lengthy calculations and multi-
page proofs.
It is evidently desirable to shorten these proofs and
to clarify the essential mathematical underpinning. It is
shown here that if one uses the original proof structure
of SSA in [9] these proofs can be significantly shortened.
A second thing we do in this paper is eliminate the
aforementioned restriction of [6] in the 2–space case.
Those authors point out that the obvious extension is
necessarily false; we find a more general formulation that
makes the extension possible.
Finally, we go back to the Maassen–Uffink inequality
above and relate the x-space entropy to the p-space en-
tropy, but this time with an auxiliary quantum system,
thereby promoting the Maassen–Uffink inequality to a
truly quantum one.
The notation in this field is not uniform and we be-
gin by defining our terms. We have three Hilbert spaces
(degrees of freedom) H1, H2 and H3 and their tensor
products H123 = H1⊗H2⊗H3, H12 = H1⊗H2 etc. We
have density matrices ρ123, ρ12, etc. (i.e., non-negative
operators of trace one). If ρ123 is defined on H123 then
there is a natural ρ12 on H12 given by ρ12 = Tr3 ρ123,
where Tr3 is the partial trace over H3. In general, en-
tropy is defined by S(ρ) = −TrH ρ ln ρ, where H is one
of the above spaces. If ρ and H are understood, then
S12 = S(ρ12) and S1 = S(ρ1). Conditional entropy is
defined by
S(1|2) = S12 − S2 = S(ρ12)− S(ρ2)
and, as shown in [9], this is a concave function of ρ12.
This concavity is mathematically equivalent to SSA, and
we use it in the Lemma below.
A classical measurement onH1 is defined by a sequence
of operators A1, A2, etc. on H1 such that
∑
j A
∗
jAj = 1.
We will need two measurements in our discussion, so we
will need a second sequence B1, B2, etc. on H1 such
that
∑
k B
∗
kBk = 1. At the end of this paper we will let
the indices j and k be continuous and the sums become
integrals, but for simplicity we stay with sums for now.
2A measurement A is called ‘rank-one’ if every Aj is a
rank-one matrix, namely, Aj = |aj〉〈aj |.
The classical/quantum conditional entropy corre-
sponding to A and ρ is
H(1A|2) =−
∑
j
Tr2
(
Tr1Ajρ12A
∗
j
)
ln
(
Tr1Ajρ12A
∗
j
)
− S(ρ2) .
Note where the summation sits. (This is not the condi-
tional entropy of
∑
j AjρA
∗
j .) The quantity H(1
B|2) is
defined similarly.
2–space theorem.— For any ρ12,
H(1A|2) +H(1B|2) ≥ S(1|2)− 2 ln c1 , (1)
where c1 = supj,k
√
Tr1BkA∗jAjB
∗
k.
In the special case that ρ12 = ρ1⊗ρ2 is a product state
one easily sees that (1) reduces to the 1–space theorems
in [6, 7, 12], which extend the Maassen–Uffink theorem.
Coles et al. [6] prove (1) if either A or B is rank-one
and if c1 is replaced by the smaller, better, number
c∞ = sup
j,k
√
‖BkA∗jAjB
∗
k‖∞
They state correctly that the theorem cannot hold for
general A and B in the c∞ version. However, if either A
or B is rank-one, then c1 = c∞. So our improvement of
Coles et al. consists of eliminating the rank-one condition
and using c1.
The norm ‖BkA
∗
jAjB
∗
k‖∞ is the largest eigen-
value of BkA
∗
jAjB
∗
k, which is also the largest eigen-
value of
√
B∗kBkA
∗
jAj
√
B∗kBk, and thus coincides with
‖
√
A∗jAj
√
B∗kBk‖
2
∞. This shows that our c
2
∞ is, indeed,
the same as that in [6, Eq. (57)].
3–space theorem.[6]— For any ρ123,
H(1A|2) +H(1B|3) ≥ −2 ln c∞ .
We shall prove the 3–space theorem by first proving the
following theorem which is not conveniently expressed in
the H(·|·) notation.
2 1
2
–space theorem.— For any ρ12,
H(1A|2)−
∑
k
Tr12 BkρB
∗
k lnBkρB
∗
k−S(ρ12) ≥ −2 ln c∞ .
This looks like another 2–space theorem but, as we
explain below, it is really the 3-space theorem applied to
a pure state (i.e., rank-one) ρ123. Thus our name.
The key inequality behind the proofs of all our three
theorems is the three operator generalization of the
Golden–Thompson inequality from [13], which was also
the key ingredient in the proof of SSA in [9]. It states
that for non-negative operators X,Y, Z,
Tr elnX−lnY+lnZ ≤
∫ ∞
0
dt TrX(Y + t)−1Z(Y + t)−1.
(2)
Note that in the special case Z = 1 this reduces to the
classical Golden–Thompson inequality. This inequality
was a byproduct of the proof of concavity of the gener-
alized Wigner–Yanase skew information.
We shall also need the Gibbs variational principle
(equivalent to the Peierls–Bogolubov inequality),
Tr ρh− S(ρ) ≥ − lnTr e−h (3)
for any density matrix ρ and any self-adjoint h, and the
Davis operator Jensen inequality [14],
∑
j
A∗j (lnKj)Aj ≤ ln

∑
j
A∗jKjAj

 (4)
for any positive operators Kj and any Aj with∑
j A
∗
jAj = 1.
Proof of the 2–space theorem. We use (4) for both A
and B to bound
H(1A|2) +H(1B|2)− S(1|2) ≥ Tr12 ρ12h− S(ρ12) , (5)
with the operator
h = ln ρ2 − ln
∑
j
A∗jAj
(
Tr1Ajρ12A
∗
j
)
− ln
∑
k
B∗kBk (Tr1Bkρ12B
∗
k) .
We do not need to invoke (4) when A and B are rank-
one; in that case (4) is an equality and the proof simplifies
further.
Thus, by (5) and (3),
H(1A|2) +H(1B|2)− S(1|2) ≥ − lnTr12 e
−h ,
and it remains to show that Tr12 e
−h ≤ c21. Now comes
the crucial step! We use (2) to bound
Tr12 e
−h ≤
∫ ∞
0
dt
∑
j,k
Tr12 Cj,k(t)
with
Cj,k(t) = A
∗
jAj
(
Tr1Ajρ12A
∗
j
)
(ρ2 + t)
−1
×B∗kBk (Tr1Bkρ12B
∗
k) (ρ2 + t)
−1
= A∗jAjB
∗
kBk Dj,k(t) .
Here, Dj,k(t) is the operator on H2 given by(
Tr1Ajρ12A
∗
j
)
(ρ2 + t)
−1 (Tr1Bkρ12B
∗
k) (ρ2 + t)
−1.
3Thus,
Tr12 Cj,k(t) =
(
Tr1A
∗
jAjB
∗
kBk
)
Tr2Dj,k(t)
≤ c21Tr2Dj,k(t) .
We next note that
∑
j Tr1Ajρ12A
∗
j =
∑
j Tr1A
∗
jAjρ12 =
ρ2, and similarly for the k sum, and obtain∑
j,k
Tr2Dj,k(t) = Tr2 ρ
2
2(ρ2 + t)
−2 .
Thus, since
∫∞
0
dt (ρ2 + t)
−2 = ρ−12 ,∫ ∞
0
dt
∑
j,k
Tr2Dj,k(t) = Tr2 ρ2 = 1 .
This completes the proof of the 2–space theorem. QED
We shall now show that the 3–space theorem is a corol-
lary of the 2 1
2
–space theorem, so that the proof of the
2 1
2
–space theorem will finish everything. We use the fol-
lowing:
Lemma.— H(1A|2) is a concave function on the set of
non-negative operators ρ12 on H12.
Proof of the Lemma.— The idea is to view the sum
over j as the trace over an auxiliary space K of a matrix
that happens to be diagonal in this space, and to apply
concavity of the conditional entropy [9] in H2 ⊗ K. The
details are as follows: Let (ej) be an orthonormal basis
of K and consider the operator Γ =
∑
j
(
Tr1 Ajρ12A
∗
j
)
⊗
|ej〉〈ej | onH2⊗K. As in the proof of the 2–space theorem
we have Γ2 = TrK Γ = ρ2 and therefore
H(1A|2) = S(Γ)− S(Γ2) .
This is the conditional entropy of Γ with respect to H2.
Since ρ12 7→ Γ is linear, the asserted concavity follows
from the fact that conditional entropy is concave, as
shown in [9, Thm. 1]. QED
Proof of the 3–space theorem.— It follows from the
Lemma that H(1A|2)+H(1B|3) is a concave function of
ρ123. Thus, for the proof we may assume that ρ123 is a
pure state (rank one). In that case S(ρ3) = S(ρ12) and,
since Bkρ123B
∗
k is pure as well,
H(1B|3) = −
∑
k
Tr12 Bkρ12B
∗
k lnBkρB
∗
k − S(ρ12) .
This reduces the inequality of the 3–space theorem to
that of the 2 1
2
–space theorem. QED
Proof of the 21
2
–space theorem.— The proof runs very
parallel to that of the 2–space theorem. Namely, the right
side in the desired inequality is bounded from below by
Tr12 ρ12h˜− S(ρ12), where now
h˜ = ln ρ2 − ln
∑
j
A∗jAj
(
Tr1Ajρ12A
∗
j
)
− ln
∑
k
B∗kBkρ12B
∗
kBk .
Here we used (4). After applying (3) as before, every-
thing is reduced to showing Tr12 e
−h˜ ≤ c2∞. The crucial
ingredient is again (2) which now leads to
Tr12 e
−h˜ ≤
∫ ∞
0
dt
∑
j,k
Tr12 C˜j,k(t)
with
C˜j,k(t) = A
∗
jAj
(
Tr1Ajρ12A
∗
j
)
(ρ2 + t)
−1
×B∗kBkρ12B
∗
kBk(ρ2 + t)
−1 .
At this point the proof diverges somewhat from that of
the 2–space theorem. Namely, by cyclicity of the trace
we write
Tr12 C˜j,k(t) = Tr12
(
BkA
∗
jAjB
∗
kD˜j,k(t)
)
with D˜j,k(t) given by(
Tr1Ajρ12A
∗
j
)
(ρ2 + t)
−1Bkρ12B
∗
k(ρ2 + t)
−1 .
Since
BkA
∗
jAjB
∗
k
(
Tr1Ajρ12A
∗
j
)
≤ c2∞
(
Tr1Ajρ12A
∗
j
)
and since (ρ2 + t)
−1Bkρ12B
∗
k(ρ2 + t)
−1 ≥ 0, we have
Tr12 C˜j,k(t) ≤ c
2
∞Tr12 D˜j,k(t) .
From here, everything is as before:∑
j,k
Tr12 D˜j,k(t) = Tr2 ρ
2
2(ρ2 + t)
−2
and∫ ∞
0
dt
∑
j,k
Tr12 D˜j,k(t) = Tr2 ρ2 = 1 . QED
At last, we turn to the continuous version, the most
important application being the position-momentum un-
certainty. We start with this case. TakeH1 to be L
2(Rd),
the square-integrable functions on Rd. The spaces H2
and H3 can be anything. The measurement A
∗
jAj is
|aj〉〈aj | where aj is the delta-function δ(x − x
′) for
some x′ in Rn. The j becomes x′, which is a continu-
ous variable, the sum
∑
j becomes the integral
∫
ddx′,
and the normalization condition
∑
j A
∗
jAj = 1 becomes∫
ddx′ δ(x− x′)δ(x′′ − x′) = δ(x− x′′). (We realize that
the delta-function is not a function, but all of this can
be made rigorous.) Similarly, the B∗kBk’s are |bk〉〈bk|
where the bk will also not be square-integrable func-
tions. They will be plane waves ei2pik·x for some k in
R
d. Again, k is a continuous index and sums are inte-
grals. The normalization condition
∑
k B
∗
kBk = 1 be-
comes
∫
ddk e−2piik·xe2piik·x
′
= δ(x− x′).
4Let ρ12 be a density matrix on the Hilbert space
L2(Rd) ⊗ H2. For every fixed x ∈ R
n we can de-
fine 〈x|ρ12|x〉 as an operator on H2. This really means
the partial trace Tr1 of A
∗
jρ12Aj . Likewise we define
〈k|ρ12|k〉 to be Tr1B
∗
kρ12Bk. We see that 〈x|ρ12|x〉 is an
operator-valued density in position space and 〈k|ρ12|k〉
is the corresponding density in momentum space.
Now we apply the 2–space theorem and infer that
H(1A|2) +H(1B|2) ≥ S(1|2) , (6)
where
H(1A|2) = −
∫
Rd
ddxTr2〈x|ρ12|x〉 ln〈x|ρ12|x〉 − S(ρ2)
and
H(1B|2) = −
∫
Rd
ddkTr2〈k|ρ12|k〉 ln〈k|ρ12|k〉 − S(ρ2) .
Here c1 = c∞ = supx,k |e
i2pik·x| = 1 and ln c1 = ln c∞ =
0. This is a generalization of the uncertainty principle
in [7], which is what (6) reduces to when ρ12 = ρ1 ⊗ ρ2
is a product state. The 3–space theorem and the 2 1
2
–
space theorem obviously generalize in a similar way for
the Fourier transform.
This example of a continuum version of our theo-
rems has the obvious generalization to positive operator-
valued measures (POVMs). The interested reader can
work this out for him/herself, but we mention here
one further specific generalization. Let us take H1 =
L2(X, dx), where X is some configuration space (e.g.,
X = Rd or X = a torus or X = a lattice) and L2(X, dx)
are the square integrable functions on X with respect to
some measure dx. The measurements A∗jAj are again
given by rank one projections corresponding to the func-
tions δx′(x) for some x
′ in X . Now let H′1 be a second
Hilbert space of the form L2(K, dk) and let U be a uni-
tary from H1 to H
′
1, which is given by a kernel U(k,x).
For each k ∈ K, we can think of U(k,x) as a function of x
and we define the B∗kBk’s to be the rank-one projections
onto these functions. In this way we obtain, as before,
operators 〈x|ρ12|x〉 and 〈k|ρ12|k〉 on H2. Now H(1
A|2)
and H(1B|2) are defined as in the Fourier transform case,
except that the integration is over the sets X and K, re-
spectively.
Generalized Fourier transform theorem.— (1) is
valid with
c1 = sup
k,x
|U(k,x)| .
An examples in which the classical entropies are simul-
taneously discrete and continuous is the following. Sup-
pose X = Z, i.e., the integers, like the sites in a tight
binding model. An L2 function is the wave function of
an itinerant electron. The second space H′1 is the square
integrable functions on K = (−1/2, 1/2), the Brillouin
zone. In this case the delta-functions, δxx′ , on X are
legitimate Kronecker deltas, and the (normalized) plane
waves onX , parametrized by k ∈ K, are e2piikx. The uni-
tary here is U(k, x) = e2piikx and therefore c1 = 1. Since
dx is counting measure on Z the expression of H(1A|2) is
a sum
∑
x∈Z, whereas dk is ordinary Lebesgue measure
on (−1/2, 1/2) and H(1B|2) is an integral
∫
K
dk.
In conclusion, we have shown several things. (1) The
entropy inequalities in [6, 10] can be proved in a few
lines essentially by imitating the original proof of strong
subadditivity of entropy. (2) We have carried at least one
of these inequalities forward by utilizing the trace norm
c1 instead of the operator norm c∞. (3) We have shown
how these inequalities, suitably interpreted, extend the
2–space entropy uncertainty principle to continuous bases
such as position and momentum.
We thank Zhihao Ma for helpful correspondence.
U.S. National Science Foundation grants PHY-1068285
(R.F.) and PHY-0965859 (E.L.) are acknowledged.
∗ c© 2012 by the authors. This paper may be reproduced,
in its entirety, for non-commercial purposes.
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Extended Quantum Conditional Entropy and Quantum Uncertainty Inequalities

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Extended quantum conditional entropy and quantum uncertainty
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Extended quantum conditional entropy and quantum uncertainty inequalities

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