Download Chain rules for quantum Rényi entropies

Chain rules for quantum Rényi entropies
Frédéric Dupuis˚
arXiv:1410.5455v3 [quant-ph] 25 Feb 2015
Faculty of Informatics
Masaryk University
Brno, Czech Republic
26 February 2015
Abstract
We present chain rules for a new definition of the quantum Rényi conditional entropy sometimes called the “sandwiched” Rényi conditional entropy. More precisely,
we prove analogues of the equation HpAB|Cq “ HpA|BCq ` HpB|Cq, which holds
as an identity for the von Neumann conditional entropy. In the case of the Rényi
entropy, this relation no longer holds as an equality, but survives as an inequality of
the form Hα pAB|Cq ě Hβ pA|BCq ` Hγ pB|Cq, where the parameters α, β, γ obey the
β
γ
α
“ β´1
` γ´1
and pα ´ 1qpβ ´ 1qpγ ´ 1q ą 1; if pα ´ 1qpβ ´ 1qpγ ´ 1q ă 1,
relation α´1
the direction of the inequality is reversed.
1 Introduction
The Shannon entropy is one of the central concepts in information theory: it quantifies the
amount of uncertainty contained in a random variable, and is used to characterize a wide
range of information theoretical tasks. However, it is primarily useful for making asymptotic statements about problems involving repeated experiments, such as i.i.d. sources or
channels. If one is interested not only in asymptotic rates, but in how quickly we can approach these rates with increasing block sizes, the natural quantities that arise are Rényi
entropies. Likewise, in the case of “one-shot” problems, such as those arising in cryptographic settings, the (smooth) min- and max-entropies are usually the relevant quantities. (For instance, these quantities are relevant in scenarios such as privacy amplification
[RK05; Ren05], decoupling [Dup09; DBWR14], and state merging [Ber08; DBWR14].)
All of these information theoretical problems have a quantum counterpart, and it is
therefore desirable to construct quantum versions of these entropic quantities. To do this
systematically, one can use the fact that each of these entropies can be defined via a notion
of divergence between two probability distributions; the entropy of a distribution p is then
derived from the divergence between p and the uniform distribution. For example, the
Shannon entropy can be defined in this way from the Kullback-Leibler divergence. One
can then generalize the divergence quantities to quantum states and define quantum
entropies in this way. However, these quantum counterparts to the divergences are not
unique: the fact that quantum states can fail to commute leads to multiple non-equivalent
definitions that all reduce to the classical version in the commutative case. In the case of
˚
Part of this work was performed while the author was at the Institute for Computer Science, Aarhus
University, Aarhus, Denmark.
1
the Rényi divergence, this phenomenon has led to multiple non-equivalent definitions
being in use. One of them [OP93] has been in use for several years and is defined as
follows: let ρ and σ be two density operators. Then, their α-Rényi-divergence is given by
D̃α pρ}σq :“
1
Trrρα σ 1´α s
log
.
α´1
Trrρs
This definition has found many applications, most notably in hypothesis testing [ON00;
Aud+07; Hay06], as well as for proving the fully quantum asymptotic equipartition property [TCR09; Tom12]. However, a new definition has recently been proposed in [ML+13;
WWY14] which seems to possess better properties than the traditional definition in certain parameter regimes. It is defined as:
ˆ
”´ 1´α 1´α ¯α ı˙
1
1
Dα pρ}σq :“
log
Tr σ 2α ρσ 2α
.
α´1
Trrρs
(See Section 2.2 below for a more complete definition.) This new definition is sometimes
called the “sandwiched” Rényi divergence, σ being the bread and ρ the meat in the expression above. The corresponding conditional Rényi entropy is then given by:
Hα pA|Bqρ :“ ´ min Dα pρAB }idA σB q
σB
for a bipartite state ρAB . This new definition has already found a large number of applications: it characterizes the strong converse exponent in hypothesis testing [MO14a]
(at least in some parameter regimes) and in classical-quantum channel coding [MO14b],
and can be used to prove strong converses for a variety of information theoretical problems [WWY14; CMW14; TWW14]. Furthermore, because of this wide applicability, the
fundamental properties of Dα and Hα are being investigated: a number of properties
have been proven in [ML+13], including the data processing inequality for 1 ă α ď 2
and a duality property (see Fact 4 below, also independently proven in [Bei13]). The
data processing inequality was also proven in [Bei13] for α ą 1 and in [FL13] for the
full range α ě 12 . Moreover, in [AD13], the quantity was generalized to so-called α-zdivergences, and in [TBH14] the authors presented a duality property that involves both
the new “sandwiched” definition and the traditional definition.
The present work is in this vein. One of the most fundamental properties of the von
Neumann entropy is the so-called chain rule: given a tripartite state ρABC , we can break
down the conditional entropy HpAB|Cqρ into two parts: HpAB|Cqρ “ HpA|BCqρ `
HpB|Cqρ . While this rule no longer holds as an equality for the Rényi entropy, one can
nonetheless hope to salvage it in the form of inequalities, as was done in [DBWR14;
VDTR13] for the smooth min-/max-entropies. The result is the main theorem of this
paper:
Theorem 1. Let ρABC P DpA B Cq be a normalized density operator, and let α, β, γ P
β
γ
α
“ β´1
` γ´1
. Then, we have that
p 21 , 1q Y p1, 8q be such that α´1
Hα pAB|Cqρ ě Hβ pA|BCqρ ` Hγ pB|Cqρ
(1)
if pα ´ 1qpβ ´ 1qpγ ´ 1q ą 1, and
Hα pAB|Cqρ ď Hβ pA|BCqρ ` Hγ pB|Cqρ
(2)
if pα ´ 1qpβ ´ 1qpγ ´ 1q ă 1.
The proof is given in Section 3, where the two cases are split into Propositions 7 and 8,
and where the statements proven are slightly more general. The proof is based on the
Riesz-Thorin-type norm interpolation techniques developed in [Bei13], which are then
applied to convenient expressions for the Rényi entropy.
2
2 Preliminaries
2.1 Notation
In the table below, we summarize the notation used throughout the paper:
Symbol
A, B, C, . . .
A, B, . . .
LpA, Bq
LpAq
XAB
XAÑB
PospAq
DpAq
idA
α1 , β 1 , γ 1
α̂, β̂, γ̂
OpAÑB p|iyA |iyB q
X:
XJ
XA ě YA
}X}p
supp X
Definition
Quantum systems
Hilbert spaces corresponding to systems A, B, . . .
Set of linear operators from A to B
LpA, Aq
Operator in LpA Bq
Operator in LpA, Bq
Set of positive semidefinite operators on A
Set of positive semidefinite operators on A with unit trace
Identity operator on A
α´1 β´1 γ´1
α , β , γ
Dual parameter of α, β, γ: α1 ` α̂1 “ 2, etc.
|jyB xi|A , for computational basis vectors |iyA , |jyB
Adjoint of X.
Transpose of X with respect to the computational basis.
X ´ Y P PospAq.
p
1
TrrpX : Xq 2 s p . Note that if p ă 1, this is not a norm.
Support of X
Note in particular the use of the shorthand α1 to denote α´1
α , i.e. the inverse of the Hölder
conjugate of α; it will be used extensively. Using this shorthand, α P p 21 , 1q corresponds
to α1 P p´1, 0q and α ą 1 corresponds to α1 P p0, 1q.
Another convention used extensively throughout the paper is the implicit tensorization by the identity: if we have two operators XAB and YB , by XAB YB we mean XAB pidA YB q. We will also often omit subscripts when doing so should not create confusion.
We will also make use of the operator-vector correspondence. We will endow every
Hilbert space with its own computational basis, denoted by |iyA for the space A and so on,
and we define the linear map OpAÑB : AB Ñ LpA, Bq by its action on the computational
basis as follows: OpAÑB p|iyA |jyB q “ |jyB xi|A .
2.2 The quantum Rényi entropy
We now present the definition of the quantum Rényi divergence, first defined in [ML+13;
WWY14] and sometimes called the “sandwiched” Rényi divergence:
Definition 2 (Quantum Rényi divergence). Let ρ, σ P PospAq, and let α P r 21 , 1q Y p1, 8q.
Then, we define their Rényi α-divergence as
ˆ
„ˆ
˙α ˙
´α1
´α1
1
1
2
2
log
Tr σ ρσ
,
Dα pρ}σq :“
α´1
Trrρs
unless α ą 1 and supp ρ Ę supp σ, in which case Dα pρ}σq :“ 8.
The Rényi entropy is then defined as follows:
3
Definition 3 (Quantum Rényi entropy). Let ρAB P PospA Bq and σB P DpBq, and let
α P r 12 , 1q Y p1, 8q. Then,
Hα pA|Bqρ|σ :“ ´Dα pρAB }idA σB q,
and
Hα pA|Bqρ :“ ´
inf
ωB PDpBq
Dα pρAB }idA ωB q.
By taking the limit as α Ñ 1, we recover the von Neumann entropy; by taking the
limit α Ñ 8, we get the min-entropy [Ren05]; by choosing α “ 12 , we get the max-entropy
[KRS09]; and by choosing α “ 2 we get the collision entropy from [Ren05].
We also recall the duality property of quantum Rényi entropies proven in [ML+13,
Theorem 9] and also independently in [Bei13, Theorem 9], which is the Rényi analogue
of the duality between min- and max-entropy:
Fact 4 (Duality of Rényi entropies). Let |ψyABC P A B C be a normalized pure state, and
let ρABC :“ |ψyxψ|ABC . Then, we have that for any α P p 12 , 1q Y p1, 8q,
Hα pA|Bqρ “ ´Hα̂ pA|Cqρ ,
where
1
α
`
1
α̂
“ 2.
Note here that it is particularly convenient to rephrase the condition α1 `
the shorthand given in the table in Section 2.1: it corresponds to α1 “ ´α̂1 .
1
α̂
“ 2 using
3 Proof of the main result
The first ingredient of the proof is the following generalization of Hadamard’s three-line
theorem proven by Beigi in [Bei13] as part of his proof of the data processing inequality
of the sandwiched Rényi divergence for α ą 1:
Theorem 5 (Theorem 2 from [Bei13]). Let F : S Ñ LpAq, where S :“ tz P C : 0 ď Repzq ď
1u, be a bounded map that is holomorphic on the interior of S and continuous on the boundary.
Let 0 ă θ ă 1 and define pθ by
1
1´θ
θ
“
` .
pθ
p0
p1
For k “ 0, 1 define
Mk “ sup }F pk ` itq}pk .
tPR
Then, we have
}F pθq}pθ ď M01´θ M1θ .
Note that in [Bei13], the theorem is stated for more general norms involving a positive
operator σ (which can be chosen to be σ “ id to obtain this version) and adds a condition
that p0 ď p1 which is not necessary, as can be seen by applying the theorem with F̄ pzq :“
F p1 ´ zq and θ̄ “ 1 ´ θ.
The second ingredient is the following lemma, which gives a particularly useful expression for the Rényi entropy:
4
Lemma 6. Let |ψyABCD P A B C D be a normalized pure state, let XADÑBC “
OpADÑBC p|ψyq, and let α P p 21 , 1q Y p1, 8q, with α̂ such that α1 ` α̂1 “ 2, and let ρABCD “
|ψyxψ| and σC P DpCq. Then,
Hα pAB|Cqρ|σ
›2
›
α1 › α1
› ´α1
2
2 ›
›
“ ´ log sup ›σC XτD › ,
τD PDpDq
(3)
2
›2
›
› ´α1 › α1
2
Hα pB|Cqρ|σ “ ´ log ››σC X ›› ,
2α
› 21
›
›
α1 › α
›
Hα pA|BCqρ “ ´ log sup ›XτD 2 ›› .
τD PDpDq
(4)
(5)
2α̂
Proof. We start by proving (3) from the represention in equation (19) from [ML+13]:
#
´α1
´1
α1 |ψy
τD
if α ă 1
1 log inf τD PDpDq xψ|idAB σC
α
Hα pAB|Cqρ|σ “ ´1
1
´α1
α
α1 log supτD PDpDq xψ|idAB σC τD |ψy if α ą 1,
Putting the prefactor in the exponent, we get
1
1
1
α
´α
τD
|ψy α1
Hα pAB|Cqρ|σ “ ´ log sup xψ|idAB σC
τD PDpDq
´α1
α1
for all α P r 21 , 1q Y p1, 8q. Now, consider the vector σC2 τD2 |ψy: by Lemma 11, we have
that
˙
ˆ
1
1
1
1
´α
´α
α
OpADÑBC σC2 τD2 |ψy
J
“ σC2 XτD
α
2
and therefore
xψ|idAB σC
´α1
1
α1
τD
|ψy α1
›
ˆ ´α1
˙› 21
α1
›
›α
J
2
2
“ ››OpADÑBC σC τD |ψy ››
2
›
› 21
1
1
´α
α
›
›α
J 2 ›
“ ››σC2 XτD
›
2
by Lemma 12, from which (3) follows. We now derive (4) by expressing Hα pB|Cqρ|σ
using (3):
´Hα pB|Cqρ|σ
2
“
sup
τAD PDpADq
“
sup
τAD PDpADq
›
›2
1 › α1
› ´α1
›σ 2 XτAD α2 ›
› C
›
2
”
´α1
1
Tr X : σC XτAD α
›
› 11
1
›
›α
“ ›X : σC ´α X ›
α
›
›2
› ´α1 › α1
“ ››σC 2 X ›› ,
ı 11
α
2α
where the third line follows from Lemma 10, and (4) follows. Finally, we prove (5) via
5
duality:
Hα pA|BCqρ “ ´Hα̂ pA|Dqρ
›2
›
›
´α̂1 › α̂1
›
›
2
“ log inf ›XωD
›
ωD PDpDq
2α̂
› ´21
›
›
α1 › α
“ log inf ››XωD 2 ››
ωD PDpDq
2α̂
›
› 21
›
α1 › α
›
“ ´ log sup ›XωD 2 ›› ,
ωD PDpDq
2α̂
where we used the fact that α̂1 “ ´α1 , and invoked Lemma 9 to get the third line. This
concludes the proof.
We are now ready to prove Theorem 1. We break it down into its two cases, with
Proposition 7 below corresponding to Equation (1) and Proposition 8 further down corresponding to Equation (2). Note that we only prove the cases where α ą 1; the cases
where α ă 1 then follow by applying the duality property (Fact 4) to all the terms.
Proposition 7 (First case of Theorem 1). Let ρABC P DpA B Cq, and let α, β, γ ą 1 be
such that α11 “ β11 ` γ11 . Then, we have that
Hα pAB|Cqρ|σ ě Hβ pA|BCqρ ` Hγ pB|Cqρ|σ
for any σC P DpCq. In particular,
Hα pAB|Cqρ ě Hβ pA|BCqρ ` Hγ pB|Cqρ .
Proof. Let |ψyABCD P A B C D be a purification of ρ, let XADÑBC “ OpADÑBC p|ψyq,
and let τD P DpDq. We use Theorem 5 with the following choices:
´zγ 1
p1´zqβ 1
F pzq “ σC 2 XτD 2
1
1
1
“
“1´
p0
2β
2β̂
p1 “ 2γ
θ“
α1
.
γ1
Now, from these choices, we can show that 1 ´ θ “
α1
β1 :
1 ´ α1 {γ 1
1´θ
“
1
α
α1
1
1
“ 1´ 1
α
γ
1
“ 1.
β
6
Furthermore, we can show that pθ “ 2:
1´θ
θ
1
“
`
pθ
p0
p1
α1
α1
` 1
“
2β̂β 1 2γ γ
ˆ
˙
1
α1
1
“
`
2 β̂β 1 γ 1 γ
˙
ˆ
1
1 ` β1 1 ´ γ1
paq α
`
“
2
β1
γ1
˙
ˆ
1
α
1
1
“
` 1
1
2 β
γ
1
“ ,
2
where at line paq, we have used the fact that
´α1
1
γ
“ 1 ´ γ 1 , and that
α1
Finally, also note that F pθq “ σC2 XτD2 , and that
› ´itγ1
›
p1´itqβ 1 ›
›
2
2
›
›
M0 “ sup ›σC XτD
›
tPR
´itγ 1
due to the fact that σC 2
(6)
›
›
β1 ›
›
2 ›
›
“ ›XτD ›
1
β̂
“ 2´
p0
p0
´itβ 1
and τD 2
are unitary for all t P R. Likewise,
› ´γ1 ›
›
›
M1 “ ››σC2 X ›› .
p1
Hence, for any choice of normalized σ and τ , we have from Theorem 5 that
›
›
› α1
›
β1 › β1
α1 ›
› ´α1
›
›σ 2 Xτ 2 › ď ›Xτ 2 ›
D ›
› C
› D›
› ´γ1 › α11
›
›γ
›σ 2 X › .
C
›
›
›
›2
›2
›
β1 › β1
α1 › α1
›
› ´α1
›σ 2 Xτ 2 › ď ›Xτ 2 ›
D ›
› D›
› C
› ´γ1 › 21
›
›γ
›σ 2 X › ,
› C
›
2β̂
2
This leads to
2β̂
2
2γ
2γ
since α1 ą 0. Maximizing over τD on both sides yields:
›2
›
α1 › α1
› ´α1
2
2 ›
›
sup ›σC XτD › ď
τD PDpDq
Finally, using Lemma 6, we get that
2
›
›2
β1 › β1
›
2 ›
›
sup ›XτD ›
τD PDpDq
2β̂
› ´γ1 › 21
›γ
›
›σ 2 X › .
›
› C
Hα pAB|Cqρ|σ ě Hβ pA|BCqρ ` Hγ pB|Cqρ|σ ,
as advertised.
7
2γ
1
β
“ 1 ` β1.
We now turn to the second case of Theorem 1, corresponding to Equation (2), using
essentially the same proof:
Proposition 8 (Second case of Theorem 1). Let ρABC P DpA B Cq, and let α1 ą 0, and β 1
and γ 1 have opposite signs, and be such that α11 “ β11 ` γ11 . Then, we have that
Hα pAB|Cqρ|σ ď Hβ pA|BCqρ ` Hγ pB|Cqρ|σ
for every σC P DpCq. In particular,
Hα pAB|Cqρ ď Hβ pA|BCqρ ` Hγ pB|Cqρ .
Proof. Let |ψyABCD P A B C D be a purification of ρ, let XADÑBC “ OpADÑBC p|ψyq,
and let τD P DpDq. We split the proof into two cases: first, we assume that β 1 ą 0. We use
Theorem 5 with the following choices:
´α1
F pzq “ σC2
1 1
´z α2βγ1
α1
XτD2
1
´z α2
p0 “ 2
p1 “ 2γ
θ“
´β 1
γ1
β1
1
Now, note that F pθq “ XτD2 , and that 1 ´ θ “ αβ 1 and pθ “ 2β̂, as can be seen by a
calculation very similar to Equation (6). Furthermore, we have that:
›
›
› ´α1 ´it α1 γ1
α1 ›
α1
´it
› 2
2β 1
2 ›
XτD2
M0 “ sup ›σC
›
›
›
tPR
2
›
›
α1 ›
› ´α1
“ ››σC2 XτD2 ››
2
´itα1 γ 1
1
due to the fact that σC 2β
´itα1
are unitary for all t P R. Likewise,
› ´γ1 ›
›
›
M1 “ ››σC2 X ›› .
and τD 2
2γ
Hence, for any choice of normalized σ and τ , we have that
› β1
›
α1 › α1
› ´α1
ď ››σC2 XτD2 ››
› ´γ1 › ´β1 1
›
›γ
›σ 2 X ›
.
› C
›
›2
›2
›
›
β1 › β1
α1 › α1
›
› ´α1
›Xτ 2 › ď ›σ 2 Xτ 2 ›
D ›
› D›
› C
› ´γ1 › ´21
›γ
›
›σ 2 X › .
C
›
›
›
›
β1 ›
›
›Xτ 2 ›
› D›
2β̂
Now, since β 1 ą 0, this leads to:
2
2β̂
and therefore
›
›2
β1 › β1
›
sup ››XτD2 ›› ď
τD PDpDq
2β̂
2
2γ
2γ
›2
›
α1 › α1
› ´α1
sup ››σC2 XτD2 ››
τD PDpDq
8
2
› ´γ1 › ´21
›
›γ
›σ 2 X › .
› C
›
2γ
Using Lemma 6, we get that
Hβ pA|BCqρ ě Hα pAB|Cqρ|σ ´ Hγ pB|Cqρ|σ ,
or,
Hα pAB|Cqρ|σ ď Hβ pA|BCqρ ` Hγ pB|Cqρ|σ .
We now turn to the case where β 1 ă 0 (and therefore γ 1 ą 0). We again use Theorem 5,
but with these choices:
1
´α1
`z α2
2
F pzq “ σC
1 1
α1
`z α2γβ1
2
XτC
p0 “ 2
p1 “ 2β̂
θ“
´γ 1
β1
´γ 1
1
Now, note that F pθq “ σC2 X, that 1 ´ θ “ αγ 1 , and that pθ “ 2γ (see again Equation (6)
for a similar calculation), and that
›
›
1 1›
› ´α1 α1
α1
´it α2γβ1 ›
› 2 ´it 2
2
XτD
M0 “ sup ›σC
›
›
tPR ›
2
›
›
α1 ›
› ´α1
“ ››σC2 XτD2 ››
2
´itα1 β 1
2γ 1
´itα1
are unitary for all t P R. Likewise,
›
›
β1 ›
›
M1 “ ››XτD2 ›› .
and τD
due to the fact that σC 2
2β̂
Hence, for any choice of normalized σ and τ , we have that
› γ1
›
α1 › α1
› ´α1
2
2
ď ››σC XτD ››
›
› ´γ1
β1 › β1
›
2
›Xτ ›
.
› D›
›2
› ´γ1 › 21
›
α1 › α1
›γ
›
› ´α1
›σ 2 X › ď ›σ 2 Xτ 2 ›
D ›
›
› C
› C
›
› ´2
β1 › β1
›
›Xτ 2 › .
› D›
› ´γ1 ›
›
›
›σ 2 X ›
› C
›
2γ
Since γ 1 ą 0, this leads to
2
2
2γ
2β̂
2β̂
Moving the rightmost term to the left-hand side and maximizing over τD on both sides,
we get:
›
›
›2¸
›2
› ´γ1 › 21 ˜
β1 › β1
α1 › α1
›
› ´α1
›γ
›
2
2
2
2
›σ X ›
sup ››XτD ››
ď sup ››σC XτD ›› .
›
› C
2γ
τD PDpDq
2β̂
τD PDpDq
Using Lemma 6, we get that
Hγ pB|Cqρ|σ ` Hβ pA|BCqρ ě Hα pAB|Cqρ|σ .
This concludes the proof.
9
2
Acknowledgments
The author would like to thank Marco Tomamichel, Salman Beigi, Andreas Winter, Ciara
Morgan, and Renato Renner for fruitful discussions and comments. The author acknowledges the support of the Czech Science Foundation GA CR project P202/12/1142 and the
support of the EU FP7 under grant agreement no 323970 (RAQUEL). Part of this work
was performed while the author was at Aarhus University, where he was supported by
the Danish National Research Foundation and The National Science Foundation of China
(under the grant 61061130540) for the Sino-Danish Center for the Theory of Interactive
Computation, and also by the CFEM research centre (supported by the Danish Strategic
Research Council).
A Auxilliary lemmas
Lemma 9. Let ρAB P DpA Bq. Then, we have that
Hα pA|Bq “ ´ log
inf
σB PDpBq
›2
›
´α1 › α1
›
›Xσ 2 › ,
› B ›
2α
where X :“ OpABÑD p|ψyq for a purification |ψyABD of ρ.
Proof. First, we write
Hα pA|Bqρ|σ “ ´Dα pρAB }idA σB q
„ˆ ´α1 ´α1 ˙α 
1
log Tr σB2 ρσB2
“
1´α
›
›
› ´α1 ´α1 ›
´1
2
2
“ 1 log ››σB ρσB ››
α
α
›
›1
1
´α1 › α1
› ´α :
2
2 ›
›
“ ´ log ›σB X XσB ›
α
›
›2
´α1 › α1
›
2 ›
›
“ ´ log ›XσB ›
2α
where we have used the fact that ρAB “ X : X.
Lemma 10. Let α P r 12 , 1q Y p1, 8q. Then, for any X P PospAq, we have that
1
}X}α “ sup TrrY α Xs
Y PDpAq
if α ą 1, and
}X}α “
if α ă 1. (As usual, α1 “
α´1
α .)
inf
Y PDpAq
1
TrrY α Xs
In particular, this means that
1
1
1
}X}αα1 “ sup TrrY α Xs α1 .
Y PDpAq
Proof. This is simply a reformulation of Lemma 11 in [ML+13].
10
Lemma 11. Let |ψy P A B, and let XA P LpAq and YB P LpBq. Then, we have that
J
OpAÑB pXA YB |ψyq “ YB Opp|ψyqXA
.
Proof. This can be shown by a simple manipulation of indices; see for example [Wat11,
Section 2.4].
Lemma 12. Let |ψy P A B. Then,
xψ|ψy “ TrrOpAÑB p|ψyq: OpAÑB p|ψyqs
and therefore,
}|ψy} “ } OpAÑB p|ψyq}2 .
Proof. Again, see [Wat11, Section 2.4].
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