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Metrics of quantum states
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2011 J. Phys. A: Math. Theor. 44 195303
(http://iopscience.iop.org/1751-8121/44/19/195303)
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IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL
J. Phys. A: Math. Theor. 44 (2011) 195303 (7pp) doi:10.1088/1751-8113/44/19/195303
Metrics of quantum states
Zhi-Hao Ma1 and Jing-Ling Chen2
1 Department of Mathematics, Shanghai Jiaotong University, Shanghai, 200240,People’s Republic of China2 Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin,300071, People’s Republic of China
E-mail: [email protected]
Received 12 November 2010, in final form 17 March 2011Published 14 April 2011Online at stacks.iop.org/JPhysA/44/195303
AbstractIn this work we study metrics of quantum states, which are naturalgeneralizations of the usual trace metric and Bures metric. Some usefulproperties of the metrics are proved, such as the joint convexity and contractivityunder quantum operations. Our result has a potential application in studyingthe geometry of quantum states as well as the entanglement detection.
PACS numbers: 03.67.−a, 03.65.Ta
1. Introduction
The trace metric and the Uhlmann–Jozsa fidelity play important roles in quantum informationtheory [1]. Given two quantum states ρ and σ , the trace metric is given by
dt (ρ, σ ) = 12 tr |ρ − σ |, (1)
and the Uhlmann–Jozsa fidelity reads
F(ρ, σ ) = tr√
ρ12 σρ
12 . (2)
Moreover, based on the Uhlmann–Jozsa fidelity the Bures metric is defined as
dB(ρ, σ ) =√
2 − 2F(ρ, σ ). (3)
The trace metric and the Bures metric are two metrics frequently used in quantuminformation theory. Apparently, they look quite different in comparison with equations (1)and (3). However, a family of metrics is proposed in [2] such that these two important metricscan be expressed in a unified form. The definition given in [2] is as follows.
Definition 1. Let ρ and σ be quantum states; then, a family of metrics can be defined as
dp(ρ, σ ) = sup
(N∑
k=1
|[tr(ρPk)]1p − [tr(σPk)]
1p |p
) 1p
, (4)
1751-8113/11/195303+07$33.00 © 2011 IOP Publishing Ltd Printed in the UK & the USA 1
J. Phys. A: Math. Theor. 44 (2011) 195303 Z-H Ma and J-L Chen
where p is a fixed positive integer, the supremum is taken over all finite families {Pk : k =1, 2, . . . , N} of mutually orthogonal projectors and
∑Nk=1 Pk = I . We call the projectors Pk
attaining the supremum in equation (4) as the optimal projectors.
It has been shown in [2] that when p = 1, the metric d1(ρ, σ ) = 2dt (ρ, σ ), i.e. d1(ρ, σ )
equals two times the trace metric; and when p = 2, the metric d2(ρ, σ ) equals exactlytheBures metric. Thus, equation (4) unifies the trace metric and the Bures metric in a commonframe.
In this paper, we advance the study of the metrics dp(ρ, σ ) in [2] and also present a familyof the related metrics Dp(ρ, σ ). Some useful properties of the metrics are proved, such asthe joint convexity and contractivity under quantum operations as well as the majorizationrelation. Our result has a potential application in studying the geometry of quantum states aswell as the entanglement detection.
2. Contractivity and joint convexity of the metric dp(ρ, σ)
Let ρ be a quantum state. A positive-operator-valued measurement (POVM) is defined as aset of non-negative, Hermitian operators Ek which are complete in the sense that
∑k Ek = I ,
while a projective measurement (PM) requires that Ek are all projectors [3]. In this section,we study whether the metric dp(ρ, σ ) is contractive under quantum operations, which arecompletely positive trace preserving (CPTP) maps. We have the following theorem.
Theorem 1 (Contractivity of the metric under CPTP map). The metric dp(ρ, σ ) is contractiveunder quantum operations for all p. That is, suppose T is a CPTP map, and ρ, σ are densityoperators; then, we have the following inequality:
dp(T (ρ), T (σ )) � dp(ρ, σ ). (5)
Proof. Note that Mn becomes a Hilbert space with inner product 〈X, Y 〉 := tr(XY ∗),X, Y ∈Mn. A linear map T induces its adjoint map as 〈T (X), Y 〉 = 〈X, T ∗Y 〉. If T is a positive map,then its adjoint map T ∗ is also a positive map. The trace preserving property of T means thatT ∗ is unital, namely T ∗(I ) = I .
Now suppose Xk’s are optimal projectors for quantum states T (ρ) and T (σ), so we obtain
dp(T (ρ), T (σ )) = (∑k
|[tr(T (ρ)Xk)]1p − [tr(T (σ )Xk)]
1p |p)
1p . Let Yk = T ∗(Xk); then, Yk � 0
and∑k
Yk = ∑k
T ∗(Xk) = T ∗(∑k
(Xk)) = T ∗(I ) = I . So we have
∑k
|[tr(T (ρ)Xk)]1p − [tr(T (σ )Xk)]
1p |p =
∑k
|[tr(ρT ∗(Xk))]1p − [tr(σT ∗(Xk))]
1p |p
=∑
k
|[tr(ρYk)]1p − [tr(σYk)]
1p |p
� dp(ρ, σ )p. (6)
Thus, we have dp(T (ρ), T (σ )) � dp(ρ, σ ); this ends the proof. �
As for the joint convexity of the metric dp(ρ, σ ), we have the following theorem andleave its proof in the appendix.
Theorem 2. [dp(ρ, σ )]p is joint convex if and only if p = 1, 2; in other words, for p �= 1and 2, [dp(ρ, σ )]p is not joint convex.
2
J. Phys. A: Math. Theor. 44 (2011) 195303 Z-H Ma and J-L Chen
3. A related metrics Dp(ρ, σ)
We know that for p = 1, 2, there are operational forms for dp(ρ, σ ), i.e. d1(ρ, σ ) = tr |ρ −σ |,and d2(ρ, σ ) = √
2 − 2F(ρ, σ ); then, we can obtain the value of d1(ρ, σ ) and d2(ρ, σ )
directly from the matrix entries of ρ and σ . However, for other p, we cannot enjoy thisadvantage, since dp(ρ, σ ) is defined via taking the supremum of projectors, so a naturalquestion arises: just like p = 1, 2, can we obtain the operational form for all dp(ρ, σ )?
This problem is difficult and we would like to leave it as a future topic. What we wantto do in this paper is to introduce a new family of related metrics Dp(ρ, σ ), which have theadvantage of being easy to calculate.
Definition 2. Similar to definition 1, we define Dp(ρ, σ ) := [tr(|ρ 1p − σ
1p |p)]
1p .
Based on the definition, we have the following theorem.
Theorem 3. Let ρ and σ be quantum states; then, Dp(ρ, σ ) is a metric on S(H), whereS(H) denote the set of all quantum states on the Hilbert space H.
Proof. It is easy to show that Dp(ρ, σ ) = Dp(σ, ρ), Dp(ρ, σ ) � 0 and Dp(ρ, ρ) = 0. If
Dp(ρ, σ ) = 0, then |ρ 1p − σ
1p | = 0, so we obtain ρ = σ . Recall that the Schattern p-norm
‖.‖p for an operator y is defined as [4]: ‖y‖p = [tr(|y 1p |p)]
1p . Now we define two matrices y1
and y2, y1 := ρ1p , y2 := σ
1p . After using the triangle inequality for the Schattern p-norm ‖.‖p:
‖y1 − y2‖p � ‖y1‖p + ‖y2‖p = 2,
‖y1 − y3‖p � ‖y1 − y2‖p + ‖y2 − y3‖p,
we obtain the triangle inequality for Dp. Therefore, Dp(ρ, σ ) is a metric onS(H); theorem 3 is
proved. In addition, from (UDU+)1p = UD
1p U+, one easily knows that the metric Dp(ρ, σ )
is unitary invariant.In quantum information theory, majorization has become a powerful tool to detect
entanglement. It was proved in [5] that any separable state ρ acting on Cd ⊗ Cd is majorizedby its reduced state ρA:
ρA ρ, i.e. ∀ k � d :k∑
i=1
λ(A)i �
k∑i=1
λi,
where {λi} and{λ
(A)i
}are the decreasingly ordered eigenvalues of ρ and ρA, respectively. �
Now we give the following majorization relation for the metric Dp(ρ, σ ):
Theorem 4. If 1 � p � q, and ρ, σ are density operators, define the vectorsλ(| q
√ρ − q
√σ |q), λ(| p
√ρ − p
√σ |p) as the vectors of eigenvalues for | q
√ρ − q
√σ |q, | p
√ρ − p
√σ |p,
then the following majorization relation holds:
λ(| q√
ρ − q√
σ |q) ≺w λ(| p√
ρ − p√
σ |p). (7)
That is, λ(| q√
ρ− q√
σ |q) is majorized by λ(| p√
ρ− p√
σ |p). In particular, the following inequalityholds:
(Dq(ρ, σ ))q � (Dp(ρ, σ ))p. (8)
Proof. In [6], Ando proved that if f (t) is a non-negative operator-monotone function on[0,∞) and ‖|.‖| is a unitary invariant norm, then
‖|f (A) − f (B)‖| � ‖|f (|A − B|)‖|, A,B � 0,
3
J. Phys. A: Math. Theor. 44 (2011) 195303 Z-H Ma and J-L Chen
or in the majorization language as
λ(|f (A) − f (B)|) ≺w λ(f (|A − B|)).Since the function f (t) = t
p
q is operator-monotone for 1 � p � q, one has
λ(|ρ p
q − σp
q |) ≺w λ(|ρ − σ | p
q ). (9)
Consider the Schatten q-norm from equation (9) to obtain
tr(|ρ p
q − σp
q |q) � tr(|ρ − σ |p). (10)
Replace ρ, σ in equation (10) by ρ1p , σ
1p respectively; then we complete the proof. �
As for the convex property of Dp(ρ, σ ), similar to the proof of theorem 2, we have that(Dp(ρ, σ ))p is joint convex if and only p = 1, 2. In the following, we would like to discusswhether the metric Dp(ρ, σ ) is contractive under quantum operation.
Especially, for p = 1, we know that D1(ρ, σ ) equal d1(ρ, σ ), so it is contractive underquantum operation. And for p = 2, we know that (D2(ρ, σ ))2 = [tr(|ρ 1
2 − σ12 |2)] =
2 − 2 tr(ρ12 σ
12 ). The Uhlmann–Jozsa fidelity was widely studied and plays a key role in
quantum information theory, but it is not easy to calculate, so some alternative fidelity measureswere introduced, see [8–10]. The new fidelity introduced in [8–10] is all proved to be a goodfidelity measure. On the other hand, we know that Uhlmann–Jozsa fidelity can be rewrittenas F(ρ, σ ) = tr |ρ 1
2 σ12 |, that means, Uhlmann–Jozsa fidelity is the trace of the modulus of
the operator ρ12 σ
12 . However, tr(ρ
12 σ
12 ) is exactly the trace of the operator ρ
12 σ
12 . They only
differ from a phase factor!So this leads to the following idea: if we define another fidelity, called A-fidelity in this
paper, as
FA(ρ, σ ) = [tr(√
ρ√
σ)]2, (11)
then we ask: Can FA(ρ, σ ) be a good fidelity measure? The answer is yes. In fact, in [7], theauthor has shown that FA(ρ, σ ) has the following appealing properties.
Property1: CPTP expansive property. If ρ and σ are density matrices, � is a CPTP map, thenFA(�(ρ),�(σ)) � FA(ρ, σ ).
Property 2. When ρ = |φ〉〈φ| and σ = |ϕ〉〈ϕ| are two pure states, Uhlmann–Jozsa fidelityand A-fidelity both reduce to the inner product, that is, F(ρ, σ ) = FA(ρ, σ ) = |〈φ|ϕ〉|2.
Now we know that if p = 1 or p = 2, Dp(ρ, σ ) is joint convex and also is contractiveunder quantum operation. We will show that they are the only two cases satisfying the CPTPcontractive property.
For others p �= 1, 2, using the numerical method [11], we can get that Dp(ρ, σ ) is neitherdecreasing nor increasing under quantum operation. Therefore, we conclude as following:Dp(ρ, σ ) is contractive under quantum operation if and only p = 1, 2.
To prove that for others p, [dp(ρ, σ )]p is not joint convex, we need the following simpleexample.
Example 1. When p �= 1, 2, and let
ρ =(
0.2 00 0.8
), σ =
(0.4 00 0.6
),
we can prove that [dp(ρ, σ )]p is not joint convex. This example also shows that [Dp(ρ, σ )]p
is not joint convex for p �= 1, 2.
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J. Phys. A: Math. Theor. 44 (2011) 195303 Z-H Ma and J-L Chen
4. Conclusions and applications
Our conclusion is as follows.
(1) The metric dp(ρ, σ ) is contractive under quantum operation for all p, while the metricDp(ρ, σ ) is merely contractive under quantum operation if and only p = 1, 2.
(2) Both [dp(ρ, σ )]p and [Dp(ρ, σ )]p are joint convex if and only p = 1, 2.
Since the metrics dp(ρ, σ ) and Dp(ρ, σ ) are natural generalizations of the trace metricand the Bures metric, we wish that they can be also used to study the geometrical structure ofquantum states [13, 14], such as the volume of quantum states [15–19], which was traditionallystudied by using the trace metric and the Bures metric as tools.
Moreover, our result may be used to study quantum entanglement. We know thatthe geometrical entanglement measure is a famous entanglement measure; it shares someappealing properties [12, 20–23]. The idea of geometrical entanglement measure is based onthe following: the set of all separable states is a convex set denoted by S; if we have a state ρ,then the closer the state ρ to the set S, the less entangled it is. So the entanglement measure isdefined as the minimal distance of the state ρ to any state of S:
E(ρ) = minσ∈S
D(ρ, σ ).
Usually, we use the Bures metric d2(ρ, σ ) to obtain the geometrical entanglement measure;we wish that the metric D2(ρ, σ ) is also a good candidate for the geometrical entanglementmeasure. This interesting work will be done subsequently.
Acknowledgments
ZHM thanks Professor T Ando for his valuable discussion and great help for provingtheorems 1 and 4, and thanks for the valuable discussion with Professor E Werner, Professor QXu, and Dr Deping Ye. ZHM thanks Dr F L Zhang for his discussion and help to complete theprograms for this paper. ZHM is supported by NSF of China (10901103), partially supportedby a grant of science and technology commission of Shanghai Municipality (STCSM, no09XD1402500). JLC is supported by NSF of China (grant no 10975075) and the FundamentalResearch Funds for the Central Universities.
Appendix. proof of theorem 2
Now we give the proof of theorem 2.
Proof. For simplicity, we only discuss 2 × 2 density matrices ρ and σ ; there is no difficultyto prove the N × N case by applying the same method.
First, we give an inequality. Let P1, P2, P3 and P4 be two-dimensional discrete probabilitydistributions with Pi = {Pi1, Pi2}, where Pi1, Pi2 � 0, Pi1 + Pi2 = 1, i = 1, 2, 3, 4. Then thefollowing holds: for all 0 � λ � 1 and p = 1, 2,
2∑j=1
∣∣[λP1j + (1 − λ)P2j ]1p − [λP3j + (1 − λ)P4j ]
1p
∣∣p
� λ
2∑j=1
∣∣(P1j )1p − (P3j )
1p
∣∣p + (1 − λ)
2∑j=1
∣∣(P2j )1p − (P4j )
1p
∣∣p. (A.1)
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J. Phys. A: Math. Theor. 44 (2011) 195303 Z-H Ma and J-L Chen
For p = 1, the above inequality always holds. In fact, suppose P1 = {a1, 1 − a1},P2 = {a2, 1 − a2}, P3 = {b1, 1 − b1}, P4 = {b2, 1 − b2}; then, it easily follows from theabsolute values inequality |λ(a1 − b1) + (1 − λ)(a2 − b2)| � λ|a1 − b1| + (1 − λ)|a2 − b2|.
For p = 2, the problem reduces to prove that the function f (a, b) := |a 12 − b
12 |2 + |(1 −
a)12 − (1 − b)
12 |2 is joint convex; here 0 � a, b � 1. We can obtain the Hessian matrix of
f (a,b) as
H(f ) =(
f′′aa f
′′ab
f′′ba f
′′bb
),
where f′′aa = 1
2a− 32 b
12 + 1
2 (1−a)−32 (1−b)
12 , f
′′ab = f
′′ba = − 1
2a− 12 b− 1
2 − 12 (1−a)−
12 (1−b)−
12 ,
f′′bb = 1
2b− 32 a
12 + 1
2 (1 − b)−32 (1 − a)
12 .
The function f (a,b) is joint convex if and only if the Hessian matrix H(f ) is non-negativedefinite. However H(f ) is non-negative definite if and only if f
′′aa � 0, f
′′bb � 0, f
′′aaf
′′bb −
f′′abf
′′ba � 0, f
′′aa � 0, f
′′bb � 0 always hold, so we only need to prove f
′′aaf
′′bb − f
′′abf
′′ba � 0,
This is equivalent to the following: a− 32 b
12 (1 − a)
12 (1 − b)−
32 + a
12 b− 3
2 (1 − a)−32 (1 − b)
12 �
2a− 12 b− 1
2 (1 − a)−12 (1 − b)−
12 , and we know this holds from the Cauchy–Schwarz inequality.
Now we have proved the inequality (A.1) for p = 1, 2. We will use inequality (A.1) toprove that (dp(ρ, σ ))p is joint convex for p = 1, 2.
Suppose Xk are projectors and∑k
Xk = I ; then tr(λρ1+(1−λ)ρ2)Xk, tr(λσ1+(1−λ)σ2)Xk
and tr(ρ1Xk), tr(ρ2Xk), tr(σ1Xk), tr(σ2Xk) are all discrete probability distributions. Puttingthe four probability distributions tr(ρ1Xk), tr(ρ2Xk), tr(σ1Xk), tr(σ2Xk) in inequality (A.1),
we obtain∑j
|[λ tr(ρ1Xk) + (1 − λ) tr(ρ2Xk)]1p − [λ tr(σ1Xk) + (1 − λ) tr(σ2Xk)]
1p |p �
λ∑j
|(tr(ρ1Xk))1p − (tr(σ1Xk))
1p |p + (1 − λ)
∑j
|(tr(ρ2Xk))1p − (tr(σ2Xk))
1p |p. We take the
supremum in both sides of the above inequality, but the optimal projector for [dp(λρ1 + (1 −λ)ρ2, λσ1 +(1−λ)σ2))]p may not be the optimal projector for [dp(ρ1, σ1)]p and [dp(ρ2, σ2)]p;then we can obtain the needed result. Theorem 1 is proved. �
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