Comments on “Best Conventional Solutions to the King’s Problem”

Comments on “Best Conventional Solutions to the King’s Problem”

Gen Kimura, Hajime Tanaka, and Masanao Ozawa

Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-8579, Japan Reprint requests to G. K.; E-mail: gen@ims.is.tohoku.ac.jp

Z. Naturforsch.62a,152 – 156 (2007); received December 5, 2006

Conventional solutions of the (Mean) King’s problem without using entanglement have been in- vestigated by P. K. Aravind, Z. Naturforsch.58a, 682 (2003). We report that the upper bound for the success probability claimed is not valid in general but we give a condition for the claim to be justified.

Key words:Quantum State Retrodiction; Mean King’s Problem; Mutually Unbiased Bases.

The (Mean) King’s problem [1 – 3] is a kind of quantum estimation problem with delayed (classical) information and has been studied in detail [2, 3], re- lating with an unsolved problem on the existence of a maximal set of mutually unbiased bases (MUBs) [4].

The standard approach to solve the King’s problem is to utilize entanglement, and nowadays it is shown that the success probability is in fact 1 for anydlevel sys- tem [3]. On the other hand, Aravind [5] has considered the King’s problem without using entanglement, in or- der to elucidate the role of entanglement in this prob- lem. In this case he claimed the following upper bound of the success probabilityP(d)for anydlevel system:

P(d)≤2√

d+d−1

√d(d+1) .

For smalld, this gives the upper bounds:

d 2 3 4 5 8 9

P(d) 0.9024 0.7887 0.7000 0.6315 0.4972 0.4667

The purpose of this paper is, however, to show that this upper bound is not justified ford≥3 in general.

We also give a condition for the claim to be justified.

We begin by recalling the setting of the King’s prob- lem without entanglement, as formulated in [5] by the following steps:

(A) A physicist, Alice, prepares adlevel quantum systemS in a state of her choosing and gives it to the king.

(B) The king carries out a projective measurement with respect to one ofd+1 MUBs {|Ψ_j^µ}^d_j=1 (µ= 0,...,d) [6], and notes the output he obtained.

0932–0784 / 07 / 0300–0152 $ 06.00 c2007 Verlag der Zeitschrift f¨ur Naturforschung, T ¨ubingen·http://znaturforsch.com

(C) The physicist carries out a control measure- ment in an orthonormal basis χ={|χk}^d_k=1 on the systemS.

(D) The king reveals which of the MUBs he has measured. (Delayed classical information.)

(E) The physicist is required to correctly predict the output of the king’s measurement in step (B).

Let the physicist prepare a density operator ρ ^of the systemS in step (A). In step (B), the king is sup- posed to randomly choose one of d+1 MUBs. In other words, the probability that the king chooses the µ^{th MUB is} _d+1¹ . (Therefore, what we consider is a Bayes estimation problem with a uniform prior distri- bution.) Note that the conditional probabilityP(j|ρ,µ) to obtain an output jwith givenρ^andµis determined as P(j|ρ,µ) = Ψj^µ|ρ|Ψj^µ. It is assumed that the king’s measurement is a projective measurement with respect to one of givend+1 MUBs, so that the post measurement state in step (B) is|Ψj^µif the king chose theµth MUB and obtained the outputj. In step (C), the physicist is allowed to make measurements only on the system itself (the “conventional” solution [5]) which are described by an orthonormal basisχ={|χk}^d_k=1 (namely a measurement of a non-degenerate observ- able) [7]. The conditional probability to obtain an out- put with kgiven µ^{, j} and a basis measurementχ ^is given byP(k|µ,j,χ) =|Ψj^µ|χk|². Finally, in step (E) the physicist is required to prepare a decision func- tions:(k,µ)→s_kµ∈ {1,...,d}by which she guesses the king’s output j based on her output k obtained in step (C) and the post-information µ revealed in step (D). [Namely, she predicts the king’s output to bes_kµif she obtained the pair(k,µ)of her outputkand

the king’s MUBµ.] Based on this setting, the physi- cist’s strategy is to find a suitable input stateρ^{, a ba-} sis measurementχ, and a decision function swhich maximize her success probability P_d(ρ,χ,s) to cor- rectly predict the King’s output, where it is deter- mined asP_d(ρ,χ,s) =∑^d_µ=0d+1¹ P(ρ,χ,s,µ)with the conditional success probabilityP(ρ,χ,s,µ)given by P(ρ,χ,s,µ) = ∑^d_j=1P(j|ρ,µ)∑^d_k=1δj,skµP(k|µ,j,χ). Thus the success probabilityP_d(ρ,χ,s) with respect to the physicist’s choice(ρ,χ,s)for ad level system is given by

P_d(ρ,χ,s) = 1 d+1

∑

d µ=0

∑

d

k=1Ψs^µ_kµ|ρ|Ψs^µ_kµ|Ψs^µ_kµ|χk|². (1) Here we have two remarks onP_d(ρ,χ,s). First, since it is an affine function (and hence a convex function) with respect to the input stateρ, its maximum is attained by a pure state. Second, with fixedρ ^andχ, the optimal decision functionsmaxis given by

s_max(k,µ)≡argmax

j=1,...,d

Ψ_j^µ|ρ|Ψ_j^µ|Ψ_j^µ|χk|² . (2)

(Here argmax_j=1,...,d[F(j)] assigns a value j which maximizes the real function F on {1,...,d}.) In- deed, from (1) it is easy to see that P_d(ρ,χ,s)≤ P_d(ρ,χ,smax)for any decision functions.

Aravind claimed an upper bound of the success probability (1):

Claim 1 [5]. The success probability (1) on the guess of the king’s output, is bounded from above by

P_d(ρ,χ,s)≤P_d(|Ψj^µΨj^µ|,χ,s)≤2√

d+d−1

√d(d+1) , (3) where j∈ {1,...,d}andµ∈ {0,...,d}, for any den- sity operatorρ, orthonormal basisχ={|χk}^d_k=1, and decision function s:(k,µ)→s_kµ∈ {1,...,d}.

Claim 1 includes two statements: (i) An optimal in- put state can be taken from one of the basis vectors

|Ψj^µ of the MUBs [the first inequality in (3)]; and (ii) it is bounded from above by ²

√d+d−1

√d(d+1) [the second inequality in (3)].

Indeed (3) is true whend=2, and one can also find a suitable choice of(ρ,χ,µ)to attain the bound [8].

However, as we shall see some counter examples, it is not justified ford≥3 in general.

Counter Example 1(Cased=3). Let{|Ψ_j^µ}³_j=1 (µ = 0,...,3) denote the MUBs constructed by Ivanovi´c [9]. Letρ=|ϕϕ|, where

|ϕ=

0, i

√2,

√3+i 2√

2 T

, (4)

and letχ={|χk}³_k=1be the orthonormal basis defined by

|χ1=

√1 2,

√3i 2√

2,1−i 4

T

,

|χ2=

√i 2,

√3 2√

2,−1+i 4

_T ,

|χ2=

0,1−i 2√

2,

√3 2

_T .

Then, the success probability P₃(ρ,χ,s_max)with the optimal guess function (2) is given by

P₃(ρ,χ,s_max) =21+2√ 2+6√

6

32 0.8212. (5) This is strictly greater than Aravind’s bound ³⁺

√3

6

0.7887 ford=3.

Aravind [5] also showed that the bound (3) is at- tainable ford=4. To see that, he used the following MUBs:

|Ψ1⁰=1000 |Ψ2⁰=0100 |Ψ3⁰=0010 |Ψ4⁰=0001

|Ψ1¹=1111 |Ψ2¹=1¯11¯1 |Ψ3¹=11¯1¯1 |Ψ4¹=1¯1¯11

|Ψ1²=1ii¯1 |Ψ2¹=1¯ii1 |Ψ3¹=1i¯i1 |Ψ4¹=1¯i¯i¯1

|Ψ1³=1¯1ii |Ψ2³=11¯ii |Ψ3³=11ii¯ |Ψ4³=1¯1¯ii¯

|Ψ1⁴=1i¯1i |Ψ2⁴=1¯i1i |Ψ3⁴=1i1¯i |Ψ4⁴=1¯i¯1¯i

[The shorthand notation |Ψj^µ=abcd indicates that

|Ψj^µhas (unnormalized) form(a,b,c,d)^T ∈C⁴, and

¯

a stands for the negative of a.] However, we can also construct the following counter example in these MUBs:

Counter Example 2(Cased=4). Let{|Ψj^µ}⁴_j=1 (µ=0,...,4)be the MUBs given in the above table.

Letρ=|ϕϕ|, where

|ϕ= 1

√2(1,0,−1,0)^T, (6)

and letχ={|χk}⁴_k=1be the orthonormal basis defined by

|χ1= √

3i

2 ,9+3√ 3i 32 ,−√

3+i

32 ,3−3√ 3i 16

T

,

|χ2= √

3+i 4 ,−9√

3+9i

32 ,−

√3i 16 ,3√

3+9i 16

_T ,

|χ3=

0,5−5√ 3i 16 ,−3√

3−9i 16 ,3+√

3i 8

_T ,

|χ4=

0,−3+√ 3i 8 ,−3i

4,−1−√ 3i 4

T

.

Then the success probability is given by P₄(ρ,χ,smax) =6493+1065√

3

10240 0.8142, (7) which is also strictly greater than Aravind’s bound 0.7 ford =4. See also Fig. 1 for numerically generated counter examples.

Since the computations to obtain (5) and (7) are just elementary, we only comment on our choices of the in- put states (4) and (6). Since the optimal success prob- ability can always be attained by pure input states as mentioned before, we may assumeρ =|ϕϕ| for a unit vector |ϕwithout loss of generality. Now sup- pose we have|Ψj^µ₀⁰|ϕ|²=0 for some pair(j₀,µ0). In such a case, if the king choseµ0as his measurement basis, then it is already guaranteed (without resort to the physicist’s measurement result) that the king’s out- put is absolutely distinct from j₀. This suggests that a better strategy of the physicist is to find an input state which has as many pairs(j,µ)satisfying|Ψj^µ|ϕ|²= 0 as possible. In fact, our input states (4) and (6) satisfy the condition|Ψ_j^µ|ϕ|²=0 at(j,µ) = (2,1), (1,2), (1,3), (1,4) and(j,µ) = (2,0), (4,0), (1,1), (2,1), (2,4), (3,4), respectively.

Now we point out a gap in the proof in [5]. Indeed, it is the first inequality in Claim 1 that cannot be justi- fied, while the second inequality is verified to be true.

The crucial error lies in the choice of the decision func- tionsin [5]. Namely, Aravind considered essentially only the decision function given bys(k,µ) = f_µ⁻¹(k), where f_µ(j)≡argmax

k=1,...,d

Ψj^µ|ρ|Ψj^µ|Ψj^µ|χk|² pro- vided that f_µ(j)is bijective for eachµ. In particular, this restriction forces thatsmust also be bijective for

Fig. 1. For (a)d=3 and (b)d=4, the success probability P_d(ρ,χ,smax)is plotted for 1000 randomly generated PVM measurements. The input states are taken from (4) ford=3 and from (6) ford=4. The solid lines stand for Aravind’s bound (3). One finds several counter examples which surpass his bound.

each µ. However, this decision function is not opti- mal in general, and moreover, if we choose the input stateρ as discussed above, then the optimal decision functions_maxgiven in (2) is not injective for manyµ^. Based on this false inference, an argument to “prove”

the first inequality is given in the Appendix of [5].

We remark that the second inequality in Claim 1 is still valid. Thus Aravind’s result can now be stated in the following weaker form:

Theorem 1. Let {|Ψ_j^µ}^d_j=1 (µ=0,...,d) be a set of d+1MUBs. If the input stateρ is of the form ρ=|Ψj^µΨj^µ|, then the success probability P_d(ρ,χ,s) with respect to any basis measurementχ and a deci- sion function s is bounded from above as

P_d(ρ,χ,s)≤2√

d+d−1

√d(d+1) . (8) Aravind obtained this result by the method of La- grange multipliers, but here we give an alternate (and possibly simpler but more rigorous) proof for the reader’s convenience. We begin with the following lemma:

Lemma 1. Let L∈ B(H)be any bounded operator on a Hilbert space H of the form L=∑^m_i=1|φiφi|,

where m∈Nand|φi ∈ H(||φi||=1). Then the operator norm||L|| ≡sup_|ψ∈Hd(||ψ||=1)||L|ψ||has the follow- ing upper bound:

||L|| ≤ lim

n→∞

_m

i₁,...,i

∑

n=1 n−1

∏

k=1|φi_k|φi_k+1| _1/n

. (9)

Proof. For any|ψ ∈ H, we observe

||Lⁿ|ψ||²=

∑

^m

i₁,j1,...,in,jn=1ψ|φin n−1

k=1

∏

φi_k+1|φi_k

φi1|φj1 n−1

∏

l=1φj_l|φj_l+1

φjn|ψ

≤

m i1,j1,...,i

∑

n,jn=1

|φi₁|φj₁| n−1

∏

k=1

|φi_k|φi_k+1| n−1

∏

l=1

|φj_l|φj_l+1|

||ψ||²= m

i1,...,i

∑

2n=1 2n−1

∏

k=1

|φi_k|φi_k+1|

||ψ||²

by the Schwarz inequality. Hence we have

||Lⁿ|| ≤ ^m

i1,...,i

∑

2n=1 2n−1

∏

k=1|φi_k|φi_k+1|.

By combining this with the Gelfand formula||L||= lim_n→∞||Lⁿ||¹ⁿ, we obtain the bound (9).

Proof of Theorem 1. We may set ρ =|Ψ₁⁰Ψ₁⁰| without loss of generality. Then

P_d(ρ,χ,s_max)

= 1 d+1

1+

∑

^d

µ=1

∑

d k=1

1

d|Ψ_s^µ_max(k,µ)|χk|²

≤ 1 d+1

1+1

d

∑

d k=1

sup

|ψ

_d

µ=1

∑

|Ψ_s^µ_max(k,µ)|ψ|²

, (10)

where the supremum is over |ψ ∈ Hd such that

||ψ||=1. LetL≡∑^d_µ=1|Ψ_s(µ)^µ Ψ_s(µ)^µ |, wheresis any real function on{1,...,d}. SinceLis positive, it fol- lows that

||L||=sup

|ψ[ψ|L|ψ] =sup

|ψ

^d

µ=1

∑

|Ψ_s(µ)^µ |ψ|² , (11)

where the supremums are over |ψ ∈ Hd with

||ψ||=1. From Lemma 1 and the property [4] of MUBs|Ψi^µ|Ψj^ν|²=δµνδi j+ (1−δµν)¹_d, we have

||L|| ≤lim

n→∞

d µ1,...,µ

∑

n=1

n−1

∏

l=1|Ψ_s(µ^µl

l)|Ψ_s(µ^µl+1_l+1)| _1/n

=lim

n→∞

d√

√ d

d+d−1 ¹

n √

d+d−1

√d =

√d+d−1

√d . (12) From (10), (11) and (12), we obtain (8).

To summarize, we have constructed some counter examples to Aravind’s general bound (3) in [5]. How- ever, we have reconfirmed that it can be justified with restricted input states (Theorem 1). We shall investi- gate the correct bound which is valid for arbitrary input states in the near future.

Acknowledgements

This work was supported in part by the SCOPE project of the MIC of Japan and the Grand-in-Aid for scientific research (B) 17340021 of the JSPS. G. K. and H. T. are supported by Grant-in-Aid for JSPS Research Fellows.

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[6] Here we assumedis such that a maximal set ofd+ 1 MUBs exists.

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constructed d+1 MUBs by |Ψ_j^µ ≡∑^dk=1U^(µ)_jk |e_k, where U^(µ)_jk = ^√1_dexp_2πi

d (µ+1)(j+k−1)² (µ = 0,...,d−2),U⁽_jk^d−1)=^√1_dexp_2πi

d jk

, andU⁽_jk^d)=δjk.