On interaction-freemeasurements
NII
Keiji Matsumoto

Interaction-free measurement

Detection of the presence of something interacting withthe incoming particle, with high probability, but withalmost no interaction at all

KWHZK protocol(a) nothing is in the blackbox(b) something is present, the amplitude into thebackbox is negligible -> almost no interaction !!
The state of the object inside is almost unchanged
𝐷𝑏𝑏𝑏 = 𝑂(𝑇−1)

Dbox: distortion of the state in the box
T: times of the interaction

In this talk ….
1. Optimality of KWHZK protocol in a certain setting(adversary method,classical technique of query complexity )𝐷𝑏𝑏𝑏 ≥ 𝑂(𝑇 −1 )2. Detection of unitary operation with almost certainty,
and no distortion at all to the input at all to the input
particle.
𝐷𝑖𝑖𝑖𝑖𝑖 = 0

KWHZK is optimal : setting
w
III

B

𝑈𝜃

The blackbox operation is given as c-U
1 00 𝑈𝜃
𝑈0 = 1, 𝑈1 ≠ 1𝑐 − 𝑈𝜃 : unitary on 𝐻𝐶 ⊗ 𝐻𝐼 ⊗ 𝐻𝐵

which path = 𝐻𝐶 = 𝐂 2

KWHZK is optimal : settingw
IIIB𝑈𝜃

The blackbox operation is given as c-U
1 00 𝑈𝜃
𝑈0 = 1, 𝑈1 ≠ 1𝑐 − 𝑈𝜃 : unitary on 𝐻𝐶 ⊗ 𝐻𝐼 ⊗ 𝐻𝐵

which path = 𝐻𝐶 = 𝐂 2⊗𝑛
𝐻𝐵 = 𝐻𝐵𝐵: initial state |𝑓 >⊗𝑛
at each interaction, U acts one of 𝐻𝐵′ ‘s ,(the part interacting with the input is always |f>)

KWHZK is optimal : settingwIIIB𝑈𝜃The blackbox operation is given as c-U
1 00 𝑈𝜃
𝑈0 = 1, 𝑈1 ≠ 1𝑐 − 𝑈𝜃 : unitary on 𝐻𝐶 ⊗ 𝐻𝐼 ⊗ 𝐻𝐵𝐵

which path = 𝐻𝐶 = 𝐂 2⊗𝑛
𝐻𝐵 = 𝐻𝐵𝐵: initial state |𝑓 >⊗𝑛
at each interaction, U acts one of 𝐻𝐵′ ‘s ,(the part interacting with the input is always |f>)

KWHZK is optimal : setting𝑉𝑡w
IIIB𝑈𝜃

Do some information processing on theparticle,
𝑉𝑡 : Unitary on 𝐻𝑊 ⊗ 𝐻𝐶 ⊗ 𝐻It : t-th loop, Hw : working space
1 00 𝑈𝜃

𝑈0 = 1, 𝑈1 ≠ 1𝑐 − 𝑈𝜃 : unitary on 𝐻𝐶 ⊗ 𝐻𝐼 ⊗ 𝐻𝐵 ’which path = 𝐻𝐶 = 𝐂 2⊗𝑛
𝐻𝐵 = 𝐻𝐵𝐵: initial state |𝑓 >⊗𝑛

KWHZK is optimal : settingwI𝑉𝑡IIB𝑈𝜃

After T times
loop,
We measureother than B

I: the inner degree of the input particle
B: the box , = n copies of B’
C: which path=C2
W : working space𝑐 − 𝑈𝜃 : on 𝐻C ⊗ 𝐻I ⊗ 𝐻𝐵𝐵
1 0
𝑈0 = 1, 𝑈1 ≠ 𝑈𝐼 ⊗ 𝑈𝐵0 𝑈𝜃

𝑉𝑡 : Unitary on 𝐻𝑊 ⊗ 𝐻𝐶 ⊗ 𝐻It : t-th loop, W: working space⊗𝑛
𝐻𝐵 = 𝐻𝐵𝐵 : initial state |𝑓 >⊗𝑛

KWHZK is optimal : statement
The state of the whole systemafter the t-th loop when Uθ is true blackbox op

The distortion of the blackbox

Can distinguish U0=1 and U1 with prob 1-εafter T-th step

KWHZK is optimal : proofis hard to handle.
on 𝐻W ⊗ 𝐻C ⊗ 𝐻𝐼

Still can bound key quantities from safer side
RHS:Entanglement between HB and the rest

KWHZK is optimal :proofLemma

𝐶 ≔ 6 1−𝑓 𝑈1 𝑓2 −1/2

KWHZK is optimal :proof
By this lemma,

Thus by

We have the assertion

Conjecture 1
By KWHZK-like protocol,
Cannot distinguish two different operations that are notidentity operation(can distinguish nothing or something,but cannot detect what it is)Complete distortion-free detection of unitary
Given one out of {c-Uθ}, θ∊｛1,..,k}Uθ : acting on HI only (not on HI⊗HB)
can be any non-commutative unitaries
Detect θwith arbitrarily high success probability 1-ε
without any distortion at all to the input state !

Complete distortion-free detection
commutative case
Given one out of {c-Uθ}, θ∊｛1,..,k}Uθ : acting on HI only (not on HI⊗HB)
commute with each otherDetect θwith arbitrarily high success probability 1-ε
without any distortion at all to the input state !

Use phase estimation
HI is set to common eigenvectors
which never be disturbed.
can make the error probability arbitrarily smallComplete distortion-free detection of unitary
Main idea : Reduce the problem to commutative caseby preprocessing
Here we treat the following special case only:
general case is similar−1𝑎𝑒𝑈1 =0𝑒0−1𝑏U2: arbitrary 2x2

Observation: let X be one of Pauli matrices−1𝑎𝑒
𝑈1 𝑋𝑋1 𝑋 =00𝑒 −1𝑏𝑒 −1𝑏00𝑒 −1𝑎= 𝑒 −1(𝑎+𝑏) 𝟏

Complete distortion-free detection of unitary−1𝑎𝑒𝑈1 =0𝑒0−1𝑏U2: arbitrary 2x2

Observation: let X be one of Pauli matrices−1𝑎𝑒
𝑈1 𝑋𝑋1 𝑋 =00

𝑒 −1𝑏𝑒 −1𝑏00𝑒 −1𝑎= 𝑒 −1(𝑎+𝑏) 𝟏

So if U2XU2X≠U1XU1X, the problem is reduced toThe commutative case.
But what if U2XU2X=U1XU1X, ?

Complete distortion-free detection of unitary−1𝑎𝑒𝑈1 =0𝑋𝜆 ≔0𝑒 −1𝜆2𝑒0−1𝑏U2: arbitrary 2x2

𝑒 −1𝜆10𝑈1 𝑋𝜆 𝑈1 𝑋𝜆 = 𝑒 −1(𝑎+𝑏) 𝟏

Lemma : if U1 and U2 does not commute, there is atleast one 𝜆⃑ such that U2XU2X ≠c1Since 𝑈1 𝑋𝜆 𝑈1 𝑋𝜆 ≠ 𝑈2 𝑋𝜆 𝑈2 𝑋𝜆 , we are done.Complete distortion-free detection of unitary
If dim ≧３….

If more than 2 unitaries….
1. Repeat 1 vs the rest many times
2. Since the state keeps to be a pure state all the process,one can make one side of the error zerothus can avoid the accumulation of the error probability

Yet to find out more …
1. For KWHZK protocol, optimization of the constant
2. For complete distortion-free unitary detection, tradeoff between # of using the backbox and the error
probability

Thanks for attentions, if you are still awake
1. Optimality of KWHZK protocol in a certain setting
𝐷𝑏𝑏𝑏 ≥ 𝑂(𝑇 −1 )2. Dtection of unitary operation with almost certainty,
and no distortion to the input at all to the input
particle.
𝐷𝑖𝑖𝑖𝑖𝑖 = 0Conclusion