Compression for quantum population coding

0.25 MB·

by  Yuxiang Yang

1
Compression for quantum population coding
Yuxiang Yanga, Ge Baia, Giulio Chiribellaa,b and Masahito Hayashic,d
aDepartment of Computer Science, The University of Hong Kong
b Canadian Institute for Advanced Research, CIFAR Program in Quantum Information Science
cGraduate School of Mathematics, Nagoya University
d Centre for Quantum Technologies, National University of Singapore
Email: [email protected], [email protected], [email protected] & [email protected]
7
1
0 Abstract
2
We study the compression of arbitraryparametric families of n identically prepared finite-dimensionalquantum
b
states, in a setting that can be regarded as a quantum analogue of population coding. For a family with f free
e
F parameters, we propose an asymptotically faithful protocol that requires a memory of overall size (f/2)logn. Our
constructionusesaquantumversionoflocalasymptoticnormalityand,asanintermediatestep,solvestheproblemof
4
2 theoptimalcompressionofnidenticallyprepareddisplacedthermalstates.Ourprotocolachievestheultimatebound
predictedby quantumShannontheory.In addition, we explorethe minimum requirementfor quantummemory:On

] the one hand, the amountof quantum memory used by our protocolcan be made arbitrarily small compared to the
h
overall memory cost; on the other hand, any protocol using only classical memory cannot be faithful.
p
-
nt Index Terms
a
u population coding, compression, quantum system, local asymptotic normality, identically prepared state
q
[
I. INTRODUCTION
3
v Many problems in quantum information theory involve a source that prepares multiple copies of the same
2 quantum state. This is the case, for example, of quantum tomography [1], quantum cloning [2], [3], and quantum
7
state discrimination [4]. The state prepared by the source is generally unknown to the agent who has to carry out
3
3 the task. Instead, the agent knows that the state belongs to some parametric family of density matrices ρ ,
θ θ Θ
0 with the parameter θ varying in the set Θ. Also, it is promised that all the particles emitted by the so{urc}e ∈are
.
1 independently prepared in the same quantum state ρ : when the source is used n times, it generates n quantum
θ
0 particles in the tensor product state ρ n.
7 ⊗θ
1 How much information is contained in the n-particle state ρ⊗θn? One way to address this question is to quantify
: the minimum amount of memory needed to store the state. Solving this problem requires an optimization over
v
i all possible compression protocols. When the number of copies is large, it is tempting to use a classical protocol,
X
wherein the parameterθ is estimated and the estimate is stored in a classicalmemory. However,this type of storage
r
a is generally not faithful, as shown in [5] for several examples of pure state families. In order to achieve a faithful
storage, a non-zero amount of quantum memory is generally required. It is important to stress that the problem of
storingthen-copystates ρ n,θ Θ in aquantummemoryisdifferentfromthestandardproblemofquantumdata
{ ⊗θ ∈ }
compression [6], [7], [8]. In our scenario, the mixed state ρ is not regarded as the average state of an information
θ
source, but, instead, as a physicalencoding of the parameter θ. The goal of compressionis to preservethe encoding
of the parameter θ, by storing the state ρ n into a memory and retrieving it with high fidelity for all possible values
⊗θ
of θ. To stress the difference with standard quantum compression, we refer to our scenario as compression for
quantum population coding. The expression“quantumpopulation coding” refers to the encoding of the parameterθ
into the many-particle state ρ n. We choose this expression in analogy with (classical) population coding, whereby
⊗θ
a parameter is encoded into the population of n individuals [9]. The typical example of population coding arises
in computational neuroscience, where the population consists of neurons and the parameter represents an external
stimulus.
The compression for quantum population coding has been studied by Plesch and Buzˇek [10] in the case where
ρ is a pure qubit state and no error is tolerated (see also [11] for a prototype experimental implementation). A
θ
first extension to mixed states, higher dimensions, and non-zero error was proposed by some of us in [12]. The
2
protocol therein was proven to be optimal under the assumption that the decoding operation must satisfy a suitable
conservation law. Later, a new protocol that reaches the ultimate information-theoretic bound was found for qubit
states [13]. The classical version of the problem was addressed in [14]. However, finding the optimal protocol for
arbitrary parametric families of quantum states has remained as an open problem so far.
In thispaper,weprovidethegeneraltheoryforthe compressionofn-tensorproductstateinaquantumparametric
statefamily. Weconsidertwo categoriesofstatefamilies: families offinite-dimensionalstatesanddisplacedthermal
familiesofinfinite-dimensionalstates.Thesetwocategoriesofstatefamiliesturnouttobeconnectedbythequantum
version of local asymptotic normality (Q-LAN)[15], [16], [17], [18], which reduces n-tensor product of a finite-
dimensional state locally to a displaced thermal state. As the first step, we discuss this kind of compression for
the thermal states family, which can be regarded as the quantum extension of the Gaussian distribution. In the next
step, employing Q-LAN, we reduce the problem of compressing generic finite-dimensional states to the case of
displaced thermal states. Unlike previous works, our protocol does not require any assumption on the symmetry
of the state family. In addition, an intriguing feature of this protocol is that the ratio between the size of quantum
memory and the size of classical memory can be made arbitrarily close to but not equal to zero. Such a feature is
not an incidence but an essence: for identically prepared displaced thermal states and qudit states, we show that
any compression protocol using only classical memory must have non-vanishing error.
The rest of the paper is structured as follows. In Section III we study the compression of displaced thermal
states. In Section IV we propose a protocol for the compression of identically prepared finite-dimensional states.
Optimality of the protocols is proven later in Section VI. In Section V we show that it is necessary to use quantum
memory to achieve faithful compression. Finally, we conclude the paper by some discussions in Section VII.
II. MAIN RESULT.
The main result of our work is the optimal compression of identically prepared quantum states. We consider
two major categories of states: finite dimensional (i.e. qudit) states and displaced thermal states. The key question
addressed here is how much memory is needed at least to encode the states, in a way that they can be recovered
with an error vanishing in the number of input copies. The memory cost essentially depends on the (sub)family
from which the states are drawn. For instance, the memory cost for diagonal qudit states (i.e. classical probability
distributions) should be less than the cost for general qudit states. As a consequence, we need to define the state
subfamilies being considered before stating the main result.
We begin by introducing the parameterization for d(<∞)-dimensional non-degenerate quantum systems, whose
states can be generated by rotating a fixed state ρ (µ) with spectrum µ, i.e.
0
ρ =U ρ (µ)U† (1)
θ ξ 0 ξ
where
ξR T +ξI T
U =exp i ∑ j,k j,k j,k k,j (2)
ξ " 1 j<k d √µj−µk !#
≤ ≤
T =iE iE T =E +E (3)
j,k j,k k,j k,j j,k k,j
−
is the exponential form of a SU(d) element, E is a d d matrix with entry (j,k) equal to 1 and other entries
j,k
×
equal to 0. Therefore, any non-degenerate qudit state can be parameterized as ρ , where θ = (µ,ξ) Rd2 1
θ −
∈
is a vector of parameters with µ = (µ ,µ ,...,µ ) being its spectrum, ordered as µ > > µ > 0, and
1 2 d 1 1 d 1
ξ = (...,ξR ,ξI ,...)(1 j < k d) being the ro−tation parameters. A qudit state (sub)fam·i·l·y will−be denoted
j,k j,k ≤ ≤
dasisc{uρs⊗θsni}oθn∈,ΘwewiatshsuΘm=e ΘΘ1=×[Θa2,a×]··f·o×r aΘ,da2−1 wRh.eWreeΘcaililsathceomrapnogneenotfothfeµi-athfreceomclpaosnseicnatlopfaθra.mFoetresrimifpitlsicirtayngoef
i i ′i i ′i∈
is [µ,µ] for µ >µ and a component of ξ a free quantum parameter if its range is [ξ,ξ] for ξ >ξ. Otherwise
i ′i ′i i i ′i ′i i
the range set of the parameter contains only one value and the parameter is fixed. We denote by f (f ) the number
c q
of free classical (quantum) parameters of the (sub)family.
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Next we introduce displaced thermal states, which are a type of states frequently encountered in quantum optics.
Displaced thermal states can be regarded as coherent states subject to Gaussian additive noise. A coherent state,
describing the state of a laser under ideal conditions, is parametrized as
α =e−|α2|2 ∑∞ αk k α= α eiT T [0,2π),
| i √k!| i | | ∈
k=0
where k is the photon number basis. Gaussian additive noise is described by the quantum channel (completely
{| i}
positive trace preserving map)
Nβ(ρ)=Z d2µ 1π−ββ e−(1−ββ)|µ|2 DµρD†µ (4)
(cid:18) (cid:19)
where D =exp(µaˆ† µ¯aˆ) is the displacement operator and β [0,1) is a suitable parameter. When applied to an
µ
− ∈
input coherent state, the Gaussian additive noise outputs the displaced thermal state
ρ =D ρ(thm)D† α= α eiT T [0,2π), (5)
α,β α β α | | ∈
with
∞
ρ(thm)=N ( 0 0 )=(1 β)∑βi i i . (6)
β β | ih | − |ih |
i=0
ΘWe coΘnsibdeeirnag (thsuebp)afarammileyteorfspnaicdee.nStiicmailllayrparsetphaereqduddiitspstlaatceedcatshee,rtmhearlesataretesth,rdeeenroetaeldpbaryam{ρe⊗αte,nβr}s(fβo,αr)∈thΘewdiisthplaΘce=d
β α
×
thermal state family: the thermal parameter β, the strength of the displacement α , and the phase T =argα. β
| |
is a classical parameter, specifying the probability distribution of the eigenvalues, while α and T are quantum
| |
parameters, determining the eigenstates of the density matrix. Each of the three parameters is free if its range is
[a,b] for a<b, otherwise we assume that the range set contains only a single value and the parameter is fixed.
Again, we denote by f (f ) the number of free classical (quantum) parameters of the (sub)family.
c q
Acompressionprotocolconsistsoftwocomponents:theencoder,whichcompressestheinputstateintoamemory,
and the decoder, which recovers the state from the memory. A protocol is thus represented by a couple of quantum
channels (completely positive trace-preserving linear maps) (E,D) characterizing the encoder and the decoder,
respectively.We focus on faithful compression protocols, whose error vanishesin the large n limit. As a measure of
error, we choose the supremum of the trace distance between the original state and the recovered state D E(ρ n)
◦ ⊗θ
1
ε:=sup ρ n D E(ρ n) . (7)
θ Θ2k ⊗θ − ◦ ⊗θ k1
∈
The main result of our work is the following:
Theorem 1. Let {ρ⊗θn}θ=(µ,ξ)∈Θ be the state (sub)family of n identical displaced thermal states or non-degenerate
qudit states with f free classical parameters and f free quantum parameters. For any δ (0,2/9), the state
c q
∈
family can be compressed into [(1/2+δ)f +(1/2)f ]logn classical bits and (f δ)logn qubits with an error
c q q
ε=O n δ/2 +O n κ(δ) , where κ(δ) is the error of Q-LAN [18] given by Eq. (26). The compression is optimal,
− −
in the sense that any compression protocol requiring a memory of size [(f + f )/2 δ]logn with δ >0 cannot be
c q ′ ′
(cid:0) (cid:1) (cid:0) (cid:1) −
faithful.
Theorem 1 states that to encode each free parameter a memory of size (1/2+δ)logn is required. When the
parameter is classical, the required memory is fully classical; when the parameter is quantum, a quantum memory
of δlogn qubits is required. Note that Theorem 1 solves the compression of several important (sub)families:
The full model family of qudits (f =d 1 and f =d(d 1)).
c q
• − −
The classical qudit subfamily (f =d 1 and f =0) of d-dimensional classical probability distributions can
c q
• −
be compressed into (d/2)logn classical bits, retrieving the result of [14].
The phase-covariant qudit subfamily (f =d 1 and f =d(d 1)/2), where ξI =0 for any j, k.
• c − q − j,k
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III. COMPRESSION OF DISPLACED THERMAL STATES.
In this section, we consider the compression of identically prepared displaced thermal states ρ n for 8 different
⊗α,β
subfamilies, corresponding to cases when β, α and T are either free or fixed. The memory costs are summarized
| |
in Table I, which matches the statement of Theorem 1. For instance, in Case 5 we have f =1 (T is a quantum
q
parameter)and f =1 (β is a classicalparameter), andtherefore the memory costsare δlogn qubits and(1+δ)logn
c
bits. On the other hand, the errors for all cases satisfy the unified bound
ε=O n δ/2 .
−
(cid:16) (cid:17)
TABLE I
COMPRESSIONRATEFORDIFFERENTSTATEFAMILIES.HEREδ>0ISANARBITRARYPOSITIVECONSTANT.
Case displacement α= αeiT thermal parameter β quantum bits classical bits
| |
0 fixed fixed 0 0
1 fixed free 0 (1/2+δ)logn
2 free fixed 2δlogn logn
3 free free 2δlogn (3/2+δ)logn
4 T free; α fixed fixed δlogn (1/2)logn
| |
5 T free; α fixed free δlogn (1+δ)logn
| |
6 α free; T fixed fixed δlogn (1/2)logn
| |
7 α free; T fixed free δlogn (1+δ)logn
| |
We now show details of the compression protocol for each case. Since Case 0 (all parameters are fixed) is
essentially trivial, we start from Case 1, where only β is a free parameter.
A. Case 1: free β; fixed α.
An important property for n identically displaced thermal states is that they are equivalent, up to an unitary
transformation, to a product of thermal states where the displacement appears only on one mode, namely
(n 1)
U ρ nU† =ρ ρ(thm) ⊗ − , (8)
BS ⊗α,β BS √nα,β⊗ β
(cid:16) (cid:17)
where U is a unitary operator, which can be physically implemented through a suitable arrangement of beam
BS
splitters [19], [20]. Usingthis property,wecanconstructaprotocolthatseparatelycompressesthe displacedthermal
state ρ and the n 1 thermal modes. For the compressionof the thermal modes, we have the following lemma.
√nα,β
−
Lemma1(Compressionofidenticallypreparedthermalstates). Foranyx>0,thereexistsaprotocol E(thm),D(thm)
n,x n,x
compressing n copies of a thermal state ρ(thm) into (1/2+x)logn classical bits with error (cid:16) (cid:17)
β
ε =O n x . (9)
thm −
The proof of the above lemma can be found in Append(cid:0)ix. W(cid:1)e emphasize that no quantum memory is required
to encode thermal states. For any δ>0, we can then construct the compression protocol as follows:
Encoder.
•
1) First perform on each input copy the displacement operation D , defined by
α
−
D ( ):=D D†
µ · µ· µ
where D =exp(µaˆ† µ¯aˆ) is thedisplacementoperator.Thedisplacementoperationtransforms eachinput
µ
−
copy ρ into the thermal state ρ(thm).
α,β β
2) Thenapplythe thermalstateencoderE(thm) onthe n-modestateandtheoutcomeis encodedin a classical
n,δ
memory.
Decoder.
•
1) First use the thermal state decoder D(thm) to recover the n copies of the thermal state ρ(thm) from the
n,δ β
classical memory.
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2) Perform the displacement operation D on each mode.
α
Note that the input state can be regarded as the state of n optical modes with each mode in a displaced thermal
state, and thus the compression protocol can be regarded as a sequence of operations on the n-mode system. We
shall use the term “mode” throughout this section.
The above protocol requires only (1/2+δ)logn bits for any δ>0 by Lemma 1. The error is also the same as
in Lemma 1, for the displacement operations are unitary.
B. Case 2: unknown α, fixed β.
Next we study the more involved case when the displacement α is unknown. Let us first look at the idea how
to compress before going into details of the compression protocol. To save as much quantum memory as possible,
we may consider to estimate α and store the outcome in a classical memory. However, any estimate of α comes
at the price of disturbing the input state, and, as shown later in Section V, the distortion caused by measurements
on all input copies is so large that the state cannot be recovered faithfully. Instead of full measurements, here we
adopt a strategy of measuring only a small portion of the input copies so that we still get an estimate of α, while
the distortion can still be recovered. We then perform coherent quantum protocols based on the information gained
by the estimation. Such an idea of gently testing the input state and then performing coherent compression is key
to this work, which allows us to convert the memory cost from quantum to classical as much as possible.
For any δ>0, the protocol runs as follows:
Preprocessing. A preprocessing procedure is needed in order to store the estimate of α: Divide the range of
•
α into n zones, each labeled by a point αˆ in it, so that α α =O(n 1/2) (note that α is complex) for any
i ′ ′′ −
| − |
α,α in the same zone.
′ ′′
Encoder.
•
1) First perform the unitary channel U ( )=U U† on the input state, whereU is the unitary defined
BS · BS · BS BS
(n 1)
by Eq. (8). The output state has the form ρ ρ(thm) ⊗ − .
√nα,β⊗ β
2) Send the first and the last mode through a beam sp(cid:16)litter w(cid:17)ith transitivity n δ. The n-mode state is now
−
(n 2)
ρ ρ(thm) ⊗ − ρ .
√n−n1−δα,β⊗ β ⊗ √n1−δα,β
3) Estimate α by(cid:16) the h(cid:17)eterodyne measurement d2α′ α′ α′ on the last mode, which yields an estimate
{ | ih |}
αˆ =α/√n1 δ with the probability distribution
′ −
Q(αˆ α)=(1 β)exp[ (1 β)n1 δ αˆ α 2]. (10)
−
| − − − | − |
Encode the label of the zone αˆ containing αˆ in a classical memory.
∗
4) Displace the first mode with D .
√n n1 δαˆ
5) The state of the first mode then−go−es−thr∗ough a truncation in the photon number basis, described by the
channel P defined as
n2δ
P (ρ)=P ρP +(1 Tr[P ρ]) 0 0 (11)
n2δ n2δ n2δ n2δ
− | ih |
where
n2δ
P = ∑ m m . (12)
n2δ
| ih |
m=0
The output state on the first mode is encoded in a quantum memory.
Decoder.
•
1) First read αˆ and perform the displacement operation D on the state of the quantum memory.
∗ √n n1 δαˆ
2) Torecoverthedistortioncausedbytheestimationofα,aqua−nt−um∗amplifier[21]Aγn withγn=1/(1 n−δ)
−
is applied to the output state. A quantum amplifier is a device that increases the intensity of quantum
light while preserving its phase information, defined by the following quantum channel
Aγ(ρ)=Tr ecosh−1(√γ)(a†b† ab)(ρ 0 0 )ecosh−1(√γ)(ab a†b†) (13)
B − B −
⊗| ih |
h i
where a and b are the annihilation operators of the input mode and the ancillary mode B, respectively.
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3) Prepare the other (n 1) modes in the thermal state ρ(thm).
− β
4) Perform on all the n modes the inverse of the unitary channel U
BS
U 1(ρ)=U† ρU . (14)
B−S BS BS
The memory cost consists of two parts: the cost of encoding the (rounded) value αˆ of the estimate which is logn
∗
bits and the cost of encoding the first mode (in a displaced thermal state) which is 2δlogn qubits. Overall, the
protocol requires 2δlogn qubits and logn classical bits.
On the other hand, the error of the protocol can be split into three terms as
ε Q(dαˆ α)
≤Zαˆ α>n 1/2+3δ/4 |
| − | −
1
+ sup sup Aγn(ρ ) ρ + P (ρ ) ρ .
2 α,β αˆ :αˆ α n 1/2+3δ/4 √n−n1−δα,β − √nα,β 1 n2δ √n−n1−δ(α−αˆ∗),β − √n−n1−δ(α−αˆ∗),β 1
∗| ∗− |≤ − n(cid:13) (cid:13) (cid:13) (cid:13) o
(cid:13) (cid:13) (cid:13) (cid:13)
Now we bound them one by one. F(cid:13)irst, from Eq. (10) we get(cid:13) (cid:13) (cid:13)
Q(dαˆ α)= (1 β)exp[ (1 β)n1−δ αˆ α 2]=e−Ω(nδ/2).
Zαˆ α>n 1/2+3δ/4 | Zαˆ α>n 1/2+3δ/4 − − − | − |
| − | − | − | −
Second, explicit calculation shows that, when the input state is a displaced thermal state ρ , the output state of
α,β
the amplifier is
βγ+4(1 β)γ(γ 1)
Aγ ρ =ρ β = − − . (15)
α,β α,β′ ′ γ+(1 β)(γ 1)
− −
(cid:0) (cid:1)
We thus have
1
2sαu,βpαˆ :αˆ αsupn 1/2+3δ/4 Aγn(ρ√n−n1−δα,β)−ρ√nα,β 1 =O(n−δ)
∗| ∗− |≤ − (cid:13) (cid:13)
having used Eq. (19). Finally, the error of Pn2δ c(cid:13)(cid:13)an be bounded using the(cid:13)(cid:13)following lemma, whose proof can be
found in Appendix.
Lemma 2 (Photon number truncation of displaced thermal states.). Define the channel P as
N
P (ρ)=P ρP +(1 Tr[P ρ]) 0 0 (16)
N N N N
− | ih |
where
N
P = ∑ k k . (17)
N
| ih |
k=0
When N =Ω α 2+x , P satisfies
N
| |
(cid:0) (cid:1) ε(ρ )= 1 P ρ ρ =βΩ(Nx/8)+e Ω(Nx/4) (18)
α,β 2 N α,β − α,β 1 −
for any 0 β<1. (cid:13) (cid:0) (cid:1) (cid:13)
(cid:13) (cid:13)
≤
In our case we are using the projector P in Eq. (12), and the displacement is √n n1 δ(α αˆ ). Since
n2δ − − − ∗
(n n1 δ) α αˆ 2 =O n3δ/2 , by Lemma 2 we obtain the bound
− ∗
− | − |
1sup (cid:0) sup(cid:1) P (ρ ) ρ =βΩ(nδ/12)+e Ω(nδ/6)
2 α,β αˆ :αˆ α n 1/2+3δ/4 n2δ √n−n1−δ(α−αˆ∗),β − √n−n1−δ(α−αˆ∗),β 1 −
∗| ∗− |≤ − (cid:13) (cid:13)
(cid:13) (cid:13)
Summarizing the above bounds for e(cid:13)ach term of the error, we get (cid:13)
ε=O n δ .
−
(cid:16) (cid:17)
7
C. Case 3: unknown α, unknown β.
Case 3 can be treated in a similar way as Case 2, except that we also have to estimate and encode β. Luckily,
unlike α, the thermal parameter β can be estimated freely (i.e. without distortion of the input state), and thus its
estimation strategy is simpler.
In detail, for any δ>0 we can then construct the protocol for displaced thermal states with unknown β and
unknown α as follows:
Preprocessing.Divide the range of α into n zones,each labeled by a point αˆ in it, so that α α =O(n 1/2)
i ′ ′′ −
• | − |
for any α,α in the same zone.
′ ′′
Encoder.
•
1) First perform the unitary channel U ( )=U U† on the input state, whereU is the unitary defined
BS · BS · BS BS
(n 1)
by Eq. (8). The output state has the form ρ ρ(thm) ⊗ − .
√nα,β⊗ β
2) Estimate β with the von Neumann measurement of(cid:16)the ph(cid:17)oton number on the n 1 copies of ρ(thm) and
− β
ˆ
denote by β the maximum likelihood estimate of β. Note that these copies will not be disturbed since
they are diagonal in the photon number basis.
3) Next, send the first and the last mode through a beam splitter with transitivity n δ. The n-mode state is
−
(n 2)
now ρ ρ(thm) ⊗ − ρ .
√n−n1−δα,β⊗ β ⊗ √n1−δα,β
4) Estimate α by the(cid:16)hetero(cid:17)dyne measurement d2α′ α′ α′ on the last mode, which yields an estimate
{ | ih |}
αˆ =α/√n1 δ with the probability distribution Q(αˆ α) (10). Encode the label of the zone αˆ containing
′ − | ∗
αˆ in a classical memory.
5) Displace the first mode with D .
√n n1 δαˆ
− − − ∗ (n 2)
6) Preparethen-thmodeinthethermalstateρ(thm).Then-modestateisnowρ ρ(thm) ⊗ −
βˆ √n−n1−δ(α−αˆ∗),β⊗ β ⊗
ρ(thm). (cid:16) (cid:17)
βˆ
7) The state of the first mode then goes through a truncation in the photon number basis, described by the
channel P defined by Eq. (16). The output state on the first mode is encoded in a quantum memory.
n2δ
8) Meanwhile, use the thermal state encoder E(thm) (see Lemma 1) to compress the remaining n 1 modes
n 1,δ −
and encode the output state in a classical me−mory.
Decoder.
•
1) First read αˆ and perform the displacement D on the state of the quantum memory.
∗ √n n1 δαˆ
2) Apply a quantum amplifier Aγn with γn=1/(1 −n−−δ)∗to the state.
3) Then use the thermal state decoder D(thm) to re−cover the other (n 1) modes in the thermal state ρ(thm)
n 1,δ − β
from the memory. −
4) Finally, perform the inverse of the unitary channel U .
BS
The memory cost consists of three parts: the cost of encoding the (rounded) value αˆ of the estimate which is logn
∗
bits, the cost of encoding the first mode (displaced thermal state) which is 2δlogn qubits, and the cost of encoding
the other modes (thermal states) which is (1/2+δ)logn bits. Overall, the protocol requires 2δlogn qubits and
(3/2+δ)logn classical bits.
On the other hand, the error of the protocol can be split into several terms as
1 (n 1) n
ε sup D(thm) E(thm) dβˆP(dβˆ β)ρ(thm) ρ(thm) ⊗ − ρ(thm) ⊗
≤ 2 n,δ ◦ n,δ Z | βˆ ⊗ β − β
β (cid:13) (cid:20) (cid:21) (cid:13)1
1(cid:13) (cid:16) (cid:17) (cid:16) (cid:17) (cid:13)
+2(cid:13)(cid:13)sαu,βpαˆ :αˆ αsupn 1/2+3δ/4 PN(num)(ρ√n−n1−δ(α−αˆ∗),β)−ρ√n−n1−δ(α−αˆ∗),β 1+(cid:13)(cid:13) Aγn(ρ√n−n1−δα,β)−ρ√nα,β 1
∗| ∗− |≤ − n(cid:13) (cid:13) (cid:13) (cid:13) o
(cid:13) (cid:13) (cid:13) (cid:13)
+ Q(dαˆ α(cid:13)) (cid:13) (cid:13) (cid:13)
Zαˆ α>n 1/2+3δ/4 |
| − | −
ˆ
where P(dββ) denotes the probability distribution of the estimate for the input value β. The latter three terms are
|
the errors of estimation and compression of the displaced thermal state ρ of the first mode, which have been
√nα,β
8
bounded in the previous subsection as O(n δ). We now bound the first term ε , which is the error of compressing
− β
and estimating thermal states. ε can be further split into two terms as
β
1 n n
ε sup D(thm) E(thm) ρ(thm) ⊗ ρ(thm) ⊗ + P(dβˆ β)ρ(thm) ρ(thm) .
β≤ 2 n,δ ◦ n,δ β − β Z | βˆ − β
β (cid:26)(cid:13) (cid:20) (cid:21) (cid:13)1 (cid:13) (cid:13)1(cid:27)
(cid:13) (cid:16) (cid:17) (cid:16) (cid:17) (cid:13) (cid:13) (cid:13)
On the right hand side o(cid:13)f the inequality, the first error term is bou(cid:13)nded(cid:13)by Lemma 1, while the s(cid:13)econd error term
(cid:13) (cid:13) (cid:13) (cid:13)
is bounded by the following property of the maximum likelihood estimate [22]
l
ˆ
P(dββ) erfc
Z|βˆ−β|≥l/√nFβ | ≤ (cid:18)√2(cid:19)
where F =(β2+1)/[β(1 β)3] is the Fisher information of β and erfc(x):=(2/π) ∞e s2ds is the complementary
β − x −
error function. ε can thus be bounded as R
β
1
ε O(n δ)+ sup sup ρ(thm) ρ(thm) +e Ω(nδ)
β≤ − 2 β βˆ:βˆ β n (1+δ)/2 βˆ − β 1 −
| − |≤ − (cid:13) (cid:13)
=O(n−δ) (cid:13)(cid:13) (cid:13)(cid:13)
having used the tail property of complementary error function and the property
2 α β
ρ(thm) ρ(thm) = | − |+O( α β2). (19)
α − β 1 (1 α)2 | − |
(cid:13) (cid:13) −
Summarizing the above bounds f(cid:13)or each term o(cid:13)f the error, we get
(cid:13) (cid:13)
ε=O n δ .
−
(cid:16) (cid:17)
D. Case 4: fixed α , free T, fixed β and Case 6: free α , fixed T, fixed β.
| | | |
In Case4andCase6,thedisplacementαis partially known.Suchaknowledgeallowsustoreducetheamountof
memory. Since the protocols for these two cases are very similar. The protocol for Case 4 runs as in the following:
Preprocessing. Divide the range of T into n 1/2 zones, each labeled by a point Tˆ in it, so that T T =
− i ′ ′′
• | − |
O(n 1/2) for any T ,T in the same zone.
− ′ ′′
Encoder.
• (n 1)
1) First perform the unitary channel U ( ) on the input state to transform it into ρ ρ(thm) ⊗ − .
BS · √nα,β⊗ β
2) Next, send the first and the last mode through a beam splitter with transitivity n δ/2. Th(cid:16)e n-mo(cid:17)de state
−
(n 2)
is now ρ ρ(thm) ⊗ − ρ .
3) Estimate√Tn−bn1y−δt/2hαe,βh⊗et(cid:16)eroβdyn(cid:17)e measu⊗rem√en1n−tδ/2dα2,βα α α on the last mode, which yields an estimate Tˆ
′ ′ ′
| ih |
which is the phase of α. Encode the label of the zone Tˆ containing Tˆ in a classical memory.
′ ∗
45)) DThisepnlasceendthethfierssttamteoodfetwheithfirDst−m√no−dne1−δt/h2αrˆo∗uwgihthaαˆt∗ru:n=ca|αti|oeniTˆ∗c.hannel P defined in (16) and encode the
nδ
output state in a quantum memory.
Decoder.
•
1) First read αˆ and perform the displacement D on the state of the quantum memory.
∗ √n n1 δ/2αˆ
2) Then apply the quantum amplifier Aγn with γn=−1/−(1 ∗ n−δ/2).
3) Prepare the other (n 1) modes in the thermal state ρ−(thm).
− β
4) Finally, perform U 1 on the output of D(thm) and the quantum memory.
B−S n 1
−
Theprotocolfor Case6 works in the sameway exceptthat α is estimatedinsteadofT. Forboth casesthe memory
| |
costconsists of three parts: the costof encodingthe (rounded) valueαˆ of the estimate which is (1/2)logn bits, the
∗
cost of encoding the first mode (displaced thermal state) which is δlogn qubits, and the cost of encoding the other
modes (thermal states) which is (1/2+δ)logn bits. Overall, the protocol requires δlogn qubits and (1+2δ)logn
classical bits. The error can be bounded as previous as
ε=O n δ/2 .
−
(cid:16) (cid:17)
9
E. Case 5: fixed α , unknown T, unknown β and Case 7: unknown α , fixed T, unknown β.
| | | |
Case 5 and Case 7 can be treated in the same way as Case 4 and Case 6, adding additional steps to estimate
and encode β. The protocol for Case 5 runs as follows:
Preprocessing. Divide the range of T into n 1/2 zones, each labeled by a point Tˆ in it, so that T T =
− i ′ ′′
• | − |
O(n 1/2) for any T ,T in the same zone.
− ′ ′′
Encoder.
• (n 1)
1) First perform the unitary channel U ( ) on the input state to transform it into ρ ρ(thm) ⊗ − .
BS · √nα,β⊗ β
2) Estimate β with the von Neumann measurement of the photon number on the n 1 c(cid:16)opies o(cid:17)f ρ(thm).
− β
ˆ
Denote by β the maximum likelihood estimate of β.
3) Next, send the first and the last mode through a beam splitter with transitivity n δ/2. The n-mode state
−
(n 2)
is now ρ ρ(thm) ⊗ − ρ .
4) Estimate√Tn−bn1y−δt/2hαe,βh⊗et(cid:16)eroβdyn(cid:17)e measu⊗rem√en1n−tδ/2dα2,βα α α on the last mode, which yields an estimate Tˆ
′ ′ ′
| ih |
which is the phase of α. Encode the label of the zone Tˆ containing Tˆ in a classical memory.
′ ∗
5) Displace the first mode with D−√n−n1−δ/2αˆ∗ with αˆ∗:=|α|eiTˆ∗. (n 2)
6) Preparethen-thmodeinthethermalstateρ(thm).Then-modestateisnowρ ρ(thm) ⊗ −
βˆ √n−n1−δ/2(α−αˆ∗),β⊗ β ⊗
ρ(thm). (cid:16) (cid:17)
βˆ
7) Then send the state of the first mode through a truncation channel P defined in (16) and encode the
nδ
output state in a quantum memory.
8) Meanwhile, use the thermal state encoder E(thm) (see Lemma 1) to compress the remaining n 1 modes
n 1,δ −
and encode the output state in a classical me−mory.
Decoder.
•
1) First read αˆ and perform the displacement D on the state of the quantum memory.
∗ √n n1 δ/2αˆ
2) Then apply the quantum amplifier Aγn with γn=−1/−(1 ∗ n−δ/2).
3) Meanwhile, use the thermal state decoder D(thm) to rec−over the other (n 1) modes in the thermal state
n 1,δ −
ρ(thm) from the memory. −
β
4) Finally, perform U 1 on the output of D(thm) and the quantum memory.
B−S n 1
−
The protocol for Case 7 works in the same way except that α is estimated instead of T. For both cases the
| |
memory costconsistsof three parts:the costof encodingthe (rounded)valueαˆ ofthe estimate which is (1/2)logn
∗
bits, the cost of encoding the first mode (displaced thermal state) which is δlogn qubits, and the cost of encoding
the other modes (thermal states) which is (1/2+δ)logn bits. Overall, the protocol requires δlogn qubits and
(1+2δ)logn classical bits. The error can be bounded as previous as
ε=O n δ/2 .
−
(cid:16) (cid:17)
IV. COMPRESSION OF IDENTICALLY PREPARED FINITE DIMENSIONAL SYSTEMS.
In this section, we study the compression of d(<∞)-dimensional non-degeneratequantum systems, based on the
results on displaced thermal states and quantum local asymptotic normality. We show that, just as for displaced
thermal states, each free parameter of a qudit subfamily requires (1/2+δ)logn memory for any δ>0. Details of
the compression protocol are introduced in the following.
A. The compression protocol
To construct a compression protocol, we need the following techniques:
Quantum local asymptotic normality (Q-LAN). The quantum version of local asymptotic normality has been
•
derived in several different contexts [16], [17], [18]. Here we use the result of [18], which states that n
identical copies of a qudit state can be approximated by a classical-quantum Gaussian state in a sufficiently
10
small neighborhood of a point θ Θ for large n. Explicitly, for a fixed point θ =(µ ,ξ ) and for a fixed
0 0 0 0
∈
x (0,1), we define the following neighborhood
∈
Θn,x(θ0)= θ=θ0+δθ/√n, δθ ∞ n2x , (20)
| k k ≤
where δθ is the max vector norm defin(cid:8)ed as δθ :=max(δθ). Q-LA(cid:9)N states that every n-fold product
∞ ∞ i i
k k k k
state ρ n with θ in the neighborhood Θ (θ ) can be approximated by a classical-quantum Gaussian state.
⊗θ n,x 0
Specifically, the stated ρ n is approximated by the Gaussian state
⊗θ
G =N ρ , (21)
n,θ δµ,Iµ0 O αj,k,βj,k
j<k
where N is the multivariate normal distribution with mean δµ and covariance matrix I (equal to the
δµ,Iµ0 µ0
inverse of the quantum Fisher information of the eigenvalues µ, evaluated at µ ) and ρ is the displaced
0 αj,k,βj,k
thermal state defined as
ρ =D ρ(thm)D† (22)
αj,k,βj,k αj,k βj,k αj,k
δξR +iδξI (µ )
j,k j,k 0 k
α = β = . (23)
j,k 2 (µ0)j (µ0)k j,k (µ0)j
−
where (µ0)j and (µ0)k are components of µ0p. The approximation is physically implemented by two quantum
channels T(n) and S(n), satisfying the conditions
θ0 θ0
sup T(n) ρ n G =O n κ(x) (24)
θθ∈∈ΘΘsnnu,,xx(p(θθ00,,cc))(cid:13)(cid:13)(cid:13)(cid:13)ρ⊗θθ0n−(cid:0)S⊗θθ(0n)(cid:1)(−Gn,θn),θ(cid:13)(cid:13)(cid:13)(cid:13)11=O(cid:16)(cid:16)n−−κ(x)(cid:17)(cid:17), (25)
(cid:13) (cid:13)
where 1 denotes the trace norm and (cid:13) (cid:13)
k·k
1 z η 1 x 2y 2 9η
κ(x)=min − − , − − y, − (26)
2 2 − 24
(cid:26) (cid:27)
under the constraints 1>z>1+x/2, y>0, η>0 and η>x y. We note that κ(x)>0 when x [0,2/9).
− ∈
Quantum state tomography. State tomography is an important technique of quantum information processing
•
which is used to determine the density matrix of an unknown quantum state. The role of tomography here is
to provide a rough estimate of θ so that we can apply Q-LAN. The tomography protocol used here [23] gives
0
an estimate ρ of a qudit state ρ with confidence
θˆ θ
1
Prob ρ ρ ε 1 (n+1)3d2e nε2 (27)
2k θ− θˆk1 ≤ ≥ − −
(cid:20) (cid:21)
using n copies of the state.
For any δ (0,2/9), our compression protocol runs as follows:
∈
Encode the input state into memories. We break the encoding of ρ n into four steps:
• ⊗θ
1) Tomography. First take out n1 δ/2 copies of ρ for quantum tomography. In this way, one obtains a
− θ
neighborhood Θ (θ ) (note that 2δ/3 is not the only choice) that contains the state with confidence
n,2δ/3 0
approaching one in the large n limit. To encode the outcome of the tomography, the parameter space Θ
is discretized into a lattice of n(fc+fq)/2 points, each point corresponding to an outcome of tomography.
The point that is closest to θ is encoded in a classical memory.
0
2) Q-LAN. After the tomography step, n n1 δ/2 copies remain for compression. Define
−
−
γ =1/(1 n δ/2). (28)
n −
−
so that the number of remaining copies is n/γ . The n/γ copies are sent through the channelT (n/γn) (24)
n n θ0
which outputs the Gaussian state G defined by Eq. (21).
(n/γn),θ