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arXiv:1511.08710v2 [quant-ph] 19 Apr 2016

QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands One of the most sought-after goals in experimental quantum communication is the implementation of a quantum repeater. The performance of quantum repeaters can be assessed by comparing the attained rate with the quantum and private capacity of direct transmission, assisted by unlimited classical two-way communication. However, these quantities are hard to compute, motivating the search for upper bounds. Takeoka, Guha and Wilde found the squashed entanglement of a quantum channel to be an upper bound on both these capacities. In general it is still hard to find the exact value of the squashed entanglement of a quantum channel, but clever sub-optimal squashing channels allow one to upper bound this quantity, and thus also the corresponding capacities. Here, we exploit this idea to obtain bounds for any phase-insensitive Gaussian bosonic channel. This bound allows one to benchmark the implementation of quantum repeaters for a large class of channels used to model communication across fibers. In particular, our bound is applicable to the realistic scenario when there is a restriction on the mean photon number on the input. Furthermore, we show that the squashed entanglement of a channel is convex in the set of channels, and we use a connection between the squashed entanglement of a quantum channel and its entanglement assisted classical capacity. Building on this connection, we obtain the exact squashed entanglement and two-way assisted capacities of the d-dimensional erasure channel and bounds on the amplitude-damping channel and all qubit Pauli channels. In particular, our bound improves on the previous best known squashed entanglement upper bound of the depolarizing channel.

I.

INTRODUCTION

Optical quantum communication over long distances suffers from innate losses [1–5]. While in a classical setting the signal can be amplified at intermediate nodes to counteract this loss, this is prohibited in a quantum setting due to the no-cloning theorem [6]. This problem can be overcome by implementing a quantum repeater, allowing entanglement over larger distances [7, 8]. The successful implementation of a quantum repeater will form an important milestone in the development of a quantum network [9]. At this stage however, physical implementations perform worse than direct transmission [10, 11]. As the experimental results improve it will be necessary to evaluate whether or not an implementation has achieved a rate not possible via direct communications. This can be done by comparing the attainable rate with a quantum repeater [12–19] to the capacity of the associated quantum channel (i.e. direct transmission) for that task. For future quantum networks, arguably the two most relevant tasks are the transmission of quantum information and private classical communication. The capacity of a quantum channel for these two tasks, assuming that we allow the communicating parties to freely exchange classical communication, is given by the two-way assisted quantum and private capacity. We denote these quantities by Q2 (N ) and P2 (N ), respectively. Finding exact values for Q2 (N ) and P2 (N ), however, is highly nontrivial thus motivating the search for upper bounds for them [20]. After having shown that the

∗

[email protected]

squashed entanglement of a channel is a quantity that is such an upper bound [21], Takeoka, Guha and Wilde showed that there is a fundamental rate-loss trade-off in quantum key distribution and entanglement distillation over practical channels [22]. The squashed entanglement Esq (A; B)ρ of a bipartite state ρAB is a quantity defined as 1 Esq (A; B)ρ := inf I(A; B|E 0 ) , (1) 2 SE→E0 which was introduced by Christandl and Winter [23] as an entanglement measure for a bipartite state. The squashed entanglement can be interpreted as the environment E holding some purifying system of ρAB , and then squashing the correlations between A and B as much as possible by applying a channel SE→E 0 that minimizes the conditional mutual information I(A; B|E 0 ). Extending this idea from states to channels, Takeoka, Guha and Wilde [21, 22] defined the squashed entanglement Esq (N ) of a quantum channel as the maximum squashed entanglement that can be achieved between A and B, Esq (N ) := max Esq (A; B)ρ , |ψiAA0

(2)

where ρAB = NA0 →B (|ψi hψ|AA0 ) is the state shared between Alice and Bob after the A0 system is sent through the channel NA0 →B . They showed that Esq (N ) is an upper bound on the two two-way assisted capacities. Unfortunately, there is no known algorithm for computing the squashed entanglement of a channel. This is partially due to the fact that the dimension of E 0 is a priori unbounded and that computing the squashed entanglement of a state is already an NP-hard problem [24] and thus might even be uncomputable. However, fixing the channel in (1) in general yields an upper

2 bound on Esq (N ). Exploiting this idea of fixing a specific “squashing channel” SE→E 0 , Takeoka et al. derived upper bounds on the squashed entanglement of several channels. Notably, they used this technique to find an upper bound for the pure-loss bosonic channel. The main contribution of this paper is an upper bound applicable to all phase-insensitive Gaussian bosonic channels. We apply this bound to the pure-loss channel, the additive noise channel and the thermal channel. Additionally, we obtain results for finite-dimensional channels by using tools that we develop here. The first of these consists of a concrete squashing channel that we call the trivial squashing channel which can be connected with the entanglement-assisted capacity. This connection, first observed by Takeoka et al. (see [25]), allows us to compute the exact two-way assisted capacities of the d-dimensional erasure channel, and bounds on the amplitude damping channel and general Pauli channels. Second, the squashed entanglement of entanglement breaking channels is zero. Third, for channels that can be written as a convex sum of channels the convex sum of the squashed entanglement of each channel is an upper bound, i.e. Esq (N ) is convex on the set of channels. We combine all three of these tools to obtain bounds for the qubit depolarizing channel.

II.

III.

In this section we prove several properties of Esq (N ) that will be of general use for obtaining upper bounds on the squashed entanglement of concrete channels. First we define a squashing channel that we call the trivial squashing channel and connect it to the entanglement assisted capacity of that channel, an observation previously made in [25] by Takeoka et al. Second, we prove that the squashed entanglement of entanglement breaking channels is zero. The third property is that Esq (N ) is convex in the set of channels.

A.

The trivial squashing channel

One possible squashing channel SE→E 0 is the identity channel, which we will call the trivial squashing channel. The state on ABE 0 is pure, from which it can easily be calculated that 1 I(A; B|E) 2 1 = max (H(A|E) − H(A|BE)) |φiAA0 2 1 = max (H(AE) − H(E) |φiAA0 2

Esq (N ) ≤ max

|φiAA0

NOTATION

In this section we lay out the notation and conventions that we follow in this paper. For a quantum state ρA the von Neumann entropy of ρA is defined as H(A) = −trρA log ρA . For convenience we take all logarithms in base two and set log2 (·) ≡ log(·). For a quantum state ρ := ρAB the conditional entropy of system A given B is defined as H(A|B)ρ = H(AB)ρ − H(B)ρ . Here H(B) is computed over the state ρB = trA (ρAB ), where we denote the partial trace over system A of a state ρAB by trA (ρAB ). For a tripartite state ρABE the conditional mutual information is defined as I(A; B|E) = H(A|E) − H(A|BE). Whenever there is confusion regarding the state over which we are computing an entropic quantity we will add the state as a subscript. A quantum channel NA0 →B is a completely positive and trace preserving map [26] between linear operators on Hilbert spaces HA0 and HB . A quantum channel N can always be embedded into an isometry VAN0 →BE that takes the input to the output system B together with an auxiliary system E that we call the environment. This isometry is called the Stinespring dilation of the channel. The action of the channel is recovered by tracing out the environment: N (ρ) = trE (V ρV ∗ ). We denote the d-dimensional maximally mixed state by π. The dimension of π is implicit and should be clear from the context. Let N be a channel with input and output dimension d. Then N is unital if N (π) = π.

SOME PROPERTIES OF Esq (N )

− H(ABE) + H(BE)) 1 = max (H(B) + H(A) − H(AB)) |φiAA0 2 1 = max I(A; B) . |φiAA0 2

(3) (4)

(5) (6) (7)

The maximization in the right hand of (7), up to the 1/2 factor, characterizes the capacity of a quantum channel for transmitting classical information assisted by unlimited entanglement [27]. In other words, the squashed entanglement is bounded from above by one half the entanglement assisted capacity of the channel which we denote by CE (N ). This connection, which was first observed by Takeoka et al. (see [25]), allows us to bound the squashed entanglement for all channels for which CE (N ) is known.

B.

Entanglement breaking channels

Entanglement breaking channels have zero private and quantum capacities assisted by two-way communications. We show that the squashed entanglement of these channels is also zero, following a similar approach as was done for the squashed entanglement of separable states in [28]. In order to see this note that if an entanglement breaking channel NEB is applied to half of a bipartite state, the output is always separable and can be written as a

3 convex combination of product states, ψAB = I ⊗ NEB (|ψi hψ|AA0 ) X = λi |αi i hαi |A ⊗ |βi i hβi |B ,

(8) (9)

i

where we denote by I the identity map. A possible purification of ψAB is X√ λi |αi iA |βi iB |iiE1 |iiE2 , (10) |ψiABE1 E2 = i

where {|iiE1 } and {|iiE2 } are sets of orthonormal states. If the squashing channel consists of tracing out the E2 system, the resulting state is X

λi |αi i hαi |A ⊗ |βi i hβi |B ⊗ |ii hi|E1 ,

That is, the trivial squashing channel is the optimal squashing channel, yielding both two-way assisted capacities and the squashed entanglement of the d-dimensional erasure channel. We note that, up until now, this class of channels is the only class whose squashed entanglement has been calculated exactly. Independently of our work, in [30] the two-way assisted capacities of the ddimensional erasure channel are established by computing the entanglement flux of the channel, which is also an upper bound on P2 . A second channel we can apply the trivial isometry to γ is the qubit damping channel NAD , a channel that models energy dissipation in two-level systems. The qubit amplitude damping channel is defined as

γ NAD (ρ) :=

(11)

Ai ρA†i ,

(15)

i=0

i

where

which has zero conditional mutual information.

A0 = C.

1 X

Convexity of Esq (N ) in the set of channels

The squashed entanglement of the channel is convex in the set of channels. We prove this in the Appendix following similar ideas to the ones used in [23] to prove that the squashed entanglement isP convex in the set of states. P Hence, if N = j pj Nj with j pj = 1 and pj ≥ 0, then Esq (N ) ≤

X

pj Esq (Nj ) .

(12)

√ 1 √ 0 0 γ , A1 = 0 0 0 1−γ

(16)

with amplitude damping parameter γ ∈ [0, 1]. Since the entanglement assisted classical capacity of the amplitude damping channel is known [26] to be equal to γ CE (NAD ) = max [h(p) + h((1 − γ)p) − h(γp)] , (17) p∈{0,1}

where h(x) = −x log(x) − (1 − x) log(1 − x) is the binary entropy, we immediately find the bound

j

IV.

γ γ P2 (NAD ) ≤ Esq (NAD )≤

FINITE-DIMENSIONAL CHANNELS

To build intuition before moving to bosonic channels, let us first bound the squashed entanglement of finitedimensional channels, i.e. channels where both the input and output dimensions are finite. An illustrative example of the effectiveness of the trivial squashing channel is the d-dimensional erasure channel Epd (ρ) = (1−p)ρ+p |ei he|, where ρ is a d−dimensional state and |ei is an erasure flag orthogonal to the support of any ρ on the input [26]. It is well known that CE (Epd ) = 2(1 − p) log(d) [26] and that Q2 (Epd ) = (1 − p) log(d) [29]. In general we have Q2 (N ) ≤ P2 (N ) ≤ Esq (N ) ≤

1 CE (N ) , 2

(13)

where the first inequality holds since the squashed entanglement of a channel is an upper bound on Q2 (N ) and the second inequality follows from applying the trivial squashing channel. In the specific case of the erasure channel, we then must have that Q2 (Epd ) = P2 (Epd ) = Esq (Epd ) = (1 − p) log(d) .

(14)

1 γ CE (NAD ) . 2

(18)

A comparison of this bound with the best known lower bound, given by the reverse coherent information (RCI) γ maxp [h(p) − h(pγ)], and an upper bound P2 (NAD ) ≤ min{1, − log γ} found by Pirandola et al. [31] using an entanglement flux approach, can be seen in Figure 1. A third interesting example are d-dimensional unital channels for which the maximally entangled state on AA0 maximizes the mutual information I(A; B). For these channels the trivial squashing channel gives the following compact upper bound 1 I(A; B) 2 1 = [H(A) + H(B) − H(AB)] 2 1 = log(d) − H(E) . 2

Esq (N ) ≤

(19) (20) (21)

In particular, this bound holds for any Pauli channel, where we have that d = 2. Any Pauli channel can be written as P(ρ) = p0 ρ + p1 XρX + p2 XZρZX + p3 ZρZ ,

(22)

4 1

0.6 0.4

Takeoka et al. [21] Optimized Esq bound Pirandola et al. [30,31] RCI lower bound [32]

0.8

Rate

Rate

0.8

0.6 0.4 0.2

0.2 0 0

1

Pirandola et al. New Esq upper bound RCI lower bound

0.2

0.4

0.6

0.8

0 0

1

FIG. 1. Comparison of bounds for the amplitude damping channel. In dashed green the upper bound by Pirandola et al. [31], in solid blue the upper bound found in this paper and the dash-dotted magenta line is a lower bound given by the reverse coherent information [32].

P3 with i=0 pi = 1. Choosing without loss of generality the maximally entangled state |Φ+ iAA0 = √12 [|00i + |11i]AA0 as input on AA0 , we see that the output has a purification of the form √ + √ p0 Φ AB |00iE + p1 Ψ+ AB |01iE √ √ + p 2 Ψ− |10i + p3 Φ− |11i . (23) AB

E

AB

E

From orthogonality of the Bell states, it can be seen that the entropy of the environment coincides with the classical entropy of the probability vector p = (p0 , p1 , p2 , p3 ). P3 That is, H(E) = H(p) with H(p) ≡ − i=0 pi log pi . From this it follows that 1 Esq (P) ≤ 1 − H(p) . 2

3p log(p) + (4 − 3p) log(4 − 3p) . 8

(25)

The depolarizing channel can also be written as a convex combination of two other depolarizing channels, allowing us to use the convexity of Esq (N ) in the set of channels to improve on the upper bound in equation (25). We can compute the squashed entanglement of each individual channel and multiply it by the appropriate weight. Using this idea (see section 2 in the Appendix), we obtain the following stronger upper bound Esq (Dp ) ≤ min (1 − α) 0≤≤p

0.4

0.6

0.8

1

FIG. 2. Comparison of bounds for the depolarizing channel. The dotted red line is the upper bound by Takeoka et al. [21], the dashed blue line is the optimized squashed entanglement bound in this paper, the solid green line is the entanglement flux upper bound by Pirandola et al. [30, 31] and the magenta line is a lower bound given by the reverse coherent information [32].

p− . This bound is equal to (25) for 0 ≤ where α = 2/3− 1 p . 3 , after which it linearly goes to zero at p = 23 . See Figure 2 for a comparison of this new bound, the bound by Takeoka et al. [21, 33], the bound by Pirandola et al. [31], and the reverse coherent information [32].

V.

A.

PHASE-INSENSITIVE GAUSSIAN BOSONIC CHANNELS An upper bound on phase-insensitive channels

(24)

Hence, we also obtain that 2 − H(p) is the entanglement assisted classical capacity of a Pauli channel P. Let us now apply the bound for Pauli channels to a concrete channel, the (binary) depolarizing channel Dp . The action of this channel is Dp (ρ) ≡ (1 − p)ρ + pπ for p ∈ [0, 1]. This corresponds with the Pauli channel given p p p by p = (1 − 3p 4 , 4 , 4 , 4 ). After this identification we find that Esq (Dp ) ≤

0.2

p

γ

3 log() + (4 − 3) log(4 − 3) . 8 (26)

In this section we discuss our main result, an upper bound on the squashed entanglement of any phaseinsensitive Gaussian bosonic channel. Gaussian bosonic channels are of interest because they are used to model a large class of relevant operations on bosonic systems [34]. Phase-insensitive channels are those Gaussian bosonic channels which add equal noise in each quadrature of the bosonic systems. Imperfections in experimental setups for quantum communication with photons are modeled by phase-insensitive channels, motivating us to upper bound the squashed entanglement of all such channels. In particular this motivates the search for bounds where the input of the channel has a constraint on the mean photon number N . Any phase-insensitive channel NPI is completely characterized by its a loss/gain parameter τ and noise parameter ν. The Stinespring dilation of such a channel con2τ sists of a beamsplitter with transmissivity T = τ +ν+1 interacting with the vacuum on E1 , and a √ two-mode squeezer with squeezing parameter r = acosh( G) with the amplification G = τ +ν+1 ≥ 1 interacting with the 2 vacuum on E2 [35] (see Figure 3 and the Appendix for a detailed definition of the channel). T and G also

5 completely characterize any phase-insensitive channel. Takeoka et al. [21, 22, 33] found bounds for such channels by only considering the beamsplitter part of the Stinespring dilation. To be a valid channel, we must have that ν ≥ |1 − τ |. We further have that phase-insensitive channels are entanglement breaking whenever ν ≥ τ + 1 [36], or equivalently, G(1−T ) ≥ 1. Hence, the squashed entanglement must be zero for channels with such parameters as discussed in the tools section. Since we are interested in phase-insensitive Gaussian channels, we make the ansatz that a good squashing map will be a phase-insensitive channel. Numerical work suggests that, if only phase-insensitive isometries are considered, the pure-loss channel and the amplification channel separately have as optimal squashing isometry the balanced beamsplitter interacting with the vacuum. This motivates us to use the isometry consisting of two balanced beamsplitters at the outputs of the first beamsplitter and the two-mode squeezer (see Figure 3). Using this isometry we obtain a bound for all phase-insensitive channels with restricted mean photon number N (see Appendix for a derivation and a proof that the equation is monotonically non-decreasing as a function of N ). This equation equals g

νBE10 E20

1

+g

νBE10 E20

2

−g

νE10 E20

1

−g

νE10 E20

2

,

(27) with g(x) =

x+1 2

log( x+1 2 )−

x−1 2

log( x−1 2 ) [34] and

q 2 2 2 − = − 1+G +2N (1−T +GT (G−1))+N 2(GT −1) +(G−1+N (GT −1))Ω q 2 2 2 − νE10 E20 2 = − 1+G +2N (1−T +GT (G−1))+N 2(GT −1) −(G−1+N (GT −1))Ω q 2 2 2 + νBE10 E20 1 = − 1+G +2N (1−T +GT (G+1))+N 2(1+GT ) +(1+G+N (1+GT ))Ω q 2 2 2 + νBE10 E20 2 = − 1+G +2N (1−T +GT (G+1))+N 2(1+GT ) −(1+G+N (1+GT ))Ω νE10 E20

1

where we have set Ω± =

p

(1 + N )2 − 4N T ± 2G(1 + N )(N T − 1) + (G + GN T )2

. (28)

As N → ∞, the bound above converges to its maximum value of 1+T G+1 1 − T 2 G log( 1−T ) − G2 − 1 T log( G−1 ) Esq (NPI ) ≤ , 2 2 1−G T (29) Rewriting the upper bound as function of the channel parameters τ and ν [34] we obtain the upper bound Esq (NPI ) ≤

ζ(1 + ν + 3τ, 1 + ν − τ ) − τ ζ(τ + ν + 3, τ + ν − 1) , 2(1 + ν + τ )(1 − τ 2 )

(30) where ζ(a, b) = ab log( ab ).

E1

E2

A0 B1

S

F10

F1 B2

G

B

F20

F2 B3

E10

E20

FIG. 3. A squashing isometry for any phase-insensitive Gaussian channel NPI taking A0 to B. The beamsplitter B1 and the two-mode squeezer S form the Stinespring dilation, while the balanced beamsplitters B2 and B3 form the squashing map. The beamsplitter B1 interacts with the vacuum on E1 and A, and the two-mode squeezer S interacts with the output of B2 and the vacuum on E2 . The squashing isometry consists of two balanced beamsplitters B2 and B3 interacting with the vacuum on F1 and F2 and the output of the beamsplitter B1 and the two-mode squeezer S.

B. Application to concrete phase-insensitive Gaussian channels with unconstrained photon input 1.

Quantum-limited phase-insensitive channels

A pure-loss channel has G = 1. As a consequence, for pure-loss channels the bound in equation (29) reduces to 1+T ). This bound coincides with the bound found log( 1−T by Takeoka et al. In the opposite extreme we find quantum-limited amplifying channels, that is channels with T = 1 and G > 1. For these channels, the bound by Takeoka is equal to infinity while (29) is non-trivial. Concretely, it reduces G+1 to the finite value of log( G−1 ). This should be compared with the exact capacities independently found by Pirandola et al. [30, 31, 37] using an entanglement flux G approach, Q2 = P2 = log( G−1 ). 2.

Additive noise channel

An additive noise channel only adds noise to the input, without damping or amplifying the signal. For 1 an additive noise channel Nadd we have T = n+1 and 1 G = T = n + 1, where n is the noise variance. Taking the limit of equation (29) as G → T1 = n + 1 we show in the Appendix that the upper bound becomes T2 + 1 1+T 1 log( )− 2T 1−T ln 2 n2 + 2n + 2 n+2 1 = log( )− . 2n + 2 n ln 2

Esq (Nadd ) ≤

(31) (32)

6 8

Rate

6

Rate

101

Takeoka et al. [21,33] New Esq upper bound Pirandola et al. [30,31] IC lower bound [38]

4 2 0 0

10

Takeoka et al. [21,33] New Esq upper bound Pirandola et al. [30,31,37] RCI lower bound [32]

0

10-1

0.2

0.4

0.6

0.8

1

0

5

This should be compared with the upper bound indepenn−1 dently found by Pirandola et al. [30, 31, 37], ln(2) − log n 1 and the coherent information IC (Nadd ) = − ln(2) − log n which is a lower bound on P2 (N ) [38]. See Figure 4 for a comparison of these bounds.

Thermal channel

A thermal channel is similar to the pure-loss channel, but instead of the input interacting with a vacuum state on a beamsplitter of transmissivity τ , it interacts with a thermal state with mean photon number NB . For a thermal channel we have that G = (1 − η)NB + 1 and η T = (1−η)N . In Figure 5 the upper bound is plotted B +1 for NB = 1 together with two other bounds and the reverse coherent information, which is a lower bound on P2 (N ) [32]. 4.

15

20

FIG. 5. Bounds on the squashed entanglement of the thermal channel with NB = 1 as a function of the loss in dB. The red dotted line shows the upper bound by Takeoka et al. [21, 33], the dashed blue line the new bound reported in this paper, in solid green the bound by Pirandola et al. [30, 31, 37], and the dash-dotted line shows the reverse coherent information [32] which is a lower bound.

capacities. For any energy the pure-loss bound from Takeoka et al. [21, 33] and equation (86) coincide. In Figure 6 the bound from Takeoka et al. [21, 33], is shown for an average photon number of N = 0.1 [39, 40] and the two-way assisted private capacity of the pure-loss channel [30, 31, 37]. The loss-parameter runs from 0 to 2 · 10−20 , which is the expected range of losses for fiber lengths of around 1000 kilometers. In Figure 7 we

3

×10-20

2.5

1.5 1 0.5 0 0

Non-quantum limited noise for lossy channels

P2 (N ) Pirandola et al. [30,31] Finite energy bound Takeoka et al. [21,33]

2

Rate

FIG. 4. Comparison of the upper bounds mentioned in this paper for the additive noise channel. The dotted red line is the upper bound by Takeoka et al. [21], the dashed blue line is the squashed entanglement bound in this paper, the solid green line is the entanglement flux upper bound by Pirandola et al. [30, 31] and the magenta line is the coherent information of the channel which is a lower bound [38].

3.

10

Loss [dB] (=-10log10 η)

n

0.5

1

η

In experimental setups one does not measure ν, but the additional noise χ ≥ 0. We have the relation ν = 1−τ +χ where 1 − τ is the minimum amount of noise that will be introduced for a loss τ (the quantum-limited noise) [34]. The upper bound from (30) can then be rewritten as

1.5

2 ×10

-20

(33)

FIG. 6. Bound for the pure-loss channel with an average photon number of 0.1 and the secret key capacity [30, 31] as a function of η. The new bound in this paper coincides with the finite-energy variant of the bound by Takeoka et al., see [21, 22]. The loss parameter η ranges from 0 to 2 · 10−20 , which is the range of expected losses for transmissions across fibers with length ≈ 1000 km with an attenuation length of 22 km.

For low mean photon number and certain parameter ranges the finite-energy bound in equation (27) is tighter than previous upper bounds on the two-way assisted

plot the upper bound by Pirandola et al. [30, 31, 37], the finite-energy bounds of Takeoka et al. [21, 33], and equation (86) for the thermal channel with NB = 1. Note that the finite-energy bounds are zero only for η = 0, while the upper bound by Pirandola et al. [30, 31, 37] b equals zero for η ≤ NN . b +1

ζ(χ + 2 + 2τ, χ + 2 − 2τ ) − τ ζ(χ + 4, χ) . (4 + 2χ)(1 − τ 2 ) C.

Finite-energy bounds

7 1

Rate

0.8

Entanglement flux bound Pirandola et al. Finite energy bound TGW et al. New finite energy bound

0.6 0.4 0.2 0 0.5

0.6

0.7

0.8

0.9

1

η FIG. 7. Comparison of the upper bound found by Pirandola et al. [30, 31, 37] for the thermal channel with NB = 1 and the two squashed entanglement finite-energy bounds with average photon number of 0.1 as a function of the loss-parameter η [21, 33].

VI.

CONCLUSION

to the existence of an even better squashing channel for phase-insensitive Gaussian channels. Future work could investigate this intriguing avenue, especially due to its relevance to the squashed entanglement of a bipartite state as an entanglement measure. Furthermore, we have proven the exact two-way assisted capacities and the squashed entanglement of the ddimensional erasure channel, improved the previous best known upper bound on the amplitude-damping channel and derived a squashed entanglement bound for general qubit Pauli channels. In particular, our bound applies to the depolarizing channel and improves on the previous best known squashed entanglement upper bound. The only credible way to claim whether an implementation of a quantum repeater is good enough is by achieving a rate not possible by direct communication. Our bounds take special relevance in this context, especially for realistic energy constraints. VII.

ACKNOWLEDGEMENTS

In this paper we have obtained bounds on the two-way assisted capacities of several relevant channels using the squashed entanglement of a quantum channel. For practical purposes, the most relevant of the channels considered are phase-insensitive Gaussian channels. Our bound for these channels is always nonzero, even when the corresponding channel is entanglement-breaking. This points

KG, DE and SW acknowledge support from STW, Netherlands, an ERC Starting Grant and an NWO VIDI Grant. We would like to thank Mark M. Wilde and Stefano Pirandola for discussions regarding this project. We also thank Marius van Eck, Jonas Helsen, Corsin Pfister, Andreas Reiserer and Eddie Schoute for helpful comments regarding an earlier version of this paper.

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9 1.

Bounds for convex decomposition of channels

One way of obtaining bounds on the squashed entanglement is based on decomposing the channel action as a mixture of other channels actions and bounding each of them individually. Let NA0 →B be a channel such that its action can be written as the convex combination of the action of two other channels N0 and N1 ρAB = (I ⊗ N )(φAA0 ) = p(I ⊗ N0 )(φAA0 ) + (1 − p)(I ⊗ N1 )(φAA0 ) .

(34)

Then we can always purify ρAB in the following way |ρiABEF1 F2 =

√

p |ρ(0) iABE |0iF1 |0iF2 +

p 1 − p |ρ(1) iABE |1iF1 |1iF2

(35)

where 0 |ρ(0) iABE = VAN0 →BE |φiAA0

(36)

1 |ρ(1) iABE = VAN0 →BE |φiAA0 .

(37)

and

That is, |ρ(0) iABE and |ρ(1) iABE stand for the state that we obtain after applying the channel isometry to the pure input state |φiAA0 . Let us apply the following channel to |ρiABEF1 F2 |0i |1i 0 1 ρABEF1 F2 7→ trF2 (IAB ⊗ SE→E ⊗ I )(ρ ) + (I ⊗ S ⊗ P ⊗ I )(ρ ) . (38) 0 ⊗ PF 0 F2 ABEF1 F2 AB F2 ABEF1 F2 E→E F1 1 |vi

Where we denote by PF1 the projector onto the vector |vi. First we trace out F2 , then (1)

(0)

ρABEF1 = pρABE ⊗ |0i h0|F1 + (1 − p)ρABE ⊗ |1i h1|F1 .

(39)

Now, let us apply the rest of the channel. We obtain X (1) (0) 1 i 0 ρABE 0 F1 = SE→E 0 ⊗ |ii hi| F1 (ρABEF1 ) = pSE→E 0 (ρABE ) ⊗ |0i h0|F1 + (1 − p)SE→E 0 (ρABE ) ⊗ |1i h1|F1 .

(40)

i

That is, ρABE 0 F1 is a quantum-classical system. For states of this form the conditional mutual information can be simplified to I(A; B|EF1 ) = pI(A; B|E 0 )S 0

(0)

E→E 0

(ρABE )

+ (1 − p)I(A; B|E 0 )S 1

(41)

(1)

E→E 0

(ρABE )

Now we can upper bound Esq (N ) in the following way Esq (N ) ≤ max P φAA0

inf

i

i SE→E 0 ⊗|iihi|F ⊗trF2

I(A; B|E 0 F1 ))ρABEF1

(42)

1

! 0

0

= max p 0inf I(A; B|E )|ρ(0) iABE + (1 − p) 1inf I(A; B|E )|ρ(1) iABE

(43)

≤ pEsq (N1 ) + (1 − p)Esq (N2 ) .

(44)

φAA0

SE→E 0

SE→E 0

The first inequality holds by restricting the squashing channels to those channels of the form in (38). Equality (43) follows since for channels of the form (38) the resulting state is a quantum-classical state as indicated in (40), and for classical quantum states the conditional mutual information of the whole state is a convex combination of the individual conditional mutual informations as shown in (41). The last inequality follows because the state that achieves the maximum squashed entanglement might be different P for each channel. P This method generalizes easily to any number of channels, from which it follows that if N (ρ) = i pi Ni (ρ) with i pi = 1 and pi ≥ 0, then Q2 (N ) ≤ P2 (N ) ≤ Esq (N ) ≤

X i

pi Esq (Ni ) .

(45)

10 2.

Improved bound for the depolarizing channel

It is well known that the depolarizing channel becomes entanglement breaking for p ≥ 23 [41], which implies that P2 is zero in that range. For ≤ p ≤ 32 , we can write the output of the channel as a convex combination of the output of D2/3 and D . That is, there exists some 0 ≤ α ≤ 1 such that Dp (ρ) = (1 − α)D (ρ) + αD2/3 (ρ).

(46)

By expanding both sides of (46) and identifying the coefficients, we obtain α=

p− 2/3 −

(47)

which is in the range [0, 1] for 0 ≤ ≤ p. Using the decomposition of the depolarizing from (46) the action of Dp on half of a pure entangled state takes the following form, ψAB = I ⊗ Dp (|ψi hψ|AA0 )

(48)

= (1 − α) [(1 − ) |ψi hψ|AB + · π] + α

X

λi |αi i hαi |A ⊗ |βi i hβi |B .

(49)

i

Let ρAB = ((1 − ) |ψi hψ|AB + · π). A possible extension of ψAB is ψABE 0 = (1 − α)ρAB ⊗ |n + 1i hn + 1|E 0 + α

n X

λi |ψi i hψi |A ⊗ |φi i hφi |B ⊗ |ii hi|E 0 .

(50)

i=1

Since ψABE 0 is a valid extension of ρAB , this means that there exists some squashing channel SE→E 0 that takes the environment of the depolarizing channel to this particular E 0 . This is easy to see, first we can find a state |ψiABE 0 T that purifies ψABE 0 . Next, since all purifications are related by an isometry there exists some purification VE→E 0 T that takes the environment of the channel to E 0 T . After this we trace out the system T and obtain ψABE 0 . Now, ψABE 0 is a quantum-classical system. Hence, we can decompose the conditional mutual information I(A; B|E 0 ) into the sum of the mutual information conditioned on each value of E I(A : B|E 0 )ψ = (1 − α)I(A : B|E 0 )ρ + α

n X

λi I(A : B|E 0 )|ψi ihψi |A ⊗|φi ihφi |B ⊗|iihi|E0

(51)

i=1

= (1 − α)I(A : B|E 0 )ρ

(52)

Furthermore the input state that maximizes (52) is the maximally entangled state on AA0 . Hence, the following bound upper bound on Esq (Dp ) holds for 0 ≤ ≤ p Esq (Dp ) ≤ (1 − α)

3.

3 log() + (4 − 3) log(4 − 3) . 8

(53)

Squashed entanglement upper bound for any phase-insensitive Gaussian channel

In this section we discuss a proof of an upper bound for the squashed entanglement of any phase-insensitive bosonic Gaussian channel NPI . Here we use the fact that any such channel can be decomposed as √ a beamsplitter with transmissivity T concatenated with a two-mode squeezer with squeezing parameter r = acosh( G). We first show that we can restrict the input states to the class of thermal states with mean photon number N , after which the entropic quantity of interest is written as a function of N . We then show that this function is monotonically increasing, after which we take the asymptotic limit N → ∞ of the entropic quantity yielding

Esq (NPI ) ≤

1+T G+1 ) − G2 − 1 T log( G−1 ) 1 − T 2 G log( 1−T 1 − G2 T 2

.

(54)

To show this is true, we first use a different form of Esq (N ), which was proven by Takeoka et al. [21], Esq (NPI ) =

1 max inf [H(B|E 0 )ω + H(B|F )ω ] . 2 ρA0 VE→E0 F

(55)

11

E1

E2

A0 B1

S

F10

F1

G

B

F20

F2

B2

B3

E10

E20

FIG. 8. A squashing isometry for any phase-insensitive Gaussian channel NPI taking A0 to B. The beamsplitter B1 and the two-mode squeezer S form the Stinespring dilation, while the balanced beamsplitters B2 and B3 form the squashing map. The beamsplitter B1 interacts with the vacuum on E1 and A, and the two-mode squeezer S interacts with the output of B2 and the vacuum on E2 . The squashing isometry consists of two balanced beamsplitters B2 and B3 interacting with the vacuum on F1 and F2 and the output of the beamsplitter B1 and the two-mode squeezer S.

There are two differences between the characterization in (55) and the one in (2). First, the maximization runs over density operators on A0 instead of running over pure states on AA0 . Second, instead of taking the infimum over the squashing maps, it is taken over their dilations: squashing isometries VE→E 0 F that take the system E to E 0 and an auxiliary system F . The entropies are then taken on the state ω on systems BE 0 F . The total operation, which we denote by D, consists of the Stinespring dilation of the channel (B1 and S) and the squashing isometry consisting of two balanced beamsplitters (B2 and B3 ), see Figure 8. We now write H(B|E 0 ) = H(B|E1 E2 ), where the system on E 0 is the output at E10 and E20 after the total transformation D. E10 is the state after the vacuum state on E1 has interacted with the beamsplitter B1 and the balanced beamsplitter B2 . Similar statements hold also for E20 , F10 and F20 . Since the isometry consists of two balanced beamsplitters we have that H(B|E 0 ) = H(B|F10 F20 ) = H(B|F ), so that Esq (N ) ≤ H(B|E 0 ). After having found the state after the transformation we calculate the so-called symplectic eigenvalues of the states on BE10 E20 and E10 E20 , from which we can find H(B|E10 E20 ). To get an expression of the upper bound for N → ∞, we calculate for three different regimes of G and T the asymptotic behavior of the symplectic eigenvalues, after which we show that all three regimes give rise to the same form of the upper bound.

a.

Bound for finite N

A Mathematica file is included in the supplementary material to guide the reader through the calculations performed in this section. For the proof we first need to be able to calculate the entropy of a Gaussian state as a function of its covariance matrix. The entropy of an M −mode Gaussian state ρ can be calculated by finding the M symplectic eigenvalues νk ≥ 1 of the covariance matrix Γ of ρ [42]. It turns out that the 2M eigenvalues of the matrix ΩΓ are of the form ±iνk [43], where

Ω :=

M M 0 1 . −1 0

(56)

k=1

PM x−1 log( x+1 log( x−1 The entropy of the state is then k=1 g(νk ), where g(x) = x+1 2 2 )− 2 2 ) [34]. To obtain the state at the end of the isometry we determine first the optimal state for a specified mean photon number N , after which we apply the Gaussian transformations of the Stinespring dilation of the channel and the isometry, shown in Figure 8. To find the maximizing input state on A0 , we follow the same approach [21, 33] as Takeoka et al. Since the concatenation of multiple Gaussian transformations is still a Gaussian transformation, having a Gaussian state as input will always give a Gaussian state on any of the outputs. From the extremality of Gaussian states for conditional entropy [44], we get that the optimal input state is a Gaussian state.

12 To find the optimal Gaussian state, we note that the covariance matrix of all single-mode Gaussian states can be written as [45]

(1 + 2N )

cosh 2r + cos θ sinh 2r sin θ sinh 2r sin θ sinh 2r cosh 2r − cos θ sinh 2r

(57)

for some r ≥ 0 and θ ∈ R. Since the channel from A0 to BE10 E20 F10 F20 is covariant with displacements and all unitaries ˜ such that the corresponding symplectic matrices S ˜ act on the thermal state as U U SU˜ (1 + 2N ) I SU˜ T → (1 + 2N )

cosh 2r + cos θ sinh 2r sin θ sinh 2r , sin θ sinh 2r cosh 2r − cos θ sinh 2r

(58)

˜ ρU ˜ † , defining an equivalence relation. It is clear we have that H(B|E 0 )ρ = H(B|E 0 )U˜ ρU˜ † . We set ρ equivalent to U that all states with fixed N in equation (57) define an equivalence class with respect to the equivalence relation. Since H(B|E 0 )ρ = H(B|E 0 )U˜ ρU˜ † , we can set the thermal state (1 + 2N ) I to be the representative of that equivalence class, and we only have to consider thermal states for the optimization. The total system ΓA0 E10 F1 E20 F2 consists then of a thermal state ΓA0 with mean photon number N on A0 and vacuum states on all the other inputs: ΓA0 E1 F1 E2 F2 = γA0 ⊕ IE1 ⊕ IF1 ⊕ IE2 ⊕ IF2 , 1 + 2N 0 γA0 = . 0 1 + 2N

(59) (60)

The operations of the isometry are then the first beamsplitter B1 with transmissivity T on A0 and E1 √

T √0 0 T √ B1 = − 1 − T √0 0 − 1−T

√

1−T √ 0 1 − T √0 ⊕ IF1 ⊕ IE2 ⊕ IF2 , T 0 √ 0 T A0 E 0

(61)

1

the second beamsplitter B2 with transmissivity

1 2

on E1 and F1

√1 2

√1 2

0

0 √1 2 B2 = IA0 ⊕ √1 0 − 2 0 − √12

0 √1 2

0

0

√1 2

0

√1 2

⊕ IE2 ⊕ IF2 ,

(62)

E1 F 1

the two-mode squeezer S on A0 and E2 with the relation G = cosh2 (r) √

G √0 0 G 0 0 0 0 S= 0 0 0 0 √ G−1 0 √ 0 − G−1

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

√

G−1 √0 0 − G − 1 0 0 0 0 ⊕ IF2 , 0 0 0 √0 G √0 0 G A0 E 0 F 0 1

and finally the last beamsplitter B3 on E2 and F2 with transmissivity

1 2

1

(63)

13

√1 2

0

0 √1 2 B3 = IA0 ⊕ IE10 ⊕ IF1 ⊕ √1 0 − 2 0 − √12

√1 2

0 √1 2

0

0

√1 2

.

0

√1 2

(64)

E20 F2

We then have that the total symplectic transformation matrix D is

√

D = B3 SB2 B1 p

GT G(1 − T ) p 0 √0 0 GT 0 G(1 − T ) q q 1−T T − 0 0 2 2 q q T − 1−T 0 2 2 q0 q 1−T T 0 − 0 2 2 q q 1−T T = 0 0 − 2 2 q q (G−1)T (G−1)(1−T ) 0 0 2 2 q q (G−1)T (G−1)(1−T ) 0 − 0 − 2 2 q q (G−1)T (G−1)(1−T ) 0 − 0 − 2 2 q q (G−1)T (G−1)(1−T ) 0 0 2 2

√

(65) G−1 √0 0 0 − G−1 0

0 0

0 0

√1 2

0

0

0

0

0

√1 2

0

0

0

√1 2

0

0

0

0

0

√1 2

0

0

0

0

0 q

0

0

0

0

0

0

G 2

−

0 q 0

0 q

√1 2

0 q

√1 2

G 2

G 2

−

G 2

0

0

0 0 0 0 0 . 0 0 1 √ 2 0

(66)

√1 2

The covariance matrix ΓBE10 F10 E20 F20 = DΓA0 E1 F1 E2 F2 DT after the transformation is then aI −bI bI cσz −cσz dI eI −f σz f σz −bI eI dI f σz −f σz , bI cσ −f σ f σ gI hI z z z −cσz f σz −f σz hI gI

where σz =

(67)

1 0 1 0 ,I= and 0 −1 0 1 a = 2G(1 + N T ) − 1 p b = N 2 (GT (1 − T )) p c = (1 + N T ) 2 (G − 1) G d = 1 + N (1 − T ) e = N (T − 1) p f = N (G − 1) (1 − T ) T g = G + (G − 1) N T h = −(G − 1)(1 + N T ).

(68) (69) (70) (71) (72) (73) (74) (75)

The covariance matrix on the subsystems E1 E2 is then

Γ

E10 E20

dI −f σz = . −f σz gI

(76)

14 Multiplying by Ω gives 0 −d = 0 f

d 0 0 f f 0 0 −g

ΩΓE10 E20

f 0 . g 0

(77)

p Now set Ω± = (1 + N )2 − 4N T ± 2G(1 + N )(N T − 1) + (G + GN T )2 . Taking the covariance matrix corre0 0 sponding to E1 E2 we find using Mathematica the symplectic eigenvalues to be s 2 2 2 − 1 + G + 2N (1 − T + GT (G − 1)) + N (GT − 1) + (G − 1 + N (GT − 1)) Ω (78) νE10 E20 1 = − 2 s 2 2 2 − 1 + G + 2N (1 − T + GT (G − 1)) + N (GT − 1) − (G − 1 + N (GT − 1)) Ω νE10 E20 2 = − (79) . 2 The covariance matrix corresponding to BE10 E20 is

ΓBE10 E20

aI −bI cσz = −bI dI −f σz , cσz −f σz gI

(80)

so that 0 −a 0 = b 0 −c

ΩΓBE10 E20

a 0 −b 0 −c 0

0 b 0 −d 0 f

−b 0 d 0 f 0

0 −c 0 f 0 −g

−c 0 f , 0 g 0

(81)

From this the symplectic eigenvalues can be calculated to be s 2 + 2N (1 − T + GT (G + 1)) + N 2 (1 + GT )2 + (1 + G + N (1 + GT )) Ω+ 1 + G νBE10 E20 1 = − 2 s 2 + 2N (1 − T + GT (G + 1)) + N 2 (1 + GT )2 − (1 + G + N (1 + GT )) Ω+ 1 + G νBE10 E20 2 = − 2 νBE10 E20 3 = 1.

(82)

(83) (84)

We can now calculate H(B|E10 E20 ), H(B|E10 E20 ) = H(BE10 E20 ) − H(E10 E20 ) = g νBE10 E20 1 + g νBE10 E20 2 − g νE10 E20 1 − g νE10 E20 2 , where we used that g(1) = 0.

(85) (86)

15

k

Λ

ΨN AA0

A

G E10

B

E20

FIG. 9. Alice can perform a local operation Λ on one half of ΨN AA0 that yields a state on A and a classical outcome k. The state conditioned on the outcome k on systems ABE10 E2 is, up to a unitary displacement on B and E10 E20 , equal to the state 0 ρN . Alice and Bob can thus simulate any lower energy scenario.

b.

Monotonicity of the bound

For this section we restrict ourselves to the picture of calculating the squashed entanglement on the systems ABE10 E20 instead of BE10 E20 F10 F20 , where VA0 →BE10 E20 F10 F20 := V is the total isometry (see Figure 8). In this picture the optimization is over the purification of the thermal state, the two-mode squeezed vacuum state ΨN . To show monotonicity of equation (86) in N , we use that, up to a displacement on B (conditioned on a measurement outcome k at A0 ), it is 0 N0 < N ) [46]. possible to transform the state ΨN AA0 to ΨAA0 , using a local operation ΛA on Alice (where N N N † Suppose now that A performs the operation ΛA on the state ρABE 0 E 0 := trF10 F20 V Ψ V after the isometry, 1

ΛA ⊗ IBE10 E20 ρN ABE10 E20 =

Z Z

=

2

0 dk |kihk| ⊗ IA ⊗ UBk ⊗ UEk 10 E20 ρN 0 0 ABE1 E2 0

,k dk |kihk| ⊗ ρN ABE 0 E 0 . 1

(87) (88)

2

Here we used that displacement operations can always be removed by local operations [47], so that for fixed outcome k 0

0

,k N N 0 0 the state ρN ABE 0 E 0 is related to ρABE 0 E 0 := trF1 F2 V Ψ 1

1

2

2

0

by unitary displacements on B and E10 E20 . The conditional

mutual information evaluated on the state ΛA ⊗ IBE10 E20 ρN ˜N then satisfies ABE 0 E 0 = ρ 1

I(A; B|E10 E20 )ρN

0 E0 ABE1 2

2

≥ I(A; B|E10 E20 )ρ˜N Z ≥ dk I(A; B|E10 E20 )ρN 0, k Z = dk I(A; B|E10 E20 )ρN 0

(89)

= I(A; B|E10 E20 )ρN 0 .

(92)

(90) (91)

In equation (89) we used that the conditional mutual information can never increase under local operations on A [23]. 0 ,k In equation (90) we use the fact that the states ρN ABE10 E20 are flagged on the classical outcome k, and that the conditional mutual information of the whole state can not be smaller than the sum of the values of the conditional 0 ,k mutual information of the individual states [23]. In equations (91) and (92) we use the fact that all the ρN ABE 0 E 0 0

1

2

0 0 states are related to ρN ABE10 E20 by local unitaries on B and E1 E2 and that the conditional mutual information of those states thus must be equal. That is, the conditional mutual information computed over the isometry V with input state ΨN is always greater N0 than the conditional mutual information computed over the isometry V with input state Ψ if N 0 < N . This thus implies that equation (86) is a bound for all phase-insensitive Gaussian bosonic channels and all energy restrictions.

16 Expression as N → ∞

c.

To obtain an explicit form for the expression in (86) as N → ∞, we expand the eigenvalues around N = ∞ for three different regimes of G and T using Mathematica. For G = T1 we have r

G2 − 1 √ N + O (1) , G r G2 − 1 √ N + O (1) , νE10 E20 2 = G νBE10 E20 1 = 2N + O (1) , νE10 E20

1

=

(93) (94) (95)

2

G +1 + o (1) . (96) 2G Here we used the notation that f (N ) = o(h(N )) for two functions f (N ) and h(N ) if and only if ∀ > 0, ∃N 0 such that ∀N > N 0 , f (N ) ≤ h(N ). νBE10 E20

2

=

Now let us introduce the equivalence relation for two functions f (N ) and h(N ), so that f (N ) h(N ) if and only if limN →∞ |f (N ) − h(N )| = 0, i.e. we can safely replace f (N ) by h(N ) as N → ∞. For example, we have that g(N + c) g(N ) log( N2 ) + ln12 . In particular, if f (N ) = h(N ) + o(1), then g(f (N )) g(h(N )). Furthermore, this also means that if we have f (N ) = h(N ) + O(1) and limN →∞ f (N ) = limN →∞ h(N ) = ∞, then ) 1 g(f (N )) log( h(N 2 ) + ln 2 . We will call these relations the asymptotic entropic relations for short. Using these asymptotic entropic relations, we find H(BE10 E20 ) − H(E10 E20 ) = g ((νBE1 E2 )1 ) + g ((νBE1 E2 )2 ) − g ((νE1 E2 )1 ) − g ((νE1 E2 )2 ) ! ! r r 2 G2 − 1 √ G2 − 1 √ G +1 + g (2N ) − g N −g N g 2G G G r r 2 1 1 G +1 1 G2 − 1 √ G2 − 1 √ N) − N) − g + log(N ) + − log( − log( 2G ln 2 4G ln 2 4G ln 2 2 G +1 4G 1 =g + log( 2 )− 2G G −1 ln 2 G2 +1 2G

G2 +1 2G

+1

G2 +1 2G

−1

G2 −1 2G

+1 4G 1 log( )− log( ) + log( 2 )− 2 2 2 2 G −1 ln 2 2 2 2 2 (G + 1) (G + 1) (G − 1) (G − 1) 4G 1 = log( )− log( ) + log( 2 )− 4G 4G 4G 4G G −1 ln 2 2 2 (G + 1) (G − 1) 2 2 = log((G + 1) ) − log((G − 1) ) 4G 4G ! 2 2 (G + 1) (G − 1) 1 + − + + 1 log(4G) − log(G2 − 1) − 4G 4G ln 2 | {z }

=

+1

(97) (98) (99) (100) (101) (102)

(103)

0

2

2

(G + 1) (G − 1) 1 log(G + 1) − log(G − 1) − log(G2 − 1) − 2G 2G ln 2 2 2 G + 2G + 1 G − 2G + 1 1 = log(G + 1) − log(G − 1) − log(G2 − 1) − 2G 2G ln 2 G+1 1 G2 + 1 log( ) − log(G + 1) + log(G − 1) − log(G2 − 1) − = | {z } ln 2 2G G−1 =

(104) (105) (106)

0

=

G2 + 1 G+1 1 T2 + 1 1+T 1 log( )− = log( )− . 2G G−1 ln 2 2T 1−T ln 2

(107)

Here we used the asymptotic entropic relations in equations (98) and (99). Equation (100) is basic rewriting, equation (101) follows directly from the definition of g(·), and equation (102) follows from rewriting the terms. In equation

17 (103) we collect the terms proportional to log(4G), from which we can see that these terms sum up to zero. In equation (105) we expand the quadratic terms, collect corresponding terms in equation (106) and write the upper bound both as a function of G and T in the last equality. For G >

1 T

we get in the asymptotic limit that equations (78), (79), (82) and (83) become νE10 E20

1

= N (GT − 1) + O (1) ,

G−T + o (1) , νE10 E20 2 = GT − 1 νBE10 E20 1 = N (1 + GT ) + O (1) , G+T + o (1) . νBE10 E20 2 = 1 + GT

For G <

1 T

(108) (109) (110) (111)

we have G−T + o (1) , 1 − GT νE10 E20 2 = N (1 − GT ) + O (1) , νBE10 E20 1 = N (1 + GT ) + O (1) , G+T + o (1) . νBE10 E20 2 = 1 + GT νE10 E20

1

=

(112) (113) (114) (115)

For both regimes, the eigenvalues and in particular their leading terms are always positive. We see that for both G > T1 and G < T1 the absolute value of the eigenvalues are the same up to ordering, so that G−T G+T + g (N (1 + GT )) − g (N |1 − GT |)) − g − g 1 + GT |1 − GT | N (1 + GT ) 1 N |1 − GT | 1 G−T G+T + log( )+ − log( )− −g g 1 + GT 2 ln 2 2 ln 2 |1 − GT | G+T G−T 1 + GT =g −g + log( ) 1 + GT |1 − GT | |1 − GT | G+T G−T 1 + GT =g −g + log( ), 1 + GT 1 − GT 1 − GT

H(BE10 E20 )

H(E10 E20 )

(116) (117) (118) (119)

where in the first and second step we again used the asymptotic entropic relations. Equation (118) is basic algebraic rewriting of the logarithms. We can drop the absolute signs going from equation (118) to (119). To see this, note that log(−x) = log(x) + lniπ2 for x > 0, where we choose the branch cut along the negative imaginary axis, and in a −y−1 y+1 y+1 y−1 y−1 iπ similar way we find that g(−y) = −y+1 log( −y+1 log( −y−1 2 2 )− 2 2 ) = 2 log(− 2 ) − 2 log(− 2 ) = g(y) + ln 2 G−T for y ≥ 1. From this we find that −g(−y) + log(−x) = −g(y) + log(x) for x > 0, y ≥ 1. Since |1−GT | > 1 and 1+GT G−T 1+GT G−T 1+GT |1−GT | ≥ 0 for G ≥ 1, 0 ≤ T ≤ 1, we have that −g |1−GT | + log( |1−GT | ) = −g 1−GT + log( 1−GT ).

18 We can rewrite equation (119) as g +1

G+T

−g

log( 1+GT 2

+1

)−

G+T 1+GT

G−T 1 − GT

+ log(

1 + GT ) 1 − GT

G+T G−T G−T −1 +1 +1 log( 1+GT ) − 1−GT log( 1−GT ) 2 2 2 2 2 G−T G−T −1 −1 1 + GT − 1−GT log( 1−GT ) + log( ) 2 2 1 − GT (G + 1)(1 + T ) (G − 1)(1 − T ) (G − 1)(1 − T ) (G + 1)(1 + T ) log( )− log( ) = 2 (1 + GT ) 2 (1 + GT ) 2 (1 + GT ) 2 (1 + GT ) (G + 1)(1 − T ) (G + 1)(1 − T ) (G − 1)(1 + T ) (G − 1)(1 + T ) 1 + GT − log( )+ log( ) + log( ), 2 (1 − GT ) 2 (1 − GT ) 2 (1 − GT ) 2 (1 − GT ) 1 − GT

=

G+T 1+GT

G+T 1 + GT

(120)

−1

(121)

(122)

where we have used the definition of g(·) in the first equality and simplified the terms in the second step. We can expand the logarithms and collect the different terms and simplify to rewrite equation (122). Let us consider one by one the terms proportional to each logarithmic term. The terms proportional to log(G + 1) are (G + 1) (1 + T ) (G + 1) (1 − T ) − 2 (1 + GT ) 2 (1 − GT ) 2 G −1 T =− , 1 − G2 T 2

(123) (124)

the terms proportional to log(G − 1) are −

(G − 1) (1 − T ) (G − 1) (1 + T ) + 2 (1 + GT ) 2 (1 − GT ) 2 G −1 T = , 1 − G2 T 2

(125) (126)

the terms proportional to log(1 + T ) are (G + 1) (1 + T ) (G − 1) (1 + T ) + 2 (1 + GT ) 2 (1 − GT ) 1 − T2 G = , 1 − G2 T 2

(127) (128)

the terms proportional to log(1 − T ) are −

(G − 1) (1 − T ) (G + 1) (1 − T ) − 2 (1 + GT ) 2 (1 − GT ) 2 1−T G =− , 1 − G2 T 2

(129) (130)

1 the terms proportional to log( 2(1+GT ) ) = − log(1 + GT ) − 1 are

(G + 1) (1 + T ) (G − 1) (1 − T ) − 2 (1 + GT ) 2 (1 + GT ) =1,

(131) (132)

1 and finally the terms proportional to log( 2(1−GT ) ) = − log(1 − GT ) − 1 are

−

(G + 1) (1 − T ) (G − 1) (1 + T ) + 2 (1 − GT ) 2 (1 − GT ) = −1 .

(133) (134)

19 1+GT ) term, equation (122) becomes Collecting all these terms and the log( 1−GT

G2 − 1 T G2 − 1 T 1 − T2 G − log(G + 1) + log(G − 1) + log(1 + T ) 1 − G2 T 2 1 − G2 T 2 1 − G2 T 2 1 − T2 G 1 + GT − log(1 − T ) − log(1 + GT ) − 1 + log(1 − GT ) + 1 + log( ) 1 − G2 T 2 1 − GT | {z } 0 2 2 1−T G G −1 T (log(G + 1) − log(G − 1)) + (log(1 + T ) − log(1 − T )) =− 1 − G2 T 2 1 − G2 T 2 1+T G+1 1 − T 2 G log( 1−T ) − G2 − 1 T log( G−1 ) = , 2 2 1−G T

(135)

(136) (137)

where in the first equality we regrouped terms and used the fact that the sum of the last five terms equals zero. The second equality follows from rewriting the logarithm terms. 1+T Setting G = T1 , the denominator of equation (137) becomes zero. Luckily, the numerator 1 − T 2 T1 log( 1−T )− 1 1 1 1+T 1 1+T T +1 T 2 − 1 T log( T1 −1 ) = T − T log( 1−T ) − T − T log( 1−T ) = 0, also becomes zero, implying that we can use L’Hˆ opital’s rule to retrieve the limit. Differentiating the numerator from equation (137) with respect to G gives 2T G+1 1+T )+ − 2GT log( ), 1 − T 2 log( 1−T ln 2 G−1

(138)

while differentiating the denominator from equation (137) gives −2GT 2 .

(139)

so that the quotient of equation (138) and (139) gives 2T ln 2 2GT 2

1+T )− − 1 − T 2 log( 1−T Setting G =

1 T

G+1 + 2GT log( G−1 )

.

(140)

we retrieve that 1+T G+1 1 − T 2 G log( 1−T ) − G2 − 1 T log( G−1 )

lim

= lim1

1+T − 1 − T 2 log( 1−T )−

2T ln 2 2GT 2

G→ T

=

(141)

1 − G2 T 2

G→ T1

1+T T 2 − 1 log( 1−T )−

2T ln 2

G+1 + 2GT log( G−1 )

(142)

1+T + 2 log( 1−T )

(143)

2T T2 + 1 1+T 1 = log( )− . 2T 1−T ln 2

We see that for all three regimes (G = T1 , G > T1 and G < limit of N → ∞. From this we retrieve our claim that Q2 (NPI ), P2 (NPI ) ≤ Esq (NPI ) ≤ =

1 T

(144)

) equation (86), yields equation (137) in the asymptotic

H(B|E10 E20 ) + H(B|F10 F20 ) 2 1+T G+1 1 − T 2 G log( 1−T ) − G2 − 1 T log( G−1 ) 1 − G2 T 2

(145) .

(146)