Unambiguous discrimination of nonorthogonal quantum states in cavity QED

R. J. de Assis; J.S. Sales; N. G. de Almeida

arXiv:1704.03310·quant-ph·September 13, 2017

Unambiguous discrimination of nonorthogonal quantum states in cavity QED

R. J. de Assis, J.S. Sales, N. G. de Almeida

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TL;DR

This paper presents a simplified method using a three-level atom and POVM techniques to unambiguously distinguish nonorthogonal quantum states in high-Q cavity systems, enhancing quantum measurement precision.

Contribution

It introduces a novel, simplified scheme for unambiguous quantum state discrimination in cavity QED using a single atom and POVM, improving practical implementation.

Findings

01

Scheme achieves unambiguous discrimination with high efficiency.

02

Analysis aligns with current cavity QED experimental capabilities.

03

Potential for improved quantum information processing applications.

Abstract

We propose an oversimplified scheme to unambiguously discriminate nonorthogonal quantum field states inside high-Q cavities. Our scheme, which is based on positive operator-valued mea- sures (POVM) technique, uses a single three-level atom interacting resonantly with a single mode of a cavity-field and selective atomic state detectors. While the single three-level atom takes the role of the ancilla, the single cavity mode field represents the system we want to obtain information. The efficiency of our proposal is analyzed considering the nowadays achievements in the context of cavity QED.

Figures4

Click any figure to enlarge with its caption.

Tables3

Table 1. Table 1: Values for the probabilities according to our protocol to accomplish UQSD in cavity QED. To each integer m 𝑚 m , there is a minimum for the inconclusive events p i n subscript 𝑝 𝑖 𝑛 p_{in} and a corresponding maximum for the probability of success p s subscript 𝑝 𝑠 p_{s} . The table shows p i n subscript 𝑝 𝑖 𝑛 p_{in} , p b subscript 𝑝 𝑏 p_{b} , p c subscript 𝑝 𝑐 p_{c} , and p s subscript 𝑝 𝑠 p_{s} separately separately for p 1 = 0.3 subscript 𝑝 1 0.3 p_{1}=0.3 .

$m$	$κ$	$p_{i n}$	$p_{b}$	$p_{c}$	$p_{s}$
0	1.52	0.8369	0.0747	0.0884	0.1631
1	3.72	0.7011	0.0366	0.2623	0.2989
2	5.71	0.6728	0.0155	0.3117	0.3272
3	7.63	0.6627	0.0069	0.3304	0.3373
4	9.54	0.6581	0.0032	0.3387	0.3419
5	11.44	0.6556	0.0015	0.3429	0.3444
10	21.23	0.6505	0.0012	0.3483	0.3495
20	41.11	0.6505	0.0001	0.3494	0.3495
50	101.05	0.6501	0.0000	0.3499	0.3499

Table 2. Table 2: Values for the probabilities according to our protocol to accomplish UQSD in cavity QED. To each integer m 𝑚 m , there is a minimum for the inconclusive events p i n subscript 𝑝 𝑖 𝑛 p_{in} and a corresponding maximum for the probability of success p s subscript 𝑝 𝑠 p_{s} . The table shows p i n subscript 𝑝 𝑖 𝑛 p_{in} , p b subscript 𝑝 𝑏 p_{b} , p c subscript 𝑝 𝑐 p_{c} , and p b subscript 𝑝 𝑏 p_{b} separately separately for p 1 = 0.7 subscript 𝑝 1 0.7 p_{1}=0.7 .

$m$	$κ$	$p_{i n}$	$p_{b}$	$p_{c}$	$p_{s}$
0	1.45	0.7874	0.1749	0.0377	0.2126
1	5.19	0.7483	0.1693	0.0824	0.2517
2	8.54	0.7443	0.1679	0.0878	0.2557
3	12.27	0.7432	0.1674	0.0894	0.2568
4	15.80	0.7427	0.1673	0.0900	0.2573
5	19.32	0.7426	0.1671	0.0903	0.2574
10	36.91	0.7422	0.1670	0.0908	0.2578
20	72.08	0.7420	0.1670	0.0910	0.2580
50	177.58	0.7420	0.1670	0.0910	0.2580

Table 3. Table 3: Values for the probabilities according to our protocol to accomplish UQSD in cavity QED. To each integer m 𝑚 m , there is a minimum for the inconclusive events p i n subscript 𝑝 𝑖 𝑛 p_{in} and a corresponding maximum for the probability of success p s subscript 𝑝 𝑠 p_{s} . The table shows p i n subscript 𝑝 𝑖 𝑛 p_{in} , p b subscript 𝑝 𝑏 p_{b} , p c subscript 𝑝 𝑐 p_{c} , and p s subscript 𝑝 𝑠 p_{s} separately separately for p 1 = 0.5 subscript 𝑝 1 0.5 p_{1}=0.5 .

$m$	$κ$	$p_{i n}$	$p_{b}$	$p_{c}$	$p_{s}$
0	1.47	0.8123	0.1248	0.0629	0.1877
1	4.50	0.7356	0.1039	0.1605	0.2644
2	7.55	0.7252	0.0989	0.1759	0.2748
3	10.55	0.7221	0.0957	0.1822	0.2779
4	13.70	0.7209	0.0956	0.1835	0.2791
5	16.50	0.7101	0.0950	0.1849	0.2799
10	31.52	0.7191	0.0941	0.1868	0.2809
20	61.50	0.7189	0.0938	0.1873	0.2811
50	151.50	0.7188	0.0930	0.1882	0.2812

Equations54

ρ_{A S} = ∣ a_{0} ⟩ ⟨ a_{0} ∣ \otimes ρ_{S},

ρ_{A S} = ∣ a_{0} ⟩ ⟨ a_{0} ∣ \otimes ρ_{S},

U = k k^{'} \sum ∣ a_{k} ⟩ ⟨ a_{k^{'}} ∣ \otimes M_{k k^{'}}

U = k k^{'} \sum ∣ a_{k} ⟩ ⟨ a_{k^{'}} ∣ \otimes M_{k k^{'}}

M_{k k^{'}} = l l^{'} \sum u_{k l k^{'} l^{'}} ∣ s_{l} ⟩ ⟨ s_{l^{'}} ∣,

M_{k k^{'}} = l l^{'} \sum u_{k l k^{'} l^{'}} ∣ s_{l} ⟩ ⟨ s_{l^{'}} ∣,

B_{00} = k \sum M_{k}^{†} M_{k} = I .

B_{00} = k \sum M_{k}^{†} M_{k} = I .

ρ_{A S, m} = (∣ a_{m} ⟩ ⟨ a_{m} ∣ \otimes I) ρ_{A S} (∣ a_{m} ⟩ ⟨ a_{m} ∣ \otimes I)

ρ_{A S, m} = (∣ a_{m} ⟩ ⟨ a_{m} ∣ \otimes I) ρ_{A S} (∣ a_{m} ⟩ ⟨ a_{m} ∣ \otimes I)

ρ_{A S, m} = ∣ a_{m} ⟩ ⟨ a_{m} ∣ \otimes M_{m} ρ_{S} M_{m}^{†} .

ρ_{A S, m} = ∣ a_{m} ⟩ ⟨ a_{m} ∣ \otimes M_{m} ρ_{S} M_{m}^{†} .

ρ_{S, m} = \frac{M _{m} ρ _{S} M _{m}^{†}}{p _{m}},

ρ_{S, m} = \frac{M _{m} ρ _{S} M _{m}^{†}}{p _{m}},

p_{m} = Tr (M_{m}^{†} M_{m} ρ_{S})

p_{m} = Tr (M_{m}^{†} M_{m} ρ_{S})

H = ℏ g_{1} (σ_{ba} a + σ_{ab} a^{†}) + ℏ g_{2} (σ_{c b} a + σ_{b c} a^{†}),

H = ℏ g_{1} (σ_{ba} a + σ_{ab} a^{†}) + ℏ g_{2} (σ_{c b} a + σ_{b c} a^{†}),

U ∣ a, 0 ⟩ = α ∣ a, 0 ⟩ - i β sin (g_{1} t) ∣ a, 1 ⟩ + β cos (g_{1} t) ∣ b, 0 ⟩,

U ∣ a, 0 ⟩ = α ∣ a, 0 ⟩ - i β sin (g_{1} t) ∣ a, 1 ⟩ + β cos (g_{1} t) ∣ b, 0 ⟩,

U ∣ a, 1 ⟩ = α cos (g_{1} t_{c}) ∣ a, 1 ⟩ - i β (\frac{2 g _{1}}{2 g _{1}^{2} + g _{2}^{2}}) sin (2 g_{1}^{2} + g_{2}^{2} t) ∣ a, 2 ⟩ - i α sin (g_{1} t) ∣ b, 0 ⟩ + + β cos (2 g_{1}^{2} + g_{2}^{2} t) ∣ b, 1 ⟩ - i β (\frac{g _{2}}{2 g _{1}^{2} + g _{2}^{2}}) sin (2 g_{1}^{2} + g_{2}^{2} t) ∣ c, 0 ⟩,

U ∣ a, 1 ⟩ = α cos (g_{1} t_{c}) ∣ a, 1 ⟩ - i β (\frac{2 g _{1}}{2 g _{1}^{2} + g _{2}^{2}}) sin (2 g_{1}^{2} + g_{2}^{2} t) ∣ a, 2 ⟩ - i α sin (g_{1} t) ∣ b, 0 ⟩ + + β cos (2 g_{1}^{2} + g_{2}^{2} t) ∣ b, 1 ⟩ - i β (\frac{g _{2}}{2 g _{1}^{2} + g _{2}^{2}}) sin (2 g_{1}^{2} + g_{2}^{2} t) ∣ c, 0 ⟩,

U ∣ a, 2 ⟩ = α {1 + (\frac{2 g _{1}^{2}}{2 g _{1}^{2} + g _{2}^{2}}) [cos (2 g_{1}^{2} + g_{2}^{2} t) - 1]} ∣ a, 2 ⟩ - i α (\frac{2 g _{1}}{2 g _{1}^{2} + g _{2}^{2}}) sin (2 g_{1}^{2} + g_{2}^{2} t) ∣ b, 1 ⟩ + + β cos (2 g_{2} t) ∣ b, 2 ⟩ + α (\frac{2 g _{1} g _{2}}{2 g _{1}^{2} + g _{2}^{2}}) [cos (2 g_{1}^{2} + g_{2}^{2} t) - 1] ∣ c, 0 ⟩ - i β sin (2 g_{2} t) ∣ c, 1 ⟩,

U ∣ a, 2 ⟩ = α {1 + (\frac{2 g _{1}^{2}}{2 g _{1}^{2} + g _{2}^{2}}) [cos (2 g_{1}^{2} + g_{2}^{2} t) - 1]} ∣ a, 2 ⟩ - i α (\frac{2 g _{1}}{2 g _{1}^{2} + g _{2}^{2}}) sin (2 g_{1}^{2} + g_{2}^{2} t) ∣ b, 1 ⟩ + + β cos (2 g_{2} t) ∣ b, 2 ⟩ + α (\frac{2 g _{1} g _{2}}{2 g _{1}^{2} + g _{2}^{2}}) [cos (2 g_{1}^{2} + g_{2}^{2} t) - 1] ∣ c, 0 ⟩ - i β sin (2 g_{2} t) ∣ c, 1 ⟩,

M_{ν} =

M_{ν} =

u_{ν na n^{'}} = ⟨ ν, n ∣ U ∣ a, n^{'} ⟩ .

u_{ν na n^{'}} = ⟨ ν, n ∣ U ∣ a, n^{'} ⟩ .

M_{a} = α ∣ 0 ⟩ ⟨ 0 ∣ - i β sin (g_{1} t) ∣ 1 ⟩ ⟨ 0 ∣ + α cos (g_{1} t) ∣ 1 ⟩ ⟨ 1 ∣ - - i β (\frac{2 g _{1}}{2 g _{1}^{2} + g _{2}^{2}}) sin (2 g_{1}^{2} + g_{2}^{2} t) ∣ 2 ⟩ ⟨ 1 ∣ + α {1 + (\frac{2 g _{1}^{2}}{2 g _{1}^{2} + g _{2}^{2}}) [cos (2 g_{1}^{2} + g_{2}^{2} t) - 1]} ∣ 2 ⟩ ⟨ 2 ∣,

M_{a} = α ∣ 0 ⟩ ⟨ 0 ∣ - i β sin (g_{1} t) ∣ 1 ⟩ ⟨ 0 ∣ + α cos (g_{1} t) ∣ 1 ⟩ ⟨ 1 ∣ - - i β (\frac{2 g _{1}}{2 g _{1}^{2} + g _{2}^{2}}) sin (2 g_{1}^{2} + g_{2}^{2} t) ∣ 2 ⟩ ⟨ 1 ∣ + α {1 + (\frac{2 g _{1}^{2}}{2 g _{1}^{2} + g _{2}^{2}}) [cos (2 g_{1}^{2} + g_{2}^{2} t) - 1]} ∣ 2 ⟩ ⟨ 2 ∣,

M_{b} = β cos (g_{1} t) ∣ 0 ⟩ ⟨ 0 ∣ - i α sin (g_{1} t) ∣ 0 ⟩ ⟨ 1 ∣ + β cos (2 g_{1}^{2} + g_{2}^{2} t) ∣ 1 ⟩ ⟨ 1 ∣ - - i α (\frac{2 g _{1}}{2 g _{1}^{2} + g _{2}^{2}}) sin (2 g_{1}^{2} + g_{2}^{2} t) ∣ 1 ⟩ ⟨ 2 ∣ + β cos (2 g_{2} t) ∣ 2 ⟩ ⟨ 2 ∣,

M_{b} = β cos (g_{1} t) ∣ 0 ⟩ ⟨ 0 ∣ - i α sin (g_{1} t) ∣ 0 ⟩ ⟨ 1 ∣ + β cos (2 g_{1}^{2} + g_{2}^{2} t) ∣ 1 ⟩ ⟨ 1 ∣ - - i α (\frac{2 g _{1}}{2 g _{1}^{2} + g _{2}^{2}}) sin (2 g_{1}^{2} + g_{2}^{2} t) ∣ 1 ⟩ ⟨ 2 ∣ + β cos (2 g_{2} t) ∣ 2 ⟩ ⟨ 2 ∣,

M_{c} = - i β (\frac{g _{2}}{2 g _{1}^{2} + g _{2}^{2}}) sin (2 g_{1}^{2} + g_{2}^{2} t) ∣ 0 ⟩ ⟨ 1 ∣ + + α (\frac{2 g _{1} g _{2}}{2 g _{1}^{2} + g _{2}^{2}}) [cos (2 g_{1}^{2} + g_{2}^{2} t) - 1] ∣ 0 ⟩ ⟨ 2 ∣ - i β sin (2 g_{2} t) ∣ 1 ⟩ ⟨ 2 ∣ .

M_{c} = - i β (\frac{g _{2}}{2 g _{1}^{2} + g _{2}^{2}}) sin (2 g_{1}^{2} + g_{2}^{2} t) ∣ 0 ⟩ ⟨ 1 ∣ + + α (\frac{2 g _{1} g _{2}}{2 g _{1}^{2} + g _{2}^{2}}) [cos (2 g_{1}^{2} + g_{2}^{2} t) - 1] ∣ 0 ⟩ ⟨ 2 ∣ - i β sin (2 g_{2} t) ∣ 1 ⟩ ⟨ 2 ∣ .

E_{a} = [∣ α ∣^{2} + ∣ β ∣^{2} sin^{2} (g_{1} t)] ∣ 0 ⟩ ⟨ 0 ∣ + i sin (g_{1} t) cos (g_{1} t) (α β^{*} ∣ 0 ⟩ ⟨ 1 ∣ - α^{*} β ∣ 1 ⟩ ⟨ 0 ∣) + + [∣ α ∣^{2} cos^{2} (g_{1} t) + ∣ β ∣^{2} (\frac{2 g _{1}^{2}}{2 g _{1}^{2} + g _{2}^{2}}) sin^{2} (2 g_{1}^{2} + g_{2}^{2} t)] ∣ 1 ⟩ ⟨ 1 ∣ + + i (\frac{2 g _{1}}{2 g _{1}^{2} + g _{2}^{2}}) {1 + (\frac{2 g _{1}^{2}}{2 g _{1}^{2} + g _{2}^{2}}) [cos (2 g_{1}^{2} + g_{2}^{2} t) - 1]} (α β^{*} ∣ 1 ⟩ ⟨ 2 ∣ - α β^{*} ∣ 2 ⟩ ⟨ 1 ∣) + + ∣ α ∣^{2} {1 + (\frac{2 g _{1}^{2}}{2 g _{1}^{2} + g _{2}^{2}}) [cos (2 g_{1}^{2} + g_{2}^{2} t) - 1]}^{2} ∣ 2 ⟩ ⟨ 2 ∣,

E_{a} = [∣ α ∣^{2} + ∣ β ∣^{2} sin^{2} (g_{1} t)] ∣ 0 ⟩ ⟨ 0 ∣ + i sin (g_{1} t) cos (g_{1} t) (α β^{*} ∣ 0 ⟩ ⟨ 1 ∣ - α^{*} β ∣ 1 ⟩ ⟨ 0 ∣) + + [∣ α ∣^{2} cos^{2} (g_{1} t) + ∣ β ∣^{2} (\frac{2 g _{1}^{2}}{2 g _{1}^{2} + g _{2}^{2}}) sin^{2} (2 g_{1}^{2} + g_{2}^{2} t)] ∣ 1 ⟩ ⟨ 1 ∣ + + i (\frac{2 g _{1}}{2 g _{1}^{2} + g _{2}^{2}}) {1 + (\frac{2 g _{1}^{2}}{2 g _{1}^{2} + g _{2}^{2}}) [cos (2 g_{1}^{2} + g_{2}^{2} t) - 1]} (α β^{*} ∣ 1 ⟩ ⟨ 2 ∣ - α β^{*} ∣ 2 ⟩ ⟨ 1 ∣) + + ∣ α ∣^{2} {1 + (\frac{2 g _{1}^{2}}{2 g _{1}^{2} + g _{2}^{2}}) [cos (2 g_{1}^{2} + g_{2}^{2} t) - 1]}^{2} ∣ 2 ⟩ ⟨ 2 ∣,

E_{b} = ∣ β ∣^{2} cos^{2} (g_{1} t) ∣ 0 ⟩ ⟨ 0 ∣ - i sin (g_{1} t) cos (g_{1} t) (α β^{*} ∣ 0 ⟩ ⟨ 1 ∣ - α β^{*} ∣ 1 ⟩ ⟨ 0 ∣) + + [∣ α ∣^{2} sin^{2} (g_{1} t) + ∣ β ∣^{2} cos^{2} (2 g_{1}^{2} + g_{2}^{2} t)] ∣ 1 ⟩ ⟨ 1 ∣ - - i (\frac{2 g _{1}}{2 g _{1}^{2} + g _{2}^{2}}) sin (2 g_{1}^{2} + g_{2}^{2} t) cos (2 g_{1}^{2} + g_{2}^{2} t) (α β^{*} ∣ 1 ⟩ ⟨ 2 ∣ - α β^{*} ∣ 2 ⟩ ⟨ 1 ∣) + + [∣ α ∣^{2} (\frac{2 g _{1}^{2}}{2 g _{1}^{2} + g _{2}^{2}}) sin^{2} (2 g_{1}^{2} + g_{2}^{2} t) + ∣ β ∣^{2} cos^{2} (2 g_{2} t)] ∣ 2 ⟩ ⟨ 2 ∣,

E_{b} = ∣ β ∣^{2} cos^{2} (g_{1} t) ∣ 0 ⟩ ⟨ 0 ∣ - i sin (g_{1} t) cos (g_{1} t) (α β^{*} ∣ 0 ⟩ ⟨ 1 ∣ - α β^{*} ∣ 1 ⟩ ⟨ 0 ∣) + + [∣ α ∣^{2} sin^{2} (g_{1} t) + ∣ β ∣^{2} cos^{2} (2 g_{1}^{2} + g_{2}^{2} t)] ∣ 1 ⟩ ⟨ 1 ∣ - - i (\frac{2 g _{1}}{2 g _{1}^{2} + g _{2}^{2}}) sin (2 g_{1}^{2} + g_{2}^{2} t) cos (2 g_{1}^{2} + g_{2}^{2} t) (α β^{*} ∣ 1 ⟩ ⟨ 2 ∣ - α β^{*} ∣ 2 ⟩ ⟨ 1 ∣) + + [∣ α ∣^{2} (\frac{2 g _{1}^{2}}{2 g _{1}^{2} + g _{2}^{2}}) sin^{2} (2 g_{1}^{2} + g_{2}^{2} t) + ∣ β ∣^{2} cos^{2} (2 g_{2} t)] ∣ 2 ⟩ ⟨ 2 ∣,

E_{c} = ∣ β ∣^{2} (\frac{g _{2}^{2}}{2 g _{1}^{2} + g _{2}^{2}}) sin^{2} (2 g_{1}^{2} + g_{2}^{2} t) ∣ 1 ⟩ ⟨ 1 ∣ + + i [\frac{2 g _{1} g _{2}^{2}}{( 2 g _{1}^{2} + g _{2}^{2} ) ^{\frac{3}{2}}}] sin (2 g_{1}^{2} + g_{2}^{2} t) [cos (2 g_{1}^{2} + g_{2}^{2} t) - 1] (α β^{*} ∣ 1 ⟩ ⟨ 2 ∣ - α β^{*} ∣ 2 ⟩ ⟨ 1 ∣) + + {∣ α ∣^{2} [\frac{2 g _{1}^{2} g _{2}^{2}}{( 2 g _{1}^{2} + g _{2}^{2} ) ^{2}}] [cos (2 g_{1}^{2} + g_{2}^{2} t) - 1]^{2} + ∣ β ∣^{2} sin^{2} (2 g_{2} t)} ∣ 2 ⟩ ⟨ 2 ∣ .

E_{c} = ∣ β ∣^{2} (\frac{g _{2}^{2}}{2 g _{1}^{2} + g _{2}^{2}}) sin^{2} (2 g_{1}^{2} + g_{2}^{2} t) ∣ 1 ⟩ ⟨ 1 ∣ + + i [\frac{2 g _{1} g _{2}^{2}}{( 2 g _{1}^{2} + g _{2}^{2} ) ^{\frac{3}{2}}}] sin (2 g_{1}^{2} + g_{2}^{2} t) [cos (2 g_{1}^{2} + g_{2}^{2} t) - 1] (α β^{*} ∣ 1 ⟩ ⟨ 2 ∣ - α β^{*} ∣ 2 ⟩ ⟨ 1 ∣) + + {∣ α ∣^{2} [\frac{2 g _{1}^{2} g _{2}^{2}}{( 2 g _{1}^{2} + g _{2}^{2} ) ^{2}}] [cos (2 g_{1}^{2} + g_{2}^{2} t) - 1]^{2} + ∣ β ∣^{2} sin^{2} (2 g_{2} t)} ∣ 2 ⟩ ⟨ 2 ∣ .

E_{a} = (cos^{2} θ_{m} + sin^{4} θ_{m}) ∣ 0 ⟩ ⟨ 0 ∣ + \frac{1}{4} sin^{2} (2 θ_{m}) (∣ 0 ⟩ ⟨ 1 ∣ + ∣ 1 ⟩ ⟨ 0 ∣) + + [cos^{4} θ_{m} + (\frac{2}{2 + κ ^{2}}) sin^{2} θ_{m}] ∣ 1 ⟩ ⟨ 1 ∣,

E_{a} = (cos^{2} θ_{m} + sin^{4} θ_{m}) ∣ 0 ⟩ ⟨ 0 ∣ + \frac{1}{4} sin^{2} (2 θ_{m}) (∣ 0 ⟩ ⟨ 1 ∣ + ∣ 1 ⟩ ⟨ 0 ∣) + + [cos^{4} θ_{m} + (\frac{2}{2 + κ ^{2}}) sin^{2} θ_{m}] ∣ 1 ⟩ ⟨ 1 ∣,

E_{b}

E_{b}

E_{c}

E_{c}

p_{b} = Tr (E_{b} ρ) = \frac{q _{1}}{4} sin^{2} (2 θ_{m})

p_{b} = Tr (E_{b} ρ) = \frac{q _{1}}{4} sin^{2} (2 θ_{m})

p_{c} = Tr (E_{c} ρ) = \frac{q _{2}}{2} (\frac{κ ^{2}}{2 + κ ^{2}}) sin^{2} (θ_{m}),

p_{c} = Tr (E_{c} ρ) = \frac{q _{2}}{2} (\frac{κ ^{2}}{2 + κ ^{2}}) sin^{2} (θ_{m}),

p_{in} = q_{1} (cos^{2} θ_{m} + sin^{4} θ_{m}) + + \frac{q _{2}}{2} [1 + cos^{2} θ_{m} + (\frac{2}{2 + κ ^{2}}) sin^{2} θ_{m}] .

p_{in} = q_{1} (cos^{2} θ_{m} + sin^{4} θ_{m}) + + \frac{q _{2}}{2} [1 + cos^{2} θ_{m} + (\frac{2}{2 + κ ^{2}}) sin^{2} θ_{m}] .

p_{s} = \frac{q _{1}}{4} sin^{2} (2 θ_{m}) + \frac{q _{2}}{2} (\frac{κ ^{2}}{2 + κ ^{2}}) sin^{2} (θ_{m}),

p_{s} = \frac{q _{1}}{4} sin^{2} (2 θ_{m}) + \frac{q _{2}}{2} (\frac{κ ^{2}}{2 + κ ^{2}}) sin^{2} (θ_{m}),

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Full text

Unambiguous discrimination of nonorthogonal quantum states in cavity

QED

R. J. de Assis

Instituto de Física, Universidade Federal de Goiás, 74.001-970, Goiânia

GO, Brazil

J.S. Sales

Centro de Ciências Exatas e Tecnológicas, Universidade Estadual de Goiás, 75132-903, Anápolis, Goiás, Brazil

N. G. de Almeida

Instituto de Física, Universidade Federal de Goiás, 74.001-970, Goiânia

GO, Brazil

Abstract

We propose an oversimplified scheme to unambiguously discriminate nonorthogonal quantum field states inside high-Q cavities. Our scheme, which is based on positive operator-valued measures (POVM) technique, uses a single three-level atom interacting resonantly with a single mode of a cavity-field and selective atomic state detectors. While the single three-level atom takes the role of the ancilla, the single cavity mode field represents the system** **we want to obtain information. The efficiency of our proposal is analyzed considering the nowadays achievements in the context of cavity QED.

pacs:

42.50.−p, 42.50.Ct, 42.50.Pq, 05.30.-d, 03.65.Ta

I Introduction

Positive operator-valued measures (POVM) generalizes all possible kind of measurements povm ; jacobs14 and cannot be reduced to standard projections of the initial state onto orthogonal states spanning the initial Hilbert space alone, pertaining to the system we want to obtain information Peres93 ; Nielsen00 . In fact, although in general POVM can always be realized as standard projective measurements on an enlarged system Peres93 , they are such that the number of outputs may be larger than the dimensionality of the space of states of the system in which we are interest in. POVM is now standard in several areas of quantum mechanics, including quantum optics and quantum information, among others Nielsen00 ; Scully97 ; Lo98 ; Janos07 . In this paper we show how to accomplish POVM to unambiguously discriminate nonorthogonal field states inside high-Q cavities. The goal of unambiguous quantum state discrimination (UQSD) is to discern in which state the system was prepared Ivanovic87 ; Chefles00 ; Cheffles12 ; Gisin96 , founding many applications in several protocols Chefles00 ; Cheffles12 , mainly for quantum cryptography Bennet17 ; Bennett84 ; Kak06 ; Eckert91 . **Our scheme, employing one three-level atom interacting with a single mode of a cavity field, is very simple from the experimental point of view and can be implemented using nowadays techniques in cavity QED.

We begin by reviewing the general quantum measurement theory. Next, we present our model and results, comparing with the simple case of projective measurements. Then we present our conclusions.

II General measurements

Consider a quantum system we are interested to measure, and a second quantum system we call the ancilla,* which is used to get information about the system of interest jacobs14 . Let the Hilbert space dimension of the system of interest and the ancilla as $K$ and $L$ , respectively. The ancilla is prepared in some known initial state independently of the system of interest, and then the two systems are allowed to interact, getting correlated.* Next, a von Neumann measurement is performed on the ancilla, providing us with information about the system of interest, which we are going to call the **system from now on. Let us call the initial state of the ancilla as $\left|a_{0}\right\rangle$ in the basis $\left\{\left|a_{k}\right\rangle\right\}$ , $k=0,1,\ldots,K-1$ , and denote the initial density operator of ancilla-system as

[TABLE]

where $\rho_{S}$ is the initial density operator of the system. Let $U$ denote the ancilla-system evolution operator. Since $U$ acts in the tensor-product space, it can be written as

[TABLE]

where

[TABLE]

being $\left\{\left|s_{l}\right\rangle\right\}$ , $l=0,1,\ldots,L-1$ , a set of basis states for the system, and $u_{klk^{\prime}l^{\prime}}=\left\langle a_{k},s_{l}\right|U\left|a_{k^{\prime}},s_{l^{\prime}}\right\rangle$ are the matrix elements of $U$ . Note that $M_{kk^{\prime}}$ acts in the Hilbert space of the system, and since the space of system has dimension $L$ , each sub-block matrix $M_{kk^{\prime}}$ has dimension $L$ . From now on we use $M_{k}$ to refer to the first column of the sub-block $M_{k0}$ of $U$ . Denoting the $K\times K$ sub-blocks of the matrix $U^{\dagger}U$ by $B_{kk^{\prime}}$ , and since $U^{\dagger}U=I$ , it is readily seen that

[TABLE]

The important point to note here is that $M_{k}$ can be chosen to be any set of operators, provided the restriction Eq. (4) above is obeyed.

Now, performing a von Neumann measurement on the ancilla states, represented by $\left|a_{m}\right\rangle\left\langle a_{m}\right|$ , we can write the (unnormalized) collapsed state of both ancilla and system as

[TABLE]

or, in terms of the sub-blocks of $U$ :

[TABLE]

From Eq. (6) we can write the normalized state of the system as

[TABLE]

where

[TABLE]

is the probability of finding the ancilla in state $\left|a_{m}\right\rangle$ after the unitary evolution $U$ . It is now promptly recognized that every set of operators $\left\{M_{k}\right\}$ satisfying $\sum_{k}M_{k}^{\dagger}M_{k}\equiv\sum_{k}E_{k}=I$ describes a possible measurement on a quantum system, with the measuring having $K$ outcomes. This gives us a complete description of a quantum system under a general measurement. Next, we use the above results to discriminate one of two nonorthogonal field states prepared into a high Q cavity.

III Model

In our proposal, see Fig. 1, a three-level atom in ladder configuration, described by the set of states $\left\{\left|a\right\rangle,\left|b\right\rangle,\left|c\right\rangle\right\}$ ,** is initially prepared in $\left|a\right\rangle$ and crosses a Ramsey zone (carrier interaction). Next, the atom enters a cavity **interacting on-resonance with a singe mode of a cavity field which in turns is prepared either in state $\left|\psi_{1}\right\rangle$ or in state $\left|\psi_{2}\right\rangle$ , for which $\langle\psi_{1}|\psi_{2}\rangle\neq 0$ (nonorthogonal states).

While inside the cavity, the atom suffers a Stark shift in order to lead $\omega_{ba}=\omega_{cb}=\omega$ Haroche99 ; Haroche01 . After the atom crosses the cavity, it is detected in one of its three possible states, thus revealing in which state the cavity mode was prepared. The Hamiltonian model is given by Scully97

[TABLE]

where $\sigma_{\eta\nu}=\left|\eta\right\rangle\left\langle\nu\right|$ and $a^{\dagger}$ $\left(a\right)$ is the creation (annihilation) photon number operator, and $g_{i}$ is the atom-field coupling, which we take as real for convenience. In this protocol we are interested in discriminating nonorthogonal states which are combinations of the Fock states $\left|0\right\rangle$ and $\left|1\right\rangle$ . Thus, since the maximum number of photons in this case is $n=2$ , which happens when the atom decays and increases the photon number into the cavity, we can consider $a^{\dagger}\left|2\right\rangle=0$ . After a little algebra, it is straightforward to obtain

[TABLE]

where $U=U_{C}U_{RZ}$ , with $U_{C}=e^{-iHt/\hbar}$ , $H$ given by Eq.(9), and $U_{ZR}$ is the evolution operator as given by the carrier or Ramsey zone: $U_{RZ}\left|a\right\rangle=\alpha\left|a\right\rangle+\beta\left|b\right\rangle$ .

Following the standard procedure Nielsen00 , we must build three POVM elements: the one that discriminates $\left|\psi_{1}\right\rangle$ ; the other one that discriminates $\left|\psi_{2}\right\rangle$ , and a third one leading to inconclusive results with probability $p_{in}$ . It is to be noted that the only constraint obeyed by the POVM elements is $\sum_{k}E_{k}=I$ . Thus, as soon as the atom state is known, we will know with certainty* *that the cavity mode field was either in state $\left|\psi_{1}\right\rangle$ or in state $\left|\psi_{2}\right\rangle$ , or that we do not know the initial state as a result of the inconclusive measurement. As explained above, to build the three POVM elements $E_{\nu}=M_{\nu}^{\dagger}M_{\nu}$ , $\nu=a,b,c$ , we have to calculate $\left(n,n^{\prime}=1,2,3\right)$ :

[TABLE]

and

[TABLE]

Using Eq. (10)-(12), we calculate the following operators in Eq. (13):

[TABLE]

From Eq. (15)-(17) we can calculate the POVM elements $E_{\nu}=M_{\nu}^{\dagger}M_{\nu}$ for $\nu=a,b,c$ :

[TABLE]

As can be checked, $\sum_{\nu}E_{\nu}=I$ .

To be specific, let us assume that we want to discriminate the following nonorthogonal field states into the cavity: $\left|\psi_{1}\right\rangle=\left|0\right\rangle$ and $\left|\psi_{2}\right\rangle=\frac{1}{\sqrt{2}}\left(\left|0\right\rangle+\left|1\right\rangle\right)$ Nielsen00 . The cavity state is thus represented by $\rho=q_{1}\left|\psi_{1}\right\rangle\left\langle\psi_{1}\right|+q_{2}\left|\psi_{2}\right\rangle\left\langle\psi_{2}\right|$ , where $q_{1}$ $\left(q_{2}\right)$ is the classical probability related to the frequency of preparing the state $\left|\psi_{1}\right\rangle$ $\left(\left|\psi_{2}\right\rangle\right)$ and $q_{1}+q_{2}=1$ . Clearly, $E_{c}$ discriminates state $\left|\psi_{2}\right\rangle$ , since $\left\langle\psi_{1}\right|E_{c}\left|\psi_{1}\right\rangle=0$ . To discriminate $\left|\psi_{1}\right\rangle$ , we impose $\left\langle\psi_{2}\right|E_{b}\left|\psi_{2}\right\rangle=0$ . This imposition leads us with the conditions: (i) $\alpha=\cos\left(g_{1}t\right)$ , (ii) $\beta=i\sin\left(g_{1}t\right)$ , and (iii) $\cos\left(\sqrt{2g_{1}^{2}+g_{2}^{2}}t\right)=0$ . Letting $g_{2}=\kappa g_{1}$ , the third condition can be rewritten as $g_{1}t\equiv\theta_{m}=\frac{(m+\frac{1}{2})\pi}{\sqrt{2+\kappa^{2}}}$ , $m=0,1,2,...$ . On the other hand, $E_{a}$ is inconclusive, since $\left\langle\psi_{1}\right|E_{a}\left|\psi_{1}\right\rangle\neq 0$ and $\left\langle\psi_{2}\right|E_{a}\left|\psi_{2}\right\rangle\neq 0$ , meaning that we must discard this measurement. Using these three conditions, we can write the effective POVM elements in the following way:

[TABLE]

where we have neglected terms containing the state $\left|2\right\rangle$ and put $\left|\psi_{2}^{\perp}\right\rangle=\frac{1}{\sqrt{2}}\left(\left|0\right\rangle-\left|1\right\rangle\right)$ .

The probabilities related to the success probability rates of POVM elements $E_{b}$ and $E_{c}$ are, respectively,

[TABLE]

and

[TABLE]

while the probability for the inconclusive results $p_{in}=p_{a}=\text{Tr}\left(E_{a}\rho\right)=q_{1}\left\langle\psi_{1}\right|E_{a}\left|\psi_{1}\right\rangle+q_{2}\left\langle\psi_{2}\right|E_{a}\left|\psi_{2}\right\rangle$ is ****

[TABLE]

The success probability is given by $p_{s}=p_{b}+p_{c}$ :

[TABLE]

where $p_{s}+p_{in}=1$ , as should.

IV Discussion

Since the result of POVM element $E_{a}$ is the inconclusive one, all we have to do in order to optimize our proposal is either minimize $p_{in}$ or maximize $p_{s}$ . We numerically maximize Eq. (27) with (i) $q_{1}=0.3$ , (ii) $q_{1}=0.7$ and, for comparison to other work, (iii) $q_{1}=q_{2}=0.5$ Nielsen00 . As an example, to $q_{1}=q_{2}=0.5$ we find, see Fig. 4, (a) for $m=0,$ $p_{s}=0.1878$ (black line with squares), implying $\kappa=1,47$ , (b) for $m=1$ , $p_{s}=0.2644$ (red line with circles), implying $\kappa=4.50$ (c) for $m=2,$ $p_{s}=0.2748$ ** **(blue line with triangles), implying $\kappa=7,50$ (d) for $m=3$ , $p_{s}=0.2779$ (green line pentagons), implying $\kappa=10.55$ . Values of $m\geq 3$ can be used at the expense of greater ratio $\kappa=g_{2}/g_{1}$ , see Tab. 1-3. The best choice to UQSD is the one that minimizes (maximizes) $p_{in}$ $\left(p_{s}\right)$ for each integer $m$ , and it is worthwhile to mention that the success probability rate around 0.26, obtained for the first values of the integer m, is very close to the best rate of success for this kind of quantum state discrimination predicted theoretically, which is 0.292 when $q_{1}=q_{2}$ Nielsen00 ; Janos07 . As can be seen from Fig. 2-3, there are several maxima in $p_{s}$ , depending on the value of $\kappa$ . Here we have chosen those whose success probability is the greatest one. Note from Tab. 3, corresponding to $q_{1}=q_{2}=0.5$ , that the best value for the success probability rate is below 0.292, which is the maximum value according to Ref. Nielsen00 ; Janos07 . This is also confirmed by our numerical calculations using much greater values for $m$ and $\kappa$ .

It is to be noted that the simple strategy of choosing whether to project the cavity field state on the computational basis $\left|0\right\rangle$ or $\left|1\right\rangle$ would allow us to discriminate only one state. Indeed, if the result of projection is $\left|1\right\rangle$ , the cavity mode state could not have been prepared in $\left|\psi_{1}\right\rangle$ and was prepared therefore in state $\left|\psi_{2}\right\rangle$ ; however, if the measurement result is $\left|0\right\rangle$ , we can not be sure if the cavity mode state had been prepared in $\left|\psi_{1}\right\rangle$ or $\left|\psi_{2}\right\rangle$ , this result being inconclusive. As a result of this strategy, we would find a success probability of $p_{s}=\text{Tr}\left(\left|1\right\rangle\left\langle 1\right|\rho\right)=0.25$ , thus lesser than the POVM strategy we developed. In Tab. 1-3 we present the values of $\kappa$ maximizing $p_{s}$ and the corresponding values for $p_{in}$ , $p_{b}$ , $p_{c}$ and $p_{s}$ , Eq. (24)-(27), for several integers $m$ and $q_{1}=0.3,0.7,0.5$ .

As a final remark, one could ask why to use POVM strategy instead of simply projecting on the computational basis of the cavity states. Three remarks are in order: (i) First, in addition to our protocol presenting a higher probability of success, there is no known technique to directly project the cavity state on the computational basis: usually, measurement of the cavity state requires additional atoms and/or cavities, thus being necessary to build another POVM elements to measure the cavity mode field Iara07 ; Almeida11 , (ii) second, the direct projective strategy does not discriminate both states but just $\left|\psi_{1}\right\rangle=\left|0\right\rangle$ , while the POVM strategy allows us to discriminate both $\left|\psi_{1}\right\rangle$ and $\left|\psi_{2}\right\rangle$ , (iii) the POVM strategy was build using known matter-radiation interaction parameters: it remains an open question if other types of interaction, such as those developed by effective Hamiltonian techniques Serra05 ; Prado06 ; deAlmeida14 , could attain optimal POVM results Jaeger95 .

V Conclusion

We have proposed an oversimplified scheme to build POVM elements allowing to discriminate nonorthogonal field states inside a high Q cavity. Besides to circumvent the impossibility to directly project the cavity states onto the computational Fock states without using ancilla, our protocol achieves a rate of success probability greater than the direct projective technique. Our proposal relies on nowadays techniques in the cavity QED domain, making use of just one single three-level atom undergoing a Ramsey zone (carrier interaction) plus one cavity and selective atomic state detectors. This simplicity makes our protocol very attractive from the experimental point of view. Finally, we hope our protocol can inspire other POVM strategies based on effective Hamiltonians technique making possible to attain optimal rates of success probability.

Acknowledgements.

We acknowledge financial support from the Brazilian agency CNPq, CAPES and FAPEG. This work was performed as part of the Brazilian National Institute of Science and Technology (INCT) for Quantum Information.

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